MSc Mathematics and the Foundations of Computer Science HANDBOOK V 17... · MSc Mathematics and the...
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MSc Mathematics and the Foundations of Computer Science HANDBOOK ISSUED OCTOBER 2014
Transcript of MSc Mathematics and the Foundations of Computer Science HANDBOOK V 17... · MSc Mathematics and the...
MSc Mathematics and theFoundations of Computer
ScienceHANDBOOK
ISSUED OCTOBER 2014
Handbook for the MSc Mathematics and the Foundations of
Computer Science
Mathematical Institute
2014
Contents
1 Sources of Information 5
1.1 The Grey Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 The Proctors’ and Assessor’s Memorandum . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Statements of Provision for MFoCS Students . . . . . . . . . . . . . . . . . . . . . . 5
1.4 The Mathematical, Physical and Life Sciences Division Graduate Handbook . . . . . 6
2 Useful Contacts 6
3 Finding Your Way Around 7
3.1 The Mathematical Institute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1.1 The Andrew Wiles Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 The Department of Computer Science . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 The Lecture List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.4 Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Terminology 9
5 The University and You 10
5.1 Joint Consultative Committee with Graduates . . . . . . . . . . . . . . . . . . . . . 10
5.2 University Gazette and Oxford Blueprint . . . . . . . . . . . . . . . . . . . . . . . . 10
5.3 University Club . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
6 Guidance on Regulations 11
7 Reading Courses 12
8 Presentation of Miniprojects and Dissertations 13
8.1 Miniprojects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
8.2 Dissertations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
8.2.1 Wording permission for 3rd party material . . . . . . . . . . . . . . . . . . . . 14
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9 Regulations 15
10 Guidance on Examination Conventions 16
11 Role of the Supervisors 19
11.1 “General” supervisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
11.2 “Dissertation” supervisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
12 COURSES OFFERED IN 2014/2015 21
13 SECTION A: MATHEMATICAL FOUNDATIONS 23
13.1 Schedule I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
13.2 Algebraic Number Theory — Prof. Flynn — 16 HT . . . . . . . . . . . . . . . . . . 23
13.3 Algebraic Topology — Prof. Douglas — 16MT . . . . . . . . . . . . . . . . . . . . . 25
13.4 Analytic Number Theory —Prof. Heath-Brown—16MT . . . . . . . . . . . . . . . . 25
13.5 Analytic Topology — Dr Suabedissen — 16MT . . . . . . . . . . . . . . . . . . . . . 27
13.6 Commutative Algebra — Prof. Segal — 16HT . . . . . . . . . . . . . . . . . . . . . 27
13.7 Godel’s Incompleteness Theorems — Dr Isaacson — 16HT . . . . . . . . . . . . . . . 28
13.8 Introduction to Representation Theory — Prof Nikolov — 16 MT . . . . . . . . . . 29
13.9 Lambda Calculus and Types — Dr Vicary — 16 lectures HT . . . . . . . . . . . . . 30
13.10Lie Algebras — Prof. Ciubotaru — 16MT . . . . . . . . . . . . . . . . . . . . . . . . 32
13.11Model Theory — Prof. Zilber — 16MT . . . . . . . . . . . . . . . . . . . . . . . . . 32
13.12Modular Forms — Prof Lauder — 16MT . . . . . . . . . . . . . . . . . . . . . . . . 33
13.13Topology and Groups — Prof. Dancer — 16 MT . . . . . . . . . . . . . . . . . . . . 35
13.14Schedule II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
13.15Algebraic Geometry — Dr Berczi — 16MT . . . . . . . . . . . . . . . . . . . . . . . 36
13.16Axiomatic Set Theory — Dr Suabedissen — 16HT . . . . . . . . . . . . . . . . . . . 37
13.17Homological Algebra — Prof. Kremnitzer — 16MT . . . . . . . . . . . . . . . . . . . 37
13.18Infinite Groups — Prof. Nikolov — 16HT . . . . . . . . . . . . . . . . . . . . . . . . 38
13.19Non-Commutative Rings — Prof. Ardakov — 16HT . . . . . . . . . . . . . . . . . . 38
13.20Geometric Group Theory — Prof Papazoglou — 16MT . . . . . . . . . . . . . . . . 39
14 SECTION B: APPLICABLE THEORIES 41
14.1 Schedule I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
14.2 Applied Probability — Dept. of Statistics to allocate — 16 MT . . . . . . . . . . . . 41
14.3 Categories, Proofs and Processes — Prof Abramsky — 20 lectures + extra reading MT 42
14.4 Communication Theory — Dr Griffiths — 16 MT . . . . . . . . . . . . . . . . . . . . 44
14.5 Computer-Aided Formal Verification — Dr Abate —16MT . . . . . . . . . . . . . . 45
14.6 Concurrency — Dr Gibson-Robson — 16 lectures + extra reading HT . . . . . . . . 47
14.7 Foundations of Computer Science — Prof P Goldberg — 16 lectures MT . . . . . . . 49
14.8 Graph Theory — Prof. Riordan — 16HT . . . . . . . . . . . . . . . . . . . . . . . . 51
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14.9 Quantum Computer Science—Prof Bob Coecke —24 Lectures MT . . . . . . . . . . 52
14.10Schedule II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
14.11Automata, Logic and Games — Dr M Vanden Boom —16 lectures + reading MT . . 54
14.12Categorical Quantum Mechanics—Dr Heunen and Dr Vicary—16 Lectures HT . . . 56
14.13Combinatorics — Prof. Scott — 16MT . . . . . . . . . . . . . . . . . . . . . . . . . . 58
14.14Computational Algebraic Topology — Prof Tillmann & Prof Abramsky 14HT . . . . 59
14.15Computational Learning Theory — Prof. Worrell — 16MT . . . . . . . . . . . . . . 61
15 Computational Number Theory — Prof R Heath-Brown — Reading course TT 62
15.1 Distributional Models of Meaning — Prof B Coecke— Reading course HT . . . . . . 63
15.2 Elliptic Curves — Prof. Kim — 16HT . . . . . . . . . . . . . . . . . . . . . . . . . . 65
15.3 Finite Dimensional Normed Spaces — Dr Sanders — 16TT . . . . . . . . . . . . . . 67
15.4 Machine Learning - Prof Nando de Freitas - 24HT . . . . . . . . . . . . . . . . . . . 69
15.5 Networks — Prof. Porter — 16HT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
15.6 Probabilistic Combinatorics — Prof. McDiarmid — 16HT . . . . . . . . . . . . . . . 72
16 Theory of Data and Knowledge Bases— Prof T Lukasiewicz — 16 lectures HT 73
.1 Safety Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
.2 Action in Case of Emergency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
.3 Statement of Safety Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
.4 Statement of Health and Safety Organisation . . . . . . . . . . . . . . . . . . . . . . 74
.5 Departmental Health and Safety Committee . . . . . . . . . . . . . . . . . . . . . . . 74
.6 Code of Practice-Harassment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
.7 Smoking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
.8 Electricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
.9 Equipment rooms - Department of Computer Science . . . . . . . . . . . . . . . . . . 75
.10 Lighting - Department of Computer Science . . . . . . . . . . . . . . . . . . . . . . . 75
.11 Other Safety Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A University’s Policy on Plagiarism 76
A.1 What is plagiarism? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
A.2 Why does plagiarism matter? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
A.3 What forms can plagiarism take? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
A.4 Not just printed text! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
B Electronic Resources for Mathematics 78
C Applying for Computer Resources 82
C.1 University Policy on Intellectual Property . . . . . . . . . . . . . . . . . . . . . . . . 83
C.2 Regulations Relating to the Use of Information Technology Facilities and UniversityPolicy on Data Protection and Computer Misuse . . . . . . . . . . . . . . . . . . . . 83
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C.3 Equal Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
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1 Sources of Information
This handbook is designed as a guide for students on the Mathematics and Foundations of ComputerScience course in the Mathematical Institute (MFoCS). It does not replace the official regulationsrelating to your degree, which you will find in the Examination Regulations, but it is a less formaland more easily understood guide to being a student in the Mathematical Institute. It also containsgeneral information about the Department, people, facilities and safety.
There is an edition of this handbook on the Mathematical Institute’s website at:
http://www.maths.ox.ac.uk/members/students/postgraduate-courses
In addition to this handbook there are some important sources of information that you should makesure you are familiar with.
1.1 The Grey Book
The Examination Regulations, usually known for obvious reasons as the ”Grey Book”, is the author-atitive document on the regulations for the University degrees and examinations. You should receivea free copy of the relevant part of this book through your College at the beginning of your first term.The Grey Book defines the rules for admission to and progression through the programmes of studyand the syllabus for examinations. The regulations are available online at:
http://www.admin.ox.ac.uk/examregs/contents.shtml
1.2 The Proctors’ and Assessor’s Memorandum
The University has two Proctors, the Senior Proctor and the Junior Proctor, who are responsiblefor making sure that the University operates according to its statutes. As well as being members ofkey decision-making committees, they deal with
• University (as distinct from college) student discipline
• complaints about University matters
• the running of University examinations
They also carry out ceremonial duties, e.g. at degree ceremonies. The Assessor is the third seniorofficer, responsible particularly for student welfare and finance.
The Proctors’ and Assessor’s Memorandum is the document relating to the rules and the statutesof the University which you are expected to follow. This can be found at:
http://www.admin.ox.ac.uk/proctors/info/pam/
1.3 Statements of Provision for MFoCS Students
These detail the provisions that have been made for you by the University and the Colleges. Thestatement that is applicable to you is written by the Mathematical Institute; this can be found at:
http://www.maths.ox.ac.uk/members/students/postgraduate-courses/msc-mfocs
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1.4 The Mathematical, Physical and Life Sciences Division GraduateHandbook
The division also produce a graduate handbook, which you should make sure you are familiar with.This can be found at:
http://www.mpls.ox.ac.uk/intranet/teachingandlearning/graduateprog.html
2 Useful Contacts
Course DirectorProf Samson Abrambsky 83558 [email protected] Administrator and Exam CoordinatorMonica Kundan Finlayson 15206 [email protected] SO.16, Mathematical InstituteIT Support [email protected]
http://www.maths.ox.ac.uk/help/supportAcademic Administrator - MathsCharlotte Turner-Smith 15203 [email protected] Administrator - ComputingLeanne Carveth 73863 [email protected]/Graduate StudiesLibrarian: Whitehead LibraryCathy Hunt 73559 [email protected] Institute 73525 [email protected]
Confidential Harassment AdvisorsDr Sarah Waters 80141 [email protected] Alain Goriely 15169 [email protected] ContactThe Academic Administrator 15203 [email protected] Officer - MathsKeith Gillow 80605 [email protected] Officer - ComputingAndy Simpson 83515 [email protected] Management Team
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3 Finding Your Way Around
Your academic life in Oxford will involve two intimately connected but distinct institutions. You area member both of a College and of the University; your supervisor is a member of the MathematicalInstitute or the Department of Computer Science and probably a member of a different college.Your college will also allocate a college advisor to you.
3.1 The Mathematical Institute
The Mathematical Institute occupies a brand new building, the Andrew Wiles Building, in theRadcliffe Observatory Quarter.
An entry card system (using your University Card) controls access to the building. You will beprovided with information on how to activate your card at the Graduate Induction Day. Shouldyour card not work then please report the problem to [email protected] and include in theemail the details from the card:
Name. Category/Type/Status (e.g. Congregation, Student, Staff etc), Card Number, Card ExpiryDate/Valid Until Date
Rules governing access to the Mathematical Institute are as follows:
1. Cards are issued on a personal basis and must not be loaned or passed on to another person.
2. No-one should allow access to another person.
3. When a card is used to gain access to the building, the system keeps a record of that use fora period of approximately six months.
3.1.1 The Andrew Wiles Building
The Andrew Wiles Building houses lecture theatres and seminar rooms in which most of the uni-versity lectures in Mathematics take place. There are also a number of offices, most of which areoccupied by academic staff.
Opening Hours The building is open 24 hours a day (including weekends), except closed periodssuch as Bank Holidays out of term time. Reception is manned from 8.30am to 6.00pm.
Social area The Institute has a Common Room on the first floor and a cafeteria on the mezzaninelevel. A dedicated study room for MFoCS students is located on the ground floor and tea/coffeemaking facilities are also available on this floor.
3.2 The Department of Computer Science
MFoCS students will also need access to the Computing Laboratory. This is situated in the WolfsonBuilding, which is on the corner of Parks Road and Keble Road. To access this building, you willneed to have your University card activated. Please contact Brenda Deeley at the Department ofComputer Science to arrange this.
3.3 The Lecture List
The Mathematical Institute publishes a lecture list for Mathematical Sciences just before the begin-ning of each term, as do all other Divisions of the University. The Mathematics list can be found
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on the web at http://www.maths.ox.ac.uk/members/students/lecture-lists.
Lecture lists for other Departments in the MPLS Division can be found athttp://www.mpls.ox.ac.uk/introduction-graduate-training. All members of the University may at-tend any publicly announced University lectures or seminars.
3.4 Libraries
Whitehead Library The Whitehead Library of the Mathematical Institute holds material cov-ering mathematical topics at graduate and research level. It is primarily for the use of the graduatestudents and academic staff of the Mathematical Institute.
The library is kept locked at all times but your University card will give you 24/7 library access.
Library Holdings Books and journals are listed on SOLO, the University’s online catalogue(http://solo.bodleian.ox.ac.uk/ )
Borrowing Books may be borrowed (5 books per reader, 3 week loan).
All books must be checked-out on the SOLO automated loan system.
Journals are reference only and may not be taken out of the library except for brief photocopyingin the Institute.
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4 Terminology
Matriculation Matriculation is the formal University admission procedure and is organised byyour college.
University terms The three University ‘full’ terms are:
Michaelmas (12 October - 6 December)
Hilary (18 January - 14 March) Trinity (26 April - 20 June)
Each term lasts eight weeks, but terms simply set the periods during which formal instruction isgiven by way of lectures, seminars and tutorials. The University functions throughout the yearand as a graduate student you will need to work in vacation as well as in term time (apart fromreasonable breaks).
Note in particular that you should expect to remain in Oxford after the end of eachterm to work on miniprojects or your dissertation, and also that you should returnbefore the beginning of each term to discuss your programme with your supervisor,and for meeting the Course Director.
Subfusc The University Examination Regulations state that all members of the University arerequired to wear academic dress with subfusc clothing when attending formal University events suchas matriculation and University examinations. It consists of:
A dark suit with dark socks, or a dark skirt with black stockings or trousers with dark socks; blackshoes; plain white collared shirt; a black tie or white bow tie.
Candidates serving in HM Forces are permitted to wear uniform together with a gown. (The uniformcap is worn in the street and carried when indoors.)
You will be required to wear subfusc for your viva in September.
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5 The University and You
5.1 Joint Consultative Committee with Graduates
Graduate students views are fed into the departmental structure via the Consultative Committeewith Graduates. The committee’s operation is described in the following standing order:
“There shall be a Consultative Committee with Graduates consisting of up to six junior membersreading for higher degrees, and the Director of Graduate Studies. One or two of the junior membersshould be following an MSc by coursework. Committee members shall be elected from amongstgraduate students admitted by the Mathematical Institute, and graduate students following taughtMSc’s by coursework for which the Mathematical Institute shares some teaching responsibility. Nom-inations and self-nominations shall be invited by circulating these graduate students electronically inthe second week of Michaelmas Term. Elections shall be held electronically during the fourth weekof Michaelmas Term, with three working days being given for voting. The one MSc (coursework)student with the most votes, the three research students (DPhil or MSc by research) with the mostvotes, and the two remaining students (either MSc or DPhil) with the most votes will be elected.The Committee shall have the power to co-opt junior members such that membership is complete.The committee may operate, if necessary, without its full complement of places having been filled.The committee shall be concerned with matters such as the syllabus, teaching arrangements, li-brary facilities, office facilities, and the general aspects of examinations. It shall annually reviewexaminers reports for the taught MSc’s. The Director of Graduate Studies shall be the chairmanof the committee. The Graduate Studies Assistant or another member of Mathematical Institutestaff shall act as secretary to the committee. The Consultative Committee with Graduates reportsto the Research Committee. The Committee shall be able as of right to address a communicationdirect to the Departmental Committee, the Research Committee, or the Teaching Committee, ofthe Mathematical Institute depending on the matters involved. Unless the Chairman shall orderotherwise, the committee shall meet at 2pm on Thursday in the fifth week of each Full Term.”
5.2 University Gazette and Oxford Blueprint
The Gazette is published weekly in term time, and is the official publication for University business,regulation changes, meetings etc. It is available in all the University and College Libraries and inthe Common Room in the Mathematical Institute. Oxford Blueprint, a newsletter for Universityand college staff and students, is published in 0th, 3rd, 6th and 9th weeks of term. It contains news,interviews and features reflecting the diversity of activity across the University, and an events diaryis included.
5.3 University Club
The University Club provides a social and recreational venue intended to serve the University’sacademics, postdocs, staff, postgraduates, alumni and those who have retired from academic or staffpositions. To apply to become a member of the University Club, please visit the Club’s website:http://www.club.ox.ac.uk and fill in the online membership application form (accessible via the‘Membership’ link). Applications may take two weeks to process. Once processed your Universitycard will admit you to the club.
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6 Guidance on Regulations
The supervisory committee for the course currently consists of Prof S Abramsky (Chairman), ProfP Goldberg, Prof V Flynn, Prof P Landrock (External member), Dr J Pila, Dr R Pinch (Externalmember), Prof. J Worrell, Prof O Riordan, Prof A Mnch (ex officio). Any member of the committeemay be approached for guidance. (N.B. The committee membership may change from the beginningof the academic year 2014. You will be informed if this is the case).
The aim of this course is to provide a wide grounding over a range of mathematics and computingscience and the regulations are designed to ensure that this is achieved without too much special-isation, while giving students a good choice of options. The lecture courses are divided into twosections, and there is a requirement that students devote a significant portion of their effort to ap-plications. Within each section, courses are divided into Schedule I and Schedule II. At least fivecourses must be passed in total, two on courses from Section B and two courses at the Schedule IIlevel (these need not be distinct).
Schedule I courses are basic; the lectures may serve also as advanced undergraduate lectures, thoughstudents for an MSc should expect to read more widely around the material than would an un-dergraduate, and the written assignments set at the end of the course will be more searching thanundergraduate examination questions, and will often allow the student to develop a theme. ScheduleII courses are more advanced and can be expected to lead into areas where students may chooseto write dissertations. Some Schedule II courses will be offered as directed reading, rather than bylectures (see page 12); their content will be the equivalent of a standard course of sixteen lectures.
The form and timing of the assessment on courses is laid out in the Regulations. The intention isfor students to have the opportunity to show their understanding of material and an ability to readaround the subject, rather than for the assessment to be a “time test” of instant recall or speed ofmanipulation. Miniprojects will be set so as to allow scope in the choice of material to be submittedand in the approach taken. General guidance can be sought from supervisors at this stage, thoughnot in specific detail since work submitted is required to be that of the student alone.
While it is necessary to pass five courses, and not more than four may be offered in any one term, thenormal expectation is that students will take three or perhaps four courses in each of Michaelmasand Hilary terms. It should be noted that Schedule I courses will mainly be given in Michaelmas andHilary terms, and Schedule II courses in Hilary and Trinity terms. Many courses provide problemsheets and associated classes; completion of such work (and its marking) forms an integral part ofthe course, and students are assigned to such classes. Where this is not the case, either the lectureror the supervisor (see below) will arrange a limited amount of ‘tutorial-style’ teaching to supplementlectures. (See also ‘Reading Courses’, page 12.)
Each student is assigned a supervisor who will offer direct guidance in the first instance. In particular,students are advised to discuss with their supervisors at an early stage which range of courses theyintend to consider so that any prerequisite knowledge can be acquired. They should also discusswith their supervisors later the area in which they intend to write the dissertation so that a suitablesupervisor for that dissertation (who need not be the ‘assigned’ supervisor) can be approached todiscuss possible topics. Students give a short presentation on their dissertation topic, late in TrinityTerm or shortly thereafter. The dissertation is required to bear regard to some aspect of the coursematerial covered.
All students will be required to attend an oral examination. This will be held in the second half ofSeptember.
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7 Reading Courses
Some courses may be offered by means of directed reading rather than as a formal course of lectures- such courses are of exactly the same standing as those delivered via lectures. The following notes,both for students and those giving the course, give a general outline of what should be expected.
1. The content of a Reading Course should, in quantity and expectation, correspond to a standard16 lecture course given at the Schedule II level.
2. There should be a synopsis that states the aims and content of the course, together with thereading that will specify its content (but see (4) below).
3. The reading is guided - that is, the person offering the course should meet with the studentson a regular basis to discuss the material being read and to give clear guidance as to whatshould be achieved before the next meeting.
The exact format should be decided on a mutual basis. In practice, most people giving suchcourses will see all the students at the same time, on between four and eight occasions (eitherfour 2-hour or eight 1-hour meetings).
The majority of the time will be spent reviewing what has been read, and it is for the “lecturer”to see whether this is best spent by students presenting items, by reviewing the content brieflyand answering questions as he goes along, or by asking the students what particular aspectsthey may want explained. It is useful to conclude by discussing what is in the next “section”to be covered, with advice as to where problems may be expected or where suitable additionalreading may be found. The first meeting, of course, should include a general outline of thegoals of the course since at that stage some students may still be deciding whether to take thecourse.
4. It is permissible (i.e., without referring back to the Supervisory Committee) to vary the contentof the course to reflect the interests of those taking the course, though all students should coverthe same material for the purposes of preparing for the miniproject - but “extra” readingmay be assigned on an individual basis according to taste. This is particularly relevant whenstudents may be looking for a dissertation in the area of the reading course. Many dissertationsdo in fact arise from reading courses.
5. If relevant and viable (especially with a view to the fact that a miniproject will be set),problems or practical exercises may be set. The latter should, of course, be designed with thestudents’ prior computing experience taken into account.
6. On occasion, extra reading courses may be approved by the Supervisory Commitee duringthe year upon request. Students seeking such extra courses should discuss this with theirsupervisor, and also informally with the Chairman prior to any formal request.
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8 Presentation of Miniprojects and Dissertations
8.1 Miniprojects
Since these have to be written within a fairly short period of time, they may be either typed or handwritten, in which case it is important that they be both legible and laid out in the same way as ifthey were typed.
In either case, it should be borne in mind that these are projects, not “examination solutions”, andthe presentation should reflect this -
1. Your submission should be clearly written in sentences with appropriate punctuation, displayof formulae, appropriate use of ‘Definition’, ‘Lemma’, ‘Theorem’, ‘Proof’, etc.
2. You should begin with a brief statement of the overall goal of the project, and finish witha conclusion of what you have achieved (or needed to assume) and comment on what otherquestions your work might lead to.
3. Write on one side only, and number pages, but do not staple sheets together. Your projectshould be submitted in some sort of folder (a clear plastic sleeve folder will suffice). You mustnot write your name on your miniproject; the only identification should be your candidatenumber.
4. It is impossible to give precise guidance on length since this can vary considerably from projectto project, depending on how much calculation may be needed and whether such is routine. Itis unlikely, however, that a project can be completed in less that five pages, and it will moreoften be in the 10 - 15 page range. What is more important is that it should reflect the factthat you will be concentrating over a 2 - 3 week period on writing three or four projects andshould represent a commensurate amount of work, bearing in mind that some reading may berequired as part of the process (e.g., some projects may involve showing your understandingby extending a known result that you may not have seen before).
If you have any questions about the miniprojects (e.g., requests for clarification), thenyou should pass these to Monica Kundan Finlayson (the MFoCS Examination Coor-dinator in Room SO.16 of the Mathematical Institute), who will pass them on as ap-propriate to the relevant Assessor and/or the Chairman of Examiners, and will ensurethat any replies go to all students taking that miniproject. You must not communicatedirectly with the Assessor, nor discuss the projects with each other.
The University takes a strong stand against plagiarism. Students are strongly advised to read thenote on plagiarism and citation in Appendix A.
8.2 Dissertations
These must be typed, and it is recommended (though not obligatory) that LaTeX be used. Thusyou are advised to become familiar with this during the year if you are not already fully conversantwith another word processing package that can handle mathematical formulae (and diagrams).
The typing should follow the guidance for research thesis (see Examination Regulations). The workshould be properly and adequately referenced in the text, with the full list of references at the endof the dissertation, following any of the standard labelling conventions as mathematical papers (e.g.,numerical, or by abbreviated name). The dissertations do not need to be bound, but should bepresented in folders that ‘hold the pages together’.
The dissertation does not require a separate abstract. However, it is strongly recommended thata short abstract (of less than one page) be included at the beginning of the dissertation, separate
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from the Introduction. This abstract may, but need not, be that submitted to the Chairman ofthe Supervisory Committee for prior approval (though these will have been made available to theChairman of Examiners along with the title of the dissertation). The regulations also require youto include a certificate from your College stating that you have followed for three terms a courseof instruction in Mathematics and Foundations of Computer Science. You will need to approachsomeone at your college to provide this for you. You can get a certificate from one of the following
Tutor for Graduates
Tutor for Graduates’ Secretary
Graduate Administrator/Officer
Academic Administrator
College Secretary
Senior Tutor
In addition to the two copies delivered to Examination Schools, you should also email a copy toMonica Kundan Finlayson ([email protected]) by noon on 1 September in the yearof the examination.
You are advised to keep a third printed copy to take with you to your oral examination.
8.2.1 Wording permission for 3rd party material
If your thesis is to contain any material where copyright is held by a third party, you should consultthe information athttp://www.bodleian.ox.ac.uk/ora/oxford etheses/copyright and other legal issues
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9 Regulations
The Examination Regulations, usually known for obvious reasons as the “Grey Book”, is the author-atitive document on the regulations for the University degrees and examinations. You should receivea free copy of the relevant part of this book through your College at the beginning of your first term.The Grey Book defines the rules for admission to and progression through the programmes of studyand the syllabus for examinations. The regulations are available online at:
http://www.admin.ox.ac.uk/examregs/contents.shtml.
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10 Guidance on Examination Conventions
Each piece of work submitted is awarded a University Standardised Mark (USM) by the Examiners,with a USM of 50 or more representing a pass. To pass the course, passes must be obtained on atleast five miniprojects that include two on courses from Section B and two at the Schedule II level(these need not be distinct) and for the dissertation. The Examiners may award a distinction forexcellence throughout the examination.
The following sets out the conventions for the level of USMs awarded, and the mechanism by whicha final USM is determined. A pass requires a final USM of at least 50 and a distinction requires afinal USM of at least 70.
The Board of Examiners consists of at least three (currently four) members, with at least one (cur-rently two) being external to the University. The current Board of Examiners consists of ProfessorVictor Flynn and Dr Jonathan Barrett as internal examiners and Dr Martin Escardo (Universityof Birmigham) and Dr John Talbot (University College London) as external examiners. (Note:candidates must not under any circumstances communicate directly with examiners.)
Miniprojects are set by those giving the courses and are double-blind marked by that person andone other assessor (these two mark the work independent of each other). Each proposes a USM forthe work and a range of at most five USMs within which they would be content for the USM to lie.If there is overlap between the ranges proposed by the two assessors and neither of the ranges crossesa classification boundary the two proposed USMs are averaged and rounded to the nearest wholenumber (.5 is rounded up). In all other cases the two assessors are asked to discuss the miniprojectto agree on a final USM.
The exception is miniprojects which have a model solution and marking scheme approved by theexaminers. In such cases each script is marked by an assessor and this marking is checked indepen-dently to ensure that all parts have been marked and the part-marks have been correctly totalledand recorded.
The miniprojects which are set are submitted to the Examiners for prior vetting, and the Examinersmay moderate the marks given by assessors, in particular to achieve parity across subjects. Thepass list for each individual course is published before the beginning of the subsequent term andcandidates will be advised of the USMs awarded.
The dissertation is marked independently by the dissertation supervisor and by another assessor(who is neither the dissertation supervisor nor the candidate’s supervisor). These two marks arereconciled to produce a provisional USM following the same procedures given for miniprojects whichare double-blind marked above. Each Dissertation will also be seen by at least one Examiner.The Examiners will then seek advice from the dissertation supervisor as to his or her input intothe dissertation and the degree of originality on the part of the candidate. The assessor of thedissertation will normally be present at the oral examination, and the Examiners will determine theUSM only after the oral examination has been held, taking into account all the evidence from thedouble-marking, the supervisor’s additional input, and from the oral examination.
The University takes a strong stand against plagiarism. For further information see
http://www.ox.ac.uk/students/academic/guidance/skills/plagiarism
Criteria for the award of USMs
USMs will be awarded according to the following criteria:
70 - 100 Excellent - the candidate has demonstrated an excellent understanding of almost all thematerial covered with a commensurate quality of presentation, and has completed almost all of theassignment satisfactorily - further subdivided by;
• 90-100 The candidate has shown originality or insight that in the case of a minipro-
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ject goes beyond a basic completion of the task set, and in the case of the dissertationcontains some original work of potential publication standard
• 80-89 The work submitted shows a near-perfect completion of the task in hand,whether a miniproject or dissertation, but does not meet the additional requirementsabove, or does but has defects in presentation
• 70-79 The work submitted is of a generally high order, but may have minor errorsin content and/or deficiencies in presentation
60 - 69 Good - the candidate has demonstrated a good understanding of much of the material, andhas completed most of the assignment satisfactorily
50 - 59 Adequate - the candidate has demonstrated an understanding of the material and an abilityto apply his or her understanding that together are sufficient to pass;
and at levels that fail;
40 - 49 The work submitted, while sufficient in quantity, suffers from sufficient defects to show alack of adequate understanding or ability to apply results
30 - 39 The candidate, while attempting a significant part of the miniproject or in writing a disser-tation, has displayed a very limited knowledge or understanding at the level required for a master’sdegree
0 - 29 The candidate has either attempted only a fragment of a miniproject or has shown aninadequate grasp of basic material.
In all cases, the Examiners take account of the presentation of work.
Formative feedback
From the first term of the MSc students will attend classes and complete problem sheets which willbe marked and feedback given.
Determination of the final USM
To determine the final USM, F, the dissertation is given the weight of three miniprojects and firsta provisional USM, P, is calculated as
P = [(X + Y +A+B + C + 3D)/8]
where X,Y are the best two marks on Schedule II courses, A,B,C are the three highest othermarks on miniprojects, and D is the dissertation mark. Passes and distinctions, and the final USMF awarded, are determined by the following rules.
(i) If any of X,Y,A,B,C,D is less than 50, or if fewer than two Section B courses have beenpassed, then F = min{P, 49} and the candidate is failed.
(ii) If either
(a) D ≥ 70 , and X,Y,A,B,C ≥ 70
or
(b) D ≥ 80, X ≥ 70, Y ≥ 67 and (X + Y +A+B + C)/5 ≥ 70,
then F = P and the candidate is awarded a distinction.
(iii) In all other cases, F = min{P, 69} and the candidate is awarded a pass.
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[Note: Condition(ii)(b) permits the examiners to interpret the requirement “excellence throughoutthe examination” more broadly, to award a distinction for particular excellence on the dissertationwhere the miniprojects are not uniformly of distinction standard.]
A candidate who has failed the MSc may be admitted to and examined on the course as offered in theyear subsequent to the initial attempt. No piece of written work shall be submitted for examinationon more than one occasion. It is University policy that candidates who have initially failed an MScare not normally eligible for the award of distinction.
Late penalties
A candidate who does not submit a written assignment on a course for which he or she has entered,by noon on the Monday of the eleventh week of the relevant term, shall be deemed to have failedthe course in question.
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11 Role of the Supervisors
The term “supervisor” is used in two contexts within the MFoCS programme, first to denote your“general” supervisor who will guide you throughout your course, and second to denote the personunder whose specific guidance you will write your dissertation during the second half of the course.The purpose of the appendix is to explain more fully what you can expect.
11.1 “General” supervisor
This person is responsible for guiding you through the course. You should see him (or her) almost assoon as you arrive, and in particular in the week before teaching actually starts, to discuss the rangeof courses that you propose to take. There is no need to know exactly at this stage; there is an oppor-tunity to explore more than you will take formally, but it is important to discuss your background inmathematics and/or computer science so that your supervisor can assess whether taking particularcourses is realistic, and whether you are taking a sufficiently coherent set of courses (especially inthe first term) to ensure a passage through the remainder of the year. Your supervisor will also beable to give you guidance on material that he thinks it appropriate for you to study by yourself toprepare you for courses that you want to take, either by reading, or by attending undergraduatelecture courses that do not feature within the MFoCS list of courses. (This is especially true ofcourses given in Hilary term for which you may lack some of the prerequisites.)
Most courses come with associated problem classes. Where this does not happen, your supervisoris the person responsible for seeking alternative arrangements if needed. More generally, you shouldkeep in contact with your supervisor, and in particular let him know exactly which courses youfinally decide to take for assessment. He is also the person who can give you general guidance.
Much the same procedure will occur at the beginning of the second term in January. This time,though, you should expect to have a preliminary discussion of the general area in which you will behoping to write your dissertation. Often you will find that the most appropriate person to talk toafter that is the person who has given lectures in that area, but your supervisor may well suggestthat there are others to whom you should talk, and will provide the necessary “introductions”. Atthis time too, your supervisor will provide you with some feedback on your performance in theminiprojects on the previous term’s courses, and he will be the person who formally reports on yourwork to the Director of Graduate Studies and to your college.
Your general supervisor is also responsible for ensuring that you do find a dissertation supervisor.This may seem a hard task, but in practice there has rarely been any difficulty at this stage. Whilethen you will be working with your dissertation supervisor (primarily during Trinity term), yourgeneral supervisor will still be the person with overall responsibility for you, as before.
In assigning general supervisors, note has been taken of your background and indicated interests.Those acting as general supervisors are experienced in the MFoCS programme and it is not necessarythat they be experts in the particular direction that you plan to take; it may turn out that they willalso be your dissertation supervisor, but this is most often not the case.
Your supervisor may not help you with your miniprojects as such. However, he may answer generalquestions by directing you to appropriate reference material, but not to answer specific questionsrelating to the actual miniproject. In the case where English is not your native language, supervisorsare allowed to see a draft of miniprojects and to comment on grammar, spelling and usage only.
Your general supervisor can also read and comment on a draft of your dissertation - especially withregard to its general presentation.
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11.2 “Dissertation” supervisor
This person will actually guide you while you are preparing and writing your dissertation. Normally,students “find” their dissertation supervisor before the end of Hilary term so that they can startsome specialised reading after completing that term’s miniprojects, and before the beginning ofTrinity term. At the beginning of that term, you should prepare a “dissertation proposal” withyour intended dissertation supervisor and submit it for approval. It is normal to have around eightmeetings with your dissertation supervisor, mainly during Trinity term, but possibly continuing intoJuly, but the exact arrangements are made mutually. Your dissertation supervisor should read andcomment on your dissertation, and may give you guidance as he might for a research student.
Your dissertation supervisor should normally be based in the Mathematical Institute, the Depart-ment of Computer Science or the Department of Statistics; if you are considering a potential super-visor outside these departments or outside Oxford, you should consult the Course Director at anearly stage. Also, if work on your dissertation should require you to work out of Oxford at any pointduring Trinity term, you must consult the Course Director before making any arrangements.
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12 COURSES OFFERED IN 2014/2015
Section A
Schedule I
Algebraic Number Theory Prof Flynn HTAlgebraic Topology Prof Douglas MTAnalytic Number Theory Prof Heath-Brown MTAnalytic Topology Dr Suabedissen MTCommutative Algebra Dr Segal HTGodel’s Incompleteness Theorems Dr Isaacson HTIntroduction to Representation Theory Prof Nikolov MTLambda Calculus and Types Dr Vicary HTLie Algebras Prof Ciubotaru MTModel Theory Prof Zilber MTModular Forms Prof Lauder MTTopology and Groups Prof Bridon MT
Schedule II
Algebraic Geometry Dr Berczi MTAxiomatic Set Theory Dr Suabedissen HTHomological Algebra Prof Kremnitzer MTInfinite Groups Prof Nikolov HTNon-Commutative Rings Prof Ardakov HTGeometric Group Theory Prof Papazoglou MT
Section B
Schedule I
Applied Probability Dept. of Statistics to allocate MTCategories, Proofs and Processes Prof Abramsky MTCommunication Theory Dr Griffiths MTComputer Aided Formal Verification Dr Abate MTConcurrency Dr Gibson-Robinson HTFoundations of Computer Science Prof Goldberg MTGraph Theory Prof Riordan HTQuantum Computer Science Prof Coecke MT
Schedule II
Automata, Logic and Games Dr Vanden Boom MTCategorical Quantum Mechanics Dr Heunen/Dr Vicary HTCombinatorics Prof Scott MTComputational Algebraic Topology Prof Tillmann & Prof Abramsky HTComputational Learning Theory Prof Worrell MTComputational Number Theory * Prof Heath-Brown TTDistributional Models of Meaning* Prof Coecke HTElliptic Curves Prof Kim HTFinite Dimensional Normed Spaces Dr Sanders TTMachine Learning Prof Nando de Freitas HTNetworks Dr Porter HTProbabilistic Combinatorics Prof McDiarmid HTTheory of Data and Knowledge Bases Prof Lukasiewicz HT
*These courses are offered as directed reading courses, with syllabuses provided as in the case of
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lecture courses. There may be one or two more reading courses to be added later.
WE REGRET THAT DUE TO TIMETABLING RESTRICTIONS THERE WILL BE A NUMBEROF CLASHES BETWEEN LECTURE COURSES. PLEASE CHECK THE LECTURE TIMETABLECAREFULLY.
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13 SECTION A: MATHEMATICAL FOUNDATIONS
13.1 Schedule I
13.2 Algebraic Number Theory — Prof. Flynn — 16 HT
Prerequisites: B9a Galois Theory, Algebra 2 and Number Theory.
Recommended Prerequisites: All second-year algebra and arithmetic. Students who have nottaken Part A Number Theory should read about quadratic residues in, for example, the appendixto Stewart and Tall. This will help with the examples.
Overview An introduction to algebraic number theory. The aim is to describe the properties ofnumber fields, but particular emphasis in examples will be placed on quadratic fields, where it iseasy to calculate explicitly the properties of some of the objects being considered. In such fields thefamiliar unique factorisation enjoyed by the integers may fail, and a key objective of the course isto introduce the class group which measures the failure of this property.
Learning Outcomes Students will learn about the arithmetic of algebraic number fields. Theywill learn to prove theorems about integral bases, and about unique factorisation into ideals. Theywill learn to calculate class numbers, and to use the theory to solve simple Diophantine equations.
Synopsis
1. field extensions, minimum polynomial, algebraic numbers, conjugates, discriminants, Gaussianintegers, algebraic integers, integral basis
2. examples: quadratic fields
3. norm of an algebraic number
4. existence of factorisation
5. factorisation in Q(√d)
6. ideals, Z-basis, maximal ideals, prime ideals
7. unique factorisation theorem of ideals
8. relationship between factorisation of number and of ideals
9. norm of an ideal
10. ideal classes
11. statement of Minkowski convex body theorem
12. finiteness of class number
13. computations of class number to go on example sheets
Reading
1. I. Stewart and D. Tall, Algebraic Number Theory and Fermat’s Last Theorem. (Third Edition,Peters, 2002).
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Further Reading
1. D. Marcus, Number Fields (Springer-Verlag, New York–Heidelberg, 1977). ISBN 0-387-90279-1.
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13.3 Algebraic Topology — Prof. Douglas — 16MT
Overview Homology theory is a subject that pervades much of modern mathematics. Its basicideas are used in nearly every branch, pure and applied. In this course, the homology groups oftopological spaces are studied. These powerful invariants have many attractive applications. Forexample we will prove that the dimension of a vector space is a topological invariant and the factthat ‘a hairy ball cannot be combed’.
Learning Outcomes At the end of the course, students are expected to understand the basicalgebraic and geometric ideas that underpin homology and cohomology theory. These include thecup product and Poincare Duality for manifolds. They should be able to choose between the differenthomology theories and to use calculational tools such as the Mayer-Vietoris sequence to computethe homology and cohomology of simple examples, including projective spaces, surfaces, certainsimplicial spaces and cell complexes. At the end of the course, students should also have developeda sense of how the ideas of homology and cohomology may be applied to problems from otherbranches of mathematics.
Synopsis Chain complexes of free Abelian groups and their homology. Short exact sequences.Delta (and simplicial) complexes and their homology. Euler characteristic.
Singular homology of topological spaces. Relative homology and the Five Lemma. Homotopyinvariance and excision (details of proofs not examinable). Mayer-Vietoris Sequence. Equivalenceof simplicial and singular homology.
Degree of a self-map of a sphere. Cell complexes and cellular homology. Application: the hairy balltheorem.
Cohomology of spaces and the Universal Coefficient Theorem (proof not examinable). Cup products.Kunneth Theorem (without proof). Topological manifolds and orientability. The fundamental classof an orientable, closed manifold and the degree of a map between manifolds of the same dimension.Poincare Duality (without proof).
Reading
1. A. Hatcher, Algebraic Topology (Cambridge University Press, 2001). Chapters 3 and 4.
2. G. Bredon, Topology and Geometry (Springer, 1997). Chapters 4 and 5.
3. J. Vick, Homology Theory, Graduate Texts in Mathematics 145 (Springer, 1973).
13.4 Analytic Number Theory —Prof. Heath-Brown—16MT
Recommended Prerequisites Complex analysis (holomorphic and meromorphic functions, Cauchy’sResidue Theorem, Evaluation of integrals by contour integration, Uniformly convergent sums of holo-morphic functions). Elementary number theory (Unique Factorization Theorem).
Overview The course aims to introduce students to the theory of prime numbers, showing howthe irregularities in this elusive sequence can be tamed by the power of complex analysis. Thecourse builds up to the Prime Number Theorem which is the corner-stone of prime number theory,and culminates in a description of the Riemann Hypothesis, which is arguably the most importantunsolved problem in modern mathematics.
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Learning Outcomes Students will learn to handle multiplicative functions, to deal with Dirichletseries as functions of a complex variable, and to prove the Prime Number Theorem and simplevariants.
Synopsis Introductory material on primes.
Arithmetic functions — Mobius function, Euler function, Divisor function, Sigma function — mul-tiplicativity.
Dirichlet series — Euler products — von Mangoldt function.
Riemann Zeta-function — analytic continuation to Re(s) > 0.
Non-vanishing of ζ(s) on Re(s) = 1.
Proof of the prime number theorem.
The Riemann hypothesis and its significance.
The Gamma function, the functional equation for ζ(s), the value of ζ(s) at negative integers.
Reading
1. T.M. Apostol, Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics(Springer-Verlag, 1976). Chapters 2,3,11,12 and 13.
2. M. Ram Murty, Problems in Analytic Number Theory (Springer, 2001). Chapters 1 – 5.
3. G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers (Sixth edition, OxfordUniversity Press, 2008). Chapters 16 ,17 and 18.
4. G.J.O. Jameson, The Prime Number Theorem, LMS Student Texts 53 (Cambridge UniversityPress, 2003).
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13.5 Analytic Topology — Dr Suabedissen — 16MT
Recommended Prerequisites Part A Topology; a basic knowledge of Set Theory, includingcardinal arithmetic, ordinals and the Axiom of Choice, will also be useful.
Overview The aim of the course is to present a range of major theory and theorems, both impor-tant and elegant in themselves and with important applications within topology and to mathematicsas a whole. Central to the course is the general theory of compactness and Tychonoff’s theorem, oneof the most important in all mathematics (with applications across mathematics and in mathematicallogic) and computer science.
Synopsis Bases and initial topologies (including pointwise convergence and the Tychonoff producttopology). Separation axioms, continuous functions, Urysohn’s lemma. Separable, Lindelof and sec-ond countable spaces. Urysohn’s metrization theorem. Filters and ultrafilters. Tychonoff’s theorem.Compactifications, in particular the Alexandroff One-Point Compactification and the Stone–CechCompactification. Connectedness and local connectedness. Components and quasi-components.Totally disconnected compact spaces, Boolean algebras and Stone spaces. Paracompactness (brieftreatment).
Reading
1. S. Willard, General Topology (Addison–Wesley, 1970), Chs. 1–8.
2. N. Bourbaki, General Topology (Springer-Verlag, 1989), Ch. 1.
13.6 Commutative Algebra — Prof. Segal — 16HT
Recommended Prerequisites A thorough knowledge of the second-year algebra courses, inparticular rings, ideals and fields.
Overview Amongst the most familiar objects in mathematics are the ring of integers and thepolynomial rings over fields. These play a fundamental role in number theory and in algebraicgeometry, respectively. The course explores the basic properties of such rings.
Synopsis Modules, ideals, prime ideals, maximal ideals.Noetherian rings; Hilbert basis theorem. Minimal primes.Localization.Polynomial rings and algebraic sets. Weak Nullstellensatz.Nilradical and Jacobson radical; strong Nullstellensatz.Artin-Rees Lemma; Krull intersection theorem.Integral extensions. Prime ideals in integral extensions.Noether Normalization Lemma.Krull dimension; ‘Principal ideal theorem’; dimension of an affine algebra.
Reading
1. M. F. Atiyah and I. G. MacDonald: Introduction to Commutative Algebra, (Addison-Wesley,1969).
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13.7 Godel’s Incompleteness Theorems — Dr Isaacson — 16HT
Recommended Prerequisites This course presupposes knowledge of first-order predicate logicup to and including soundness and completeness theorems for a formal system of first-order predicatelogic (B1 Logic).
Overview The starting point is Godel’s mathematical sharpening of Hilbert’s insight that manip-ulating symbols and expressions of a formal language has the same formal character as arithmeticaloperations on natural numbers. This allows the construction for any consistent formal system con-taining basic arithmetic of a ‘diagonal’ sentence in the language of that system which is true butnot provable in the system. By further study we are able to establish the intrinsic meaning of sucha sentence. These techniques lead to a mathematical theory of formal provability which generalizesthe earlier results. We end with results that further sharpen understanding of formal provability.
Learning Outcomes Understanding of arithmetization of formal syntax and its use to establishincompleteness of formal systems; the meaning of undecidable diagonal sentences; a mathemati-cal theory of formal provability; precise limits to formal provability and ways of knowing that anunprovable sentence is true.
Synopsis Godel numbering of a formal language; the diagonal lemma. Expressibility in a formallanguage. The arithmetical undefinability of truth in arithmetic. Formal systems of arithmetic;arithmetical proof predicates. Σ0-completeness and Σ1-completeness. The arithmetical hierarchy.ω-consistency and 1-consistency; the first Godel incompleteness theorem. Separability; the Rosserincompleteness theorem. Adequacy conditions for a provability predicate. The second Godel in-completeness theorem; Lob’s theorem. Provable Σ1-completeness. Provability logic; the fixed pointtheorem. The ω-rule.
Reading
1. Lecture notes for the course.
Further Reading
1. Raymond M. Smullyan, Godel’s Incompleteness Theorems (Oxford University Press, 1992).
2. George S. Boolos and Richard C. Jeffrey, Computability and Logic (3rd edition, CambridgeUniversity Press, 1989), Chs 15, 16, 27 (pp 170–190, 268-284).
3. George Boolos, The Logic of Provability (Cambridge University Press, 1993).
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13.8 Introduction to Representation Theory — Prof Nikolov — 16 MT
Recommended Prerequisites: All second year algebra.
Overview This course gives an introduction to the representation theory of finite groups and finitedimensional algebras. Representation theory is a fundamental tool for studying symmetry by meansof linear algebra: it is studied in a way in which a given group or algebra may act on vector spaces,giving rise to the notion of a representation.
We start in a more general setting, studying modules over rings, in particular over euclidean domains,and their applications. We eventually restrict ourselves to modules over algebras (rings that carry avector space structure). A large part of the course will deal with the structure theory of semisimplealgebras and their modules (representations). We will prove the Jordan-Holder Theorem for modules.Moreover, we will prove that any finite-dimensional semisimple algebra is isomorphic to a productof matrix rings (Wedderburn’s Theorem over C).
In the later part of the course we apply the developed material to group algebras, and classify whengroup algebras are semisimple (Maschke’s Theorem).
Learning Outcomes Students will have a sound knowledge of the theory of non-commutativerings, ideals, associative algebras, modules over euclidean domains and applications. They willknow in particular simple modules and semisimple algebras and they will be familiar with examples.They will appreciate important results in the course such as the Jordan-Holder Theorem, Schur’sLemma, and the Wedderburn Theorem. They will be familiar with the classification of semisimplealgebras over C and be able to apply this.
Synopsis Noncommutative rings, one- and two-sided ideals. Associative algebras (over fields).Main examples: matrix algebras, polynomial rings and quotients of polynomial rings. Group alge-bras, representations of groups.
Modules over euclidean domains and applications such as finitely generated abelian groups, ratio-nal canonical forms. Modules and their relationship with representations. Simple and semisimplemodules, composition series of a module, Jordan-Holder Theorem. Semisimple algebras. Schur’sLemma, the Wedderburn Theorem, Maschke’s Theorem.
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13.9 Lambda Calculus and Types — Dr Vicary — 16 lectures HT
Recommended Prerequisites: There are no prerequisites, but the course will assume familiaritywith constructing mathematical proofs. Some basic knowledge of computability would be useful forone of the topics (the Models of Computation course is much more than enough), but is certainlynot necessary.
Overview As a language for describing functions, any literate computer scientist would expect tounderstand the vocabulary of the lambda calculus. It is folklore that various forms of the lambdacalculus are the prototypical functional programming languages, but the pure theory of the lambdacalculus is also extremely attractive in its own right. This course introduces the terminology andphilosophy of the lambda calculus, and then covers a range of self-contained topics studying thelanguage and some related structures. Topics covered include the equational theory, term rewrit-ing and reduction strategies, combinatory logic, Turing completeness and type systems. As such,the course will also function as a brief introduction to many facets of theoretical computer science,illustrating each (and showing the connections with practical computer science) by its relation tothe lambda calculus. There are no prerequisites, but the course will assume familiarity with con-struting mathematical proofs. Some basic knowledge of computability would be useful for one of thetopics (the Models of Computation course is much more than enough), but is certainly not necessary.
Learning Outcomes The course is an introductory overview of the foundations of computerscience with particular reference to the lambda-calculus. Students will
• understand the syntax and equational theory of the untyped lambda-calculus, and gain famil-iarity with manipulation of terms;
• be exposed to a variety of inductive proofs over recursive structures;
• learn techniques for analysing term rewriting systems, with particular reference to beta-reduction;
• see the connections between lambda-calculus and computabilty, and an example of how anundecidability proof can be constructed;
• see the connections and distinctions between lambda-calculus and combinatory logic;
• learn about simple type systems for the lambda-calculus, and how to prove a strong normal-ization result;
• understand how to deduce types for terms, and prove correctness of a principal type algorithm.
Synopsis Chapter 0 (1 lecture)
Introductory lecture. Preparation for use of inductive definitions and proofs.
Chapters 1–3 (5 lectures)
Terms, free and bound variables, alpha-conversion, substitution, variable convention, contexts, theformal theory lambda beta, the eta rule, fixed point combinators, lambda-theories. Reduction.Compatible closure, reflexive transitive closure, diamond and Church-Rosser properties for generalnotions of reduction. beta-reduction, proof of the Church-Rosser property (via parallel reduction),connection between beta-reduction and lambda beta, consistency of lambda beta. Inconsistency of
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equating all terms without beta-normal form. Reduction strategies, head and leftmost reduction.Standard reductions. Proof that leftmost reduction is normalising. Statement, without proof, ofGenericity Lemma, and simple applications.
Chapter 4 (2 lectures)
Church numerals, definability of total recursive functions. Second Recursion Theorem, Scott-CurryTheorem, undecideability of equality in lambda beta. Briefly, extension to partial functions.
Chapter 5 (2 lectures)
Untyped combinatory algebras. Abstraction algorithm, combinatory completeness, translations toand from untyped lambda-calculus, mismatches between combinary logic and lambda-calculus, ba-sis. Term algebras.
Chapters 6–8 (6 lectures)
Simple type assignment a la Curry using Hindley’s TA lambda system. Contexts and deductions.Subject Construction Lemma, Subject Reduction Theorem and failure of Subject Expansion. Briefly,a system with type invariance under equality. Informal and cursory treatment of Curry-Howardisomorphism. Tait’s proof of strong normalisation. Consequences: no fixed point combinators, poordefinability power. Pointer to literature on PCF as the obvious extension of simple types to coverall computable functions. Type substitutions and unification, Robinson’s algorithm. Principal Typealgorithm and correctness.
Syllabus Terms, formal theories lambda beta and lambda beta eta , fixed point combinators;reduction, Church-Rosser property of beta-reduction and consistency of lambda beta; reductionstrategies, standard reduction sequences, proof that leftmost reduction is normalising; Church nu-merals, definability of total recursive functions in the lambda-calculus, Second Recusion Theoremand undecidability results; combinatory algebras, combinatory completeness, basis; simple types a laCurry, type deductions, Subject Reduction Theorem, strong normalisation and consequences; typesubstitutions, unification, correctness of Principal Type Algorithm.
Reading List Essential
• Andrew Ker, lecture notes. Available online and handed out in the lectures. Comprehensivenotes on the entire course, including practice questions and class exercises.
Useful Background
• H. P. Barendregt, The Lambda Calculus, North-Holland, revised edition, 1984.
• J. R. Hindley, Basic Simple Type Theory, CUP Cambridge Tracts in Theoretical ComputerScience 42, 1997.
• J-Y. Girard, Y.Lafont and P. Taylor, Proofs and Types, CUP Cambridge Tracts in TheoreticalComputer Science 7, 1989.
• C. Hankin, Lambda Calculi, A Guide for Computer Scientists, OUP Graduate Texts in Com-puter Science, 1994.
• J. R. Hindley & J. P. Seldin, Introduction to Combinators and Lambda-Calculus (CambridgeUniversity Press, 1986).
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13.10 Lie Algebras — Prof. Ciubotaru — 16MT
Recommended Prerequisites Part B course B2a. A thorough knowledge of linear algebra andthe second year algebra courses; in particular familiarity with group actions, quotient rings andvector spaces, isomorphism theorems and inner product spaces will be assumed. Some familiaritywith the Jordan–Holder theorem and the general ideas of representation theory will be an advantage.
Overview Lie Algebras are mathematical objects which, besides being of interest in their ownright, elucidate problems in several areas in mathematics. The classification of the finite-dimensionalcomplex Lie algebras is a beautiful piece of applied linear algebra. The aims of this course are tointroduce Lie algebras, develop some of the techniques for studying them, and describe parts of theclassification mentioned above, especially the parts concerning root systems and Dynkin diagrams.
Learning Outcomes Students will learn how to utilise various techniques for working with Liealgebras, and they will gain an understanding of parts of a major classification result.
Synopsis Definition of Lie algebras, small-dimensional examples, some classical groups and theirLie algebras (treated informally). Ideals, subalgebras, homomorphisms, modules.
Nilpotent algebras, Engel’s theorem; soluble algebras, Lie’s theorem. Semisimple algebras andKilling form, Cartan’s criteria for solubility and semisimplicity, Weyl’s theorem on complete re-ducibility of representations of semisimple Lie algebras.
The root space decomposition of a Lie algebra; root systems, Cartan matrices and Dynkin diagrams.Discussion of classification of irreducible root systems and semisimple Lie algebras.
Reading
1. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts inMathematics 9 (Springer-Verlag, 1972, reprinted 1997). Chapters 1–3 are relevant and part ofthe course will follow Chapter 3 closely.
2. B. Hall, Lie Groups, Lie Algebras, and Representations. An Elementary Introduction, Gradu-ate Texts in Mathematics 222 (Springer-Verlag, 2003).
3. K. Erdmann, M. J. Wildon, Introduction to Lie Algebras (Springer-Verlag, 2006), ISBN:1846280400.
Additional Reading
1. J.-P. Serre, Complex Semisimple Lie Algebras (Springer, 1987). Rather condensed, assumesthe basic results. Very elegant proofs.
2. N. Bourbaki, Lie Algebras and Lie Groups (Masson, 1982). Chapters 1 and 4–6 are relevant;this text fills in some of the gaps in Serre’s text.
3. William Fulton, Joe Harris, Representation theory: a first course, GTM, Springer.
13.11 Model Theory — Prof. Zilber — 16MT
Recommended Prerequisites This course presupposes basic knowledge of First Order PredicateCalculus up to and including the Soundness and Completeness Theorems. A familiarity with (atleast the statement of) the Compactness Theorem would also be desirable.
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Overview The course deepens a student’s understanding of the notion of a mathematical struc-ture and of the logical formalism that underlies every mathematical theory, taking B1 Logic as astarting point. Various examples emphasise the connection between logical notions and practicalmathematics.
The concepts of completeness and categoricity will be studied and some more advanced technicalnotions, up to elements of modern stability theory, will be introduced.
Learning Outcomes Students will have developed an in depth knowledge of the notion of analgebraic mathematical structure and of its logical theory, taking B1 Logic as a starting point. Theywill have an understanding of the concepts of completeness and categoricity and more advancedtechnical notions.
Synopsis Structures. The first-order language for structures. The Compactness Theorem forfirst-order logic. Elementary embeddings. Lowenheim–Skolem theorems. Preservation theorems forsubstructures. Model Completeness. Quantifier elimination.
Categoricity for first-order theories. Types and saturation. Omitting types. The Ryll Nardzewskitheorem characterizing aleph-zero categorical theories. Theories with few types. Ultraproducts.
Reading
1. D. Marker, Model Theory: An Introduction (Springer, 2002).
2. W. Hodges, Shorter Model Theory (Cambridge University Press, 1997).
3. J. Bridge, Beginning Model Theory (Oxford University Press, 1977). (Out of print but can befound in libraries.)
Further reading
1. All topics discussed (and much more) can also be found in W. Hodges, Model Theory (Cam-bridge University Press, 1993).
13.12 Modular Forms — Prof Lauder — 16MT
Prerequisites Part A Analysis and Algebra (core material) and Prelims Group Actions (corematerial). Part A Number Theory Topology and Part B Geometry of Surfaces, Algebraic Curvesare useful but not essential.
Overview The course aims to introduce students to the beautiful theory of modular forms, one ofthe cornerstones of modern number theory. This theory is a rich and challenging blend of methodsfrom complex analysis and linear algebra, and an explicit application of group actions.
Learning Outcomes The student will learn about modular curves and spaces of modular forms,and understand in special cases how to compute their genus and dimension, respectively. They willsee that modular forms can be described explicitly via their q-expansions, and they will be familiarwith explicit examples of modular forms. They will learn about the rich algebraic structure onspaces of modular forms, given by Hecke operators and the Petersson inner product.
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Synopsis
1. Overview and examples of modular forms. Definition and basic properties of modular forms.
2. Topology of modular curves: a fundamental domain for the full modular group; fundamentaldomains for subgroups Γ of finite index in the modular group; the compact surfaces XΓ; explicittriangulations of XΓ and the computation of the genus using the Euler characteristic formula;the congruence subgroups Γ(N),Γ1(N) and Γ0(N); examples of genus computations.
3. Dimensions of spaces of modular forms: general dimension formula (proof non-examinable);the valence formula (proof non-examinable).
4. Examples of modular forms: Eisenstein series in level 1; Ramanujan’s ∆ function; some arith-metic applications.
5. The Petersson inner product.
6. Modular forms as functions on lattices: modular forms of level 1 as functions on lattices;Eisenstein series revisited.
7. Hecke operators in level 1: Hecke operators on lattices; Hecke operators on modular forms andtheir q-expansions; Hecke operators are Hermitian; multiplicity one.
Reading
1. F. Diamond and J. Shurman, A First Course in Modular Forms, Graduate Texts in Mathe-matics 228, Springer-Verlag, 2005.
2. R.C. Gunning, Lectures on Modular Forms, Annals of mathematical studies 48, PrincetonUniversity Press, 1962.
3. J.S. Milne, Modular Functions and Modular Forms:www.jmilne.org/math/CourseNotes/mf.html
4. J.-P. Serre, Chapter VII, A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer-Verlag, 1973.
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13.13 Topology and Groups — Prof. Dancer — 16 MT
Recommended Prerequisites 2nd year Topology is essential. 2nd year Group Theory is recom-mended.
Overview This course introduces the important link between topology and group theory. On theone hand, associated to each space, there is a group, known as its fundamental group. This can beused to solve topological problems using algebraic methods. On the other hand, many results aboutgroups are best proved and understood using topology. For example, presentations of groups, wherethe group is defined using generators and relations, have a topological interpretation. The endpointof the course is the Nielsen–Shreier Theorem, an important, purely algebraic result, which is provedusing topological techniques.
Synopsis Homotopic mappings, homotopy equivalence. Simplicial complexes. Simplicial approx-imation theorem.
The fundamental group of a space. The fundamental group of a circle. Application: the fundamentaltheorem of algebra. The fundamental groups of spheres.
Free groups. Existence and uniqueness of reduced representatives of group elements. The funda-mental group of a graph.
Groups defined by generators and relations (with examples). Tietze transformations.
The free product of two groups. Amalgamated free products.
The Seifert–van Kampen Theorem.
Cell complexes. The fundamental group of a cell complex (with examples). The realization of anyfinitely presented group as the fundamental group of a finite cell complex.
Covering spaces. Liftings of paths and homotopies. A covering map induces an injection betweenfundamental groups. The use of covering spaces to determine fundamental groups: the circle again,and real projective n-space. The correspondence between covering spaces and subgroups of thefundamental group. Regular covering spaces and normal subgroups.
Cayley graphs of a group. The relationship between the universal cover of a cell complex, and theCayley graph of its fundamental group. The Cayley 2-complex of a group.
The Nielsen–Schreier Theorem (every subgroup of a finitely generated free group is free) provedusing covering spaces.
Reading
1. John Stillwell, Classical Topology and Combinatorial Group Theory (Springer-Verlag, 1993).
Additional Reading
1. D. Cohen, Combinatorial Group Theory: A Topological Approach, Student Texts 14 (LondonMathematical Society, 1989), Chapters 1–7.
2. A. Hatcher, Algebraic Topology (CUP, 2001), Chapter. 1.
3. M. Hall, Jr, The Theory of Groups (Macmillan, 1959), Chapters. 1–7, 12, 17 .
4. D. L. Johnson, Presentations of Groups, Student Texts 15 (Second Edition, London Mathe-matical Society, Cambridge University Press, 1997). Chapters. 1–5, 10,13.
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5. W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory (Dover Publications,1976). Chapters. 1–4.
13.14 Schedule II
13.15 Algebraic Geometry — Dr Berczi — 16MT
Recommended Prerequisites Part A Group Theory and Introduction to Fields (B3 AlgebraicCurves useful but not essential).
Overview Algebraic geometry is the study of algebraic varieties: an algebraic variety is roughlyspeaking, a locus defined by polynomial equations. One of the advantages of algebraic geometry isthat it is purely algebraically defined and applied to any field, including fields of finite characteristic.It is geometry based on algebra rather than calculus, but over the real or complex numbers it providesa rich source of examples and inspiration to other areas of geometry.
Synopsis Affine algebraic varieties, the Zariski topology, morphisms of affine varieties. Irreduciblevarieties.
Projective space. Projective varieties, affine cones over projective varieties. The Zariski topologyon projective varieties. The projective closure of affine variety. Morphisms of projective varieties.Projective equivalence.
Veronese morphism: definition, examples. Veronese morphisms are isomorphisms onto their image;statement, and proof in simple cases. Subvarieties of Veronese varieties. Segre maps and productsof varieties, Categorical products: the image of Segre map gives the categorical product.
Coordinate rings. Hilbert’s Nullstellensatz. Correspondence between affine varieties (and morphismsbetween them) and finitely generate reduced k-algebras (and morphisms between them). Gradedrings and homogeneous ideals. Homogeneous coordinate rings.
Categorical quotients of affine varieties by certain group actions. The maximal spectrum.
Primary decomposition of ideals.
Discrete invariants projective varieties: degree dimension, Hilbert function. Statement of theoremdefining Hilbert polynomial.
Quasi-projective varieties, and morphisms of them. The Zariski topology has a basis of affine opensubsets. Rings of regular functions on open subsets and points of quasi-projective varieties. Thering of regular functions on an affine variety in the coordinate ring. Localisation and relationshipwith rings of regular functions.
Tangent space and smooth points. The singular locus is a closed subvariety. Algebraic re-formulationof the tangent space. Differentiable maps between tangent spaces.
Function fields of irreducible quasi-projective varieties. Rational maps between irreducible vari-eties, and composition of rational maps. Birational equivalence. Correspondence between dominantrational maps and homomorphisms of function fields. Blow-ups: of affine space at appoint, of sub-varieties of affine space, and general quasi-projective varieties along general subvarieties. Statementof Hironaka’s Desingularisation Theorem. Every irreducible variety is birational to hypersurface.Re-formulation of dimension. Smooth points are a dense open subset.
Reading KE Smith et al, An Invitation to Algebraic Geometry, (Springer 2000), Chapters 1–8.
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Further Reading
1. M Reid, Undergraduate Algebraic Geometry, LMS Student Texts 12, (Cambridge 1988).
2. K Hulek, Elementary Algebraic Geometry, Student Mathematical Library 20. (AmericanMathematical Society, 2003).
13.16 Axiomatic Set Theory — Dr Suabedissen — 16HT
Recommended Prerequisites This course presupposes basic knowledge of First Order PredicateCalculus up to and including the Soundness and Completeness Theorems, together with a courseon basic set theory, including cardinals and ordinals, the Axiom of Choice and the Well OrderingPrinciple.
Overview Inner models and consistency proofs lie at the heart of modern Set Theory, historicallyas well as in terms of importance. In this course we shall introduce the first and most importantof inner models, Godel’s constructible universe, and use it to derive some fundamental consistencyresults.
Synopsis A review of the axioms of ZF set theory. The recursion theorem for the set of naturalnumbers and for the class of ordinals. The Cumulative Hierarchy of sets and the consistency of theAxiom of Foundation as an example of the method of inner models. Levy’s Reflection Principle.Godel’s inner model of constructible sets and the consistency of the Axiom of Constructibility(V = L). The fact that V = L implies the Axiom of Choice. Some advanced cardinal arithmetic.The fact that V = L implies the Generalized Continuum Hypothesis.
Reading For the review of ZF set theory:
1. D. Goldrei, Classic Set Theory (Chapman and Hall, 1996).
For course topics (and much more):
1. K. Kunen, Set Theory: An Introduction to Independence Proofs (North Holland, 1983) (nowin paperback). Review: Chapter 1. Course topics: Chapters 3, 4, 5, 6 (excluding section 5).
Further Reading
1. K. Hrbacek and T. Jech, Introduction to Set Theory (3rd edition, M Dekker, 1999).
13.17 Homological Algebra — Prof. Kremnitzer — 16MT
Synopsis Chain complexes: complexes of R-modules, operations on chain complexes, long exactsequences, chain homotopies, mapping cones and cylinders (4 hours) Derived functors: delta functors,projective and injective resolutions, left and right derived functors (5 hours) Tor and Ext: Tor andflatness, Ext and extensions, universal coefficients theorems, Koszul resolutions (4 hours) Grouphomology and cohomology: definition, interpretation of H1 and H2, universal central extensions,the Bar resolution (3 hours).
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Reading Weibel, Charles An introduction to Homological algebra (see Google Books)
13.18 Infinite Groups — Prof. Nikolov — 16HT
Recommended Prerequisites A thorough knowledge of the second-year algebra courses; inparticular, familiarity with group actions, quotient rings and quotient groups, and isomorphismtheorems will be assumed. Familiarity with the Commutative Algebra course will be helpful but notessential.
Overview The concept of a group is so general that anything which is true of all groups tendsto be rather trivial. In contrast, groups that arise in some specific context often have a rich andbeautiful theory. The course introduces some natural families of groups, various questions that onecan ask about them, and various methods used to answer these questions; these involve among otherthings rings and trees.
Synopsis Free groups and their subgroups; finitely generated groups: counting finite-index sub-groups; finite presentations and decision problems; Linear groups: residual finiteness; structure ofsoluble linear groups; Nilpotency and solubility: lower central series and derived series; structuraland residual properties of finitely generated nilpotent groups and polycyclic groups; characterizationof polycyclic groups as soluble Z-linear groups; Torsion groups and the General Burnside Problem.
Reading
1. D. J. S. Robinson, A course in the theory of groups, 2nd ed., Graduate texts in Mathematics,(Springer-Verlag, 1995). Chapters 2, 5, 6, 15.
2. D. Segal, Polycyclic groups, (CUP, 2005) Chapters 1 and 2.
13.19 Non-Commutative Rings — Prof. Ardakov — 16HT
Recommended Prerequisites Prerequisites: Part A Algebra (Algebra 1 and 2 from 2015-2016).Recommended background: Introduction to Representation Theory B2a, Part B Commutative Al-gebra (from 2016 onwards).
Overview This course builds on Algebra 2 from the second year. We will look at several classesof non-commutative rings and try to explain the idea that they should be thought of as functionson ”non-commutative spaces”. Along the way, we will prove several beautiful structure theoremsfor Noetherian rings and their modules.
Learning Outcomes Students will be able to appreciate powerful structure theorems, and befamiliar with examples of non-commutative rings arising from various parts of mathematics.
Synopsis 1. Examples of non-commutative Noetherian rings: enveloping algebras, rings of differ-ential operators, group rings of polycyclic groups. Filtered and graded rings. (3 hours)
2. Jacobson radical in general rings. Jacobson’s density theorem. Artin-Wedderburn. (3 hours)
3. Ore localisation. Goldie’s Theorem. Lifting Ore sets from graded rings. (4 hours)
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4. Gabber’s Theorem on the integrability of the characteristic variety. (3 hours)
5. Hilbert polynomials. Bernstein’s Inequality for modules over the Weyl algebra. (3 hours)
Reading
1. K.R. Goodearl and R.B. Warfield, An Introduction to Noncommutative Noetherian Rings(CUP, 2004).
Further reading
1. M. Atiyah and I. MacDonald, Introduction to Commutative Algebra (Westview Press, 1994).
2. S.C. Coutinho, A Primer of Algebraic D-modlues (CUP, 1995).
3. J. Bjork, Analytic D-Modules and Applications (Springer, 1993).
13.20 Geometric Group Theory — Prof Papazoglou — 16MT
Recommended Prerequisites. The Topology & Groups course is a helpful, though not essentialprerequisite.
Overview. The aim of this course is to introduce the fundamental methods and problems ofgeometric group theory and discuss their relationship to topology and geometry.
The first part of the course begins with an introduction to presentations and the list of problemsof M. Dehn. It continues with the theory of group actions on trees and the structural study offundamental groups of graphs of groups.
The second part of the course focuses on modern geometric techniques and it provides an introductionto the theory of Gromov hyperbolic groups.
Synopsis. Free groups. Group presentations. Dehn’s problems. Residually finite groups.
Group actions on trees. Amalgams, HNN-extensions, graphs of groups, subgroup theorems forgroups acting on trees.
Quasi-isometries. Hyperbolic groups. Solution of the word and conjugacy problem for hyperbolicgroups.
If time allows: Small Cancellation Groups, Stallings Theorem, Boundaries.
Reading.
1. J.P. Serre, Trees (Springer Verlag 1978).
2. M. Bridson, A. Haefliger, Metric Spaces of Non-positive Curvature, Part III (Springer, 1999),Chapters I.8, III.H.1, III. Gamma 5.
3. H. Short et al., ‘Notes on word hyperbolic groups’, Group Theory from a Geometrical View-point, Proc. ICTP Trieste (eds E. Ghys, A. Haefliger, A. Verjovsky, World Scientific 1990)
available online at: http://www.cmi.univ-mrs.fr/ hamish/
4. C.F. Miller, Combinatorial Group Theory, notes:http://www.ms.unimelb.edu.au/ cfm/notes/cgt-notes.pdf.
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Additional Reading.
1. G. Baumslag, Topics in Combinatorial Group Theory (Birkhauser, 1993).
2. O. Bogopolski, Introduction to Group Theory (EMS Textbooks in Mathematics, 2008).
3. R. Lyndon, P. Schupp, Combinatorial Group Theory (Springer, 2001).
4. W. Magnus, A. Karass, D. Solitar,Combinatorial Group Theory: Presentations of Groups inTerms of Generators and Relations (Dover Publications, 2004).
5. P. de la Harpe, Topics in Geometric Group Theory, (University of Chicago Press, 2000).
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14 SECTION B: APPLICABLE THEORIES
14.1 Schedule I
14.2 Applied Probability — Dept. of Statistics to allocate — 16 MT
Recommended Prerequisites: Part A Probability.
Overview This course is intended to show the power and range of probability by considering realexamples in which probabilistic modelling is inescapable and useful. Theory will be developed asrequired to deal with the examples.
Synopsis Poisson processes and birth processes. Continuous-time Markov chains. Transitionrates, jump chains and holding times. Forward and backward equations. Class structure, hittingtimes and absorption probabilities. Recurrence and transience. Invariant distributions and limitingbehaviour. Time reversal. Renewal theory. Limit theorems: strong law of large numbers, stronglaw and central limit theorem of renewal theory, elementary renewal theorem, renewal theorem, keyrenewal theorem. Excess life, inspection paradox.
Applications in areas such as: queues and queueing networks - M/M/s queue, Erlang’s formula,queues in tandem and networks of queues, M/G/1 and G/M/1 queues; insurance ruin models;applications in applied sciences.
Reading
1. J. R. Norris, Markov Chains (Cambridge University Press, 1997).
2. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes (3rd edition, OxfordUniversity Press, 2001).
3. G. R. Grimmett and D. R. Stirzaker, One Thousand Exercises in Probability (Oxford Univer-sity Press, 2001).
4. S. M. Ross, Introduction to Probability Models (4th edition, Academic Press, 1989).
5. D. R. Stirzaker: Elementary Probability (2nd edition, Cambridge University Press, 2003).
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14.3 Categories, Proofs and Processes — Prof Abramsky — 20 lectures+ extra reading MT
Recommended Prerequisites Some familiarity with basic discrete mathematics: sets, functions,relations, mathematical induction. Basic familiarity with logic: propositional and predicate calculus.Some first acquaintance with abstract algebra: vector spaces and linear maps, and/or groups andgroup homomorphisms. Some familiarity with programming, particularly functional programming,would be useful but is not essential.
Overview Category Theory is a powerful mathematical formalism which has become an importanttool in modern mathematics, logic and computer science. One main idea of Category Theory is tostudy mathematical ‘universes’, collections of mathematical structures and their structure-preservingtransformations, as mathematical structures in their own right, i.e. categories - which have theirown structure-preserving transformations (functors). This is a very powerful perspective, whichallows many important structural concepts of mathematics to be studied at the appropriate levelof generality, and brings many common underlying structures to light, yielding new connectionsbetween apparently different situations.Another important aspect is that set-theoretic reasoning with elements is replaced by reasoning interms of arrows. This is more general, more robust, and reveals more about the intrinsic structureunderlying particular set-theoretic representations.Category theory has many important connections to logic. We shall in particular show how itilluminates the study of formal proofs as mathematical objects in their own right. This will involvelooking at the Curry-Howard isomorphism between proofs and programs, and at Linear Logic, aresource-sensitive logic. Both of these topics have many important applications in Computer Science.Category theory has also deeply influenced the design of modern (especially functional) programminglanguages, and the study of program transformations. One exciting recent development we will lookat will be the development of the idea of coalgebra, which allows the formulation of a notion ofcoinduction, dual to that of mathematical induction, which provides powerful principles for definingand reasoning about infinite objects.This course will develop the basic ideas of Category Theory, and explore its applications to the studyof proofs in logic, and to the algebraic structure of programs and programming languages.Remark: It is recommended that students who intend to write their MSc thesis in the QuantumGroup at the Department of Computer Science take this course and additionally the QuantumComputer Science course in Hilary term.
Learning Outcomes
• To master the basic concepts and methods of categories.
• To understand how category-theory can be used to structure mathematical ideas, with the con-cepts of functoriality, naturality and universality; and how reasoning with objects and arrowscan replace reasoning with sets and elements. To learn the basic ideas of using commutativediagrams and unique existence properties.
• To understand the connections between categories and logic, focussing on structural prooftheory and the Curry-Howard isomorphism.
• To understand how some basic forms of computational processes can be modelled with cate-gories.
Synopsis
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• Introduction to category theory. Categories, functors, natural transformations. Isomorphisms.monics and epics. Products and coproducts. Universal constructions. Cartesian closed cate-gories. Symmetric monoidal closed categories. The ideas will be illustrated with many exam-ples, from both mathematics and Computer Science.
• Introduction to structural proof theory. Natural deduction, simply typed lambda calculus, theCurry-Howard correspondence. Introduction to Linear Logic. The connection between logicand categories.
• Further topics in category theory. Algebras and coalgebras. Connections to programming(structural recursion and corecursion).
A. Categories
• Background and definition
• Monics, epics, isomorphisms
• Products and coproducts
• Limits and colimits
• Functors and natural transformations
• Universal arrows and adjunctions
• Cartesian closed categories
B. Connections with logic
• Natural deduction, lambda calculus and Curry-Howard isomorphism
• Gentzen sequent calculus and linear logic
• Symmetric monoidal closed categories and categorical semantics of linear logic
• Algebras
• Coalgebras
• Topoi
Reading List Slides will be provided. The standard reference is
• Abramsky and Tzevelekos, ”Introduction to Categories and Categorical Logic”,http://web.comlab.ox.ac.uk/people/Bob.Coecke/AbrNikos.pdf
The following books provide useful background reading.
• B.C. Pierce, Basic Category Theory for Computer Science, MIT Press (1991)
• F.W. Lawvere, S.H. Schanuel, Conceptual Mathematics, Cambridge University Press (1997)
• S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer (1998)
• M. Barr, C. Wells, Category Theory for Computer Science, 2nd ed., Prentice Hall (1995)
• J.-Y. Girard, Y. Lafont, P. Taylor, Proofs and Types, http://www.paultaylor.eu/stable/prot.pdf
Of these, the book by Pierce provides a very accessible and user-friendly first introduction tothe subject (though we will cover more topics in the course).
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14.4 Communication Theory — Dr Griffiths — 16 MT
Recommended Prerequisites: Part A Probability would be helpful, but not essential.
Overview The aim of the course is to investigate methods for the communication of informationfrom a sender, along a channel of some kind, to a receiver. If errors are not a concern we are interestedin codes that yield fast communication; if the channel is noisy we are interested in achieving bothspeed and reliability. A key concept is that of information as reduction in uncertainty. The highlightof the course is Shannon’s Noisy Coding Theorem.
Learning Outcomes
(i) Know what the various forms of entropy are, and be able to manipulate them.
(ii) Know what data compression and source coding are, and be able to do it.
(iii) Know what channel coding and channel capacity are, and be able to use that.
Synopsis Uncertainty (entropy); conditional uncertainty; information. Chain rules; relative en-tropy; Gibbs’ inequality; asymptotic equipartition and typical sequences. Instantaneous and uniquelydecipherable codes; the noiseless coding theorem for discrete memoryless sources; constructing com-pact codes.
The discrete memoryless channel; decoding rules; the capacity of a channel. The noisy codingtheorem for discrete memoryless sources and binary symmetric channels.
Extensions to more general sources and channels.
Reading
1. D. J. A. Welsh, Codes and Cryptography (Oxford University Press, 1988), Chapters 1–3, 5.
2. T. Cover and J. Thomas, Elements of Information Theory (Wiley, 1991), Chapters 1–5, 8.
Further Reading
1. R. B. Ash, Information Theory (Dover, 1990).
2. D. MacKay, Information Theory, Inference, and Learning Algorithms (Cambridge, 2003). [Canbe seen at: http://www.inference.phy.cam.ac.uk/mackay/itila. Do not infringe the copyright!]
3. G. Jones and J. M. Jones, Information and Coding Theory (Springer, 2000), Chapters 1–5.
4. Y. Suhov & M. Kelbert, Information Theory and Coding by Example (Cambridge UniversityPress, not yet published - available at the end of 2013), Relevant examples.
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14.5 Computer-Aided Formal Verification — Dr Abate —16MT
Overview This course introduces the fundamentals of computer-aided formal verification. Computer-aided formal verification aims to improve the quality of digital systems by using logical reasoning,supported by software tools, to analyse their designs. The idea is to build a mathematical model ofa system and then try to prove properties of it that validate the system’s correctness ? or at leasthelp discover subtle bugs. The proofs can be millions of lines long, so specially-designed computeralgorithms are used to search for and check them.
Learning Outcomes This course provides a survey of several major software-assisted verificationmethods, covering both theory and practical applications. The aim is to familiarise students withthe mathematical principles behind current verification technologies and give them an appreciationof how these technologies are used in industrial system design today.
Synopsis
1. Introduction.
2. Modelling sequential systems, Kripke structures.
3. Temporal logic: LTL, CTL*, and CTL.
4. Specifying systems with temporal logic.
5. Reachability calculations, model checking.
6. Binary Decision Diagrams (BDDs).
7. Algorithms over BDDs.
8. Combinational equivalence checking.
9. Symbolic model checking.
10. Propositional SAT.
11. Model Checking with SAT.
12. Abstraction Refinement.
13. Decision procedures.
14. Decision procedures in Model Checking.
15. Practical, industrial-scale hardware verification.
16. Computer-aided software verification.
Syllabus Introduction to formal hardware verification. Binary Decision Diagrams and their usein combinational equivalence checking. Modelling sequential systems; Kripke structures. Specifyingsystems with temporal logic; CTL*, CTL and LTL. Reachability and symbolic model checking. Newmodel checking approaches based on algorithms for Boolean satisfiability. Automatic abstractionrefinement. Decision procedures and their use in combination with model checking. Practical,industrial-scale hardware verification. Current approaches to computer-aided software verification.
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Reading The lectures will be supplemented with notes and pointers to published articles in thefield. The following may be helpful for reference or further reading on specific topics.
Surveys
1. Formal Verification in Hardware Design: A Survey, by C. Kern and M. R. Greenstreet, ACMTransactions on Design Automation of Systems , vol. 4 (April 1999), pp. 123-193.
2. A Survey of Automated Techniques for Formal Software Verification , by D’Silva et al., IEEETransactions on Computer-Aided Design of Integrated Circuits and Systems (TCAD), 2008(http://www.kroening.com/publications/view−publications−dkw2008.html)
Temporal Logic and Model Checking
1. From Philosophical to Industrial Logics, by M. Vardi, ICLA 2009 (http://www.cs.ox.ac.uk/teaching/courses/2014-2015/computeraidedverification/),
Binary Decision Diagrams and SAT
1. An Introduction to Binary Decision Diagrams, by Henrik Reif Andersen, Lecture Notes (Tech-nical University of Denmark, October 1997)(https://www.cs.ox.ac.uk/files/4298/bdd98.pdf).
2. Formal Hardware Verification with BDDs: An Introduction, by Alan J. Hu, IEEE PacificRim Conference on Communications, Computers, and Signal Processing (1997), pp. 677-682.(https://www.cs.ox.ac.uk/files/4309/97H1.pdf)
3. Chapter 2 in Decision Procedures, by Daniel Kroening and Ofer Strichman, Springer, 2008(http://www.decision-procedures.org/)
4. Handbook of Satisfiability, Biere, Heule, Van Maaren, Walsh, IOS Press 2009.
BDDs and Model Checking
1. Logic in Computer Science: Modelling and reasoning about systems, by Michael Huth andMark Ryan (Cambridge University Press, 2000).
2. Model Checking, by Edmund M. Clarke, Jr., Orna Grumberg, and Doron A. Peled, Secondprinting (The MIT Press, 2000).
3. Concepts, Algorithms, and Tools for Model Checking, Unpublished lecture notes by J.-P. Ka-toen, 1998. (https://www.cs.ox.ac.uk/files/4297/katoen.pdf)
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14.6 Concurrency — Dr Gibson-Robson — 16 lectures + extra readingHT
Overview Computer networks, multiprocessors and parallel algorithms, though radically differ-ent, all provide examples of processes acting in parallel to achieve some goal. All benefit fromthe efficiency of concurrency yet require careful design to ensure that they function correctly. Theconcurrency course introduces the fundamental concepts of concurrency using the notation of Com-municating Sequential Processes. By introducing communication, parallelism, deadlock, live-lock,etc., it shows how CSP represents, and can be used to reason about, concurrent systems. Studentsare taught how to design interactive processes and how to modularise them using synchronisation.One important feature of the module is its use of both algebraic laws and semantic models to reasonabout reactive and concurrent designs. Another is its use of FDR to animate designs and verify thatthey meet their specifications.
Learning Outcomes At the end of the course the student should:
• understand some of the issues and difficulties involved in Concurrency
• be able to specify and model concurrent systems using CSP
• be able to reason about CSP models of systems using both algebraic laws and semantic models
• be able to analyse CSP models of systems using the model checker FDR
Syllabus Deterministic processes: traces, operational semantics; prefixing, choice, concurrencyand communication. Nondeterminism: failures and divergences; nondeterministic choice, hiding andinterleaving. Further operators: pipes and (time permitting) sequential composition. Refinement,specification and proof. Process algebra: equational and inequational reasoning.
Synopsis
• Processes and observations of processes; point synchronisation, events, alphabets. Sequentialprocesses: prefixing, choice, nondeterminism. Operational semantics; traces; algebraic laws.[3]
• Recursion. Complete partial orders and fixed points as a means of explaining recursion; ap-proximation, limits, least fixed points; guardedness and unique fixed points. [1]
• Concurrency. Hiding. Renaming. [3]
• Non-deterministic behaviours, refusals, failures; the determinism ordering. [2]
• Hiding and divergence, the failures-divergences model. [1]
• Specification and correctness. [2]
• Communication, pipes, buffers. Sequential composition. [2]
• Case study. [2]
• + extra reading: Chapters 9 and 10 of “Understanding Concurrent Systems”
Reading List
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Course Text
• A. W. Roscoe, Understanding Concurrent Systems, Chapters 1-8, Springer 2010
Alternatives
• A. W. Roscoe, The Theory and Practice of Concurrency, Chapters 1-7, Prentice-Hall Interna-tional, 1997. (http://www.cs.ox.ac.uk/oucl/work/bill.roscoe/publications/68b.pdf.)
• C. A. R. Hoare, Communicating Sequential Processes, Prentice-Hall International, 1985, http://www.usingcsp.com.(http://www.usingcsp.com)
• S. A. Schneider, Concurrent and Real-time Systems, Chapters 1-8, Wiley, 2000. (http://www.computing.surrey.ac.uk/personal/st/S.Schneider/books/CRTS.pdf)
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14.7 Foundations of Computer Science — Prof P Goldberg — 16 lecturesMT
Overview Computer scientists need to understand what it means for a problem to be determinableby a computer, what it means for a problem to be efficiently determinable by a computer, and howto reason in a semi-automated and automated fashion about computer programs and the structuresthey manipulate. The purpose of this course is to introduce students to the theoretical foundationsof computer science. It is intended both for students who have a degree in computer science and alsofor students with a good theoretical background (e.g. a degree in mathematics) but no exposure totheoretical computer science.
Students taking this course will gain background knowledge that will be useful in the course on:
• Theory of Data & Knowledge Bases
• Automata, Logics & Games
• Software Verification
• Categories, Proofs & Processes
• Game Semantics
• Computer-Aided Formal Verification
• Lambda Calculus & Types
• Logic of Multi-Agent Information Flow
Learning Outcomes At the end of this course, the student should be able to:
1. Describe in detail what is meant by a finite state automaton, a context-free grammar, and aTuring machine, and calculate the behaviour of simple examples of these devices.
2. Design machines of these types to carry out simple computational tasks.
3. Reason about the capabilities of standard machines, and demonstrate that they have limita-tions.
4. Describe precisely what it means for a problem to be in the classes P,NP, and PSPACE, andwhat it means to be complete for a class
5. Classify problems into appropriate complexity classes, including P, NP and PSPACE, and usethis information effectively.
6. Understand the syntax and semantics of propositional logic.
7. Understand the satisfiability problem for propositional logic and its connection with NP hard-ness.
8. Understand first-order predicate logic, along with the complexity/computability of the associ-ated satisfaction and satisfiability problems.
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Syllabus Finite state machines. Reduction of non-deterministic finite automata to determinis-tic finite automata. Regular languges and their closure properties. Regular expressions. Inter-translations between regular expressions and NFA. Context-free grammars and pushdown automata.Intuitive notion of computability. Church’s Thesis. Turing machines and its expressive power. Uni-versal Turing machines. Undecidable problems. Diagonalization and the Halting Problem. Deter-ministic complexity classes. P, EXPTIME and the Hierarchy Theorem. NP and NP-completeness.Space complexity. Propositional logic. Truth tables. Propositional Logic and NP-completeness.Proof systems for Propositional Logic. Syntax and semantics of first-order logic. Complexity offirst-order logic.
Reading List
1. M. Sipser, Introduction to the Theory of Computation, PWS Publishing Company, January1997. (Primary text).
2. J. E. Savage, A. Wesley,Models of Computation-Exploring the Power of Computing,1998.
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14.8 Graph Theory — Prof. Riordan — 16HT
Recommended Prerequisites: Part A Graph Theory is recommended.
Overview Graphs (abstract networks) are among the simplest mathematical structures, but nev-ertheless have a very rich and well-developed structural theory. Since graphs arise naturally in manycontexts within and outside mathematics, Graph Theory is an important area of mathematics, andalso has many applications in other fields such as computer science.
The main aim of the course is to introduce the fundamental ideas of Graph Theory, and some of thebasic techniques of combinatorics.
Learning Outcomes The student will have developed a basic understanding of the properties ofgraphs, and an appreciation of the combinatorial methods used to analyze discrete structures.
Synopsis Introduction: basic definitions and examples. Trees and their characterization. Eulercircuits; long paths and cycles. Vertex colourings: Brooks17 theorem, chromatic polynomial. Edgecolourings: Vizing’s theorem. Planar graphs, including Euler’s formula, dual graphs. Maximumflow - minimum cut theorem: applications including Menger’s theorem and Hall’s theorem. Tutte’stheorem on matchings. Extremal Problems: Turan’s theorem, Zarankiewicz problem, Erdos-Stonetheorem.
Reading B. Bollobas, Modern Graph Theory, Graduate Texts in Mathematics 184 (Springer-Verlag, 1998)
Further Reading J. A. Bondy and U. S. R. Murty, Graph Theory: An Advanced Course, Grad-uate Texts in Mathematics 244 (SpringerVerlag, 2007).
R. Diestel, Graph Theory, Graduate Texts in Mathematics 173 (third edition, Springer-Verlag, 2005).
D. West, Introduction to Graph Theory (second edition, PrenticeHall, 2001).
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14.9 Quantum Computer Science—Prof Bob Coecke —24 Lectures MT
Prerequisites We do not assume any prior knowledge of quantum mechanics. However, a solidunderstanding of basic linear algebra (finite-dimensional vector spaces, matrices, eigenvectors andeigenvalues, linear maps etc.) is required as a pre-requisite. The course notes and the slides containan overview of this material, so we advise students with a limited background in linear algebra toconsult the course notes before the course starts.
Overview Both physics and computer science have been very dominant scientific and technologi-cal disciplines in the previous century. Quantum Computer Science aims at combining both and maycome to play a similarly important role in this century. Combining the existing expertise in bothfields proves to be a non-trivial but very exciting interdisciplinary journey. Besides the actual issueof building a quantum computer or realising quantum protocols it involves a fascinating encounterof concepts and formal tools which arose in distinct disciplines.
Remark: Students who intend to write their MSc thesis in the Quantum Group at Comlab shouldalso take the Categories, Proofs and Processes course in Michaelmas term.
This course provides an interdisciplinary introduction to the emerging field of quantum computerscience, explaining basic quantum mechanics (including finite dimensional Hilbert spaces and theirtensor products), quantum entanglement, its structure and its physical consequences (e.g. non-locality, no-cloning principle), and introduces qubits. We give detailed discussions of some key algo-rithms and protocols such as Grover’s search algorithm and Shor’s factorisation algorithm, quantumteleportation and quantum key exchange, and analyse the challenges their significance for computerscience, mathematics etc. We also provide a more conceptual semantic analysis of some of the above.Other important issues such as quantum information theory (including mixed states) will also becovered, although not in great detail. We mainly discuss the circuit model and briefly mention al-ternative computational paradigms like measurement-based quantum computing, we argue the needfor high-level methods, provide some recent results concerning a graphical language and categoricalsemantics for quantum informatics and delineate the remaining scientific challenges for the future.
Learning outcomes The student will know by the end of the course what quantum computingand quantum protocols are about, why they matter, and what the scientific prospects of the fieldare. This includes a structural understanding of some basic quantum mechanics, knowledge of im-portant algorithms such as Grover’s and Shor’s algorithm and important protocols such as quantumteleportation. The student will also know where to find more details and will be able to access these.Hence this course also offers computer science and mathematics students a first stepping-stone forresearch in the field, with a particular focus on the newly developing field of quantum computerscience semantics, to which Oxford University Computing Laboratory has provided pioneering con-tributions.
Synopsis
• Lecture 1. A taster of quantum information and computation. Supporting materials includenotes on linear algebra and the slides of this lecture will be available on the Department ofComputer Science website.
• Lectures 2 and 3. Mathematical concepts in Hilbert space and their diagrammatic represen-tation. Lecture notes are now available at on the Department of Computer Science website.
Syllabus tbc
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Reading list Lecture notes, slides and additional handouts will be provided as the course pro-gresses.
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14.10 Schedule II
14.11 Automata, Logic and Games — Dr M Vanden Boom —16 lectures+ reading MT
Recommended Prerequisites: Knowledge of first-order predicate calculus will be assumed. Fa-miliarity with the basics of Finite Automata Theory and Computability (for example, as coveredby the course, Models of Computation) and Complexity Theory would be very helpful. Candidateswho do not have the required background are expected to have taken the course, Introduction tothe Foundations of Computer Science.
Overview To introduce the mathematical theory underpinning the Computer-Aided Verificationof computing systems.
The main ingredients are:
• Automata (on infinite words and trees) as a computational model of state-based systems.
• Logical systems (such as temporal and modal logics) for specifying operational behaviour.
• Two-person games as a conceptual basis for understanding interactions between a system andits environment.
Learning Outcomes At the end of the course students will be able to:
1. Describe in detail what is meant by a Buchi automaton, and the languages recognised bysimple examples of Buchi automata.
2. Use linear-time temporal logic to describe behaviourial properties such as recurrence and pe-riodicity, and translate LTL formulas to Buchi automata.
3. Use S1S to define omega-regular languages, and give an account of the equivalence betweenS1S definability and Buchi recognisability.
4. Explain the intuitive meaning of simple modal mu-calculus formulas, and describe the corre-spondence between property-checking games and modal mu-calculus model checking.
Synopsis/Syllabus
• Automata on infinite words. Buchi automata: Closure properties. Determinization and Rabinautomata.
• Nonemptiness and Nonuniversality problems for Buchi automata.
• Linear temporal logic and alternating Buchi automata.
• Modal mu-calculus: Fundamental Theorem, decidability and finite model property. ParityGames and the Model-Checking Problem: memoryless determinacy, algorithmic issues.
• Monadic Second-order Logic and its relationship with the modal mu-calculus.
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Reading List Selected parts from:
• J. Bradfield and C. P. Stirling. Modal logics and mu-calculi. In J. Bergstra, A. Ponse, and S.Smolka, editors, Handbook of Process Algebra, pages 293-332. Elsevier, North-Holland, 2001.
• B. Khoussainov and A. Nerode. Automata Theory and its Applications. Progress in ComputerScience and Applied Logic, Volume 21. Birkhauser, 2001.
• C. P. Stirling. Modal and Temporal Properties of Processes. Texts in Computer Science.Springer-Verlag, 2001.
• W. Thomas. Languages, Automata and Logic. In G. Rozenberg and A. Salomaa, editors,Handbook of Formal Languages, volume 3. Springer-Verlag, 1997.
• M. Y. Vardi. An automata — theoretic approach to linear temporal logic. In Logics for Con-currency: Structure versus Automata, ed. F. Moller and G. Birtwistle, LNCS vol. 1043, pp.238–266, Springer–Verlag, 1996.
The only copy of the Vardi book is in the RSL on open shelves. However, the article in questionis available in pdf form online at this address:http://folli.loria.fr/cds/1998/pdf/degiacomo-nardi/vardi.pdf
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14.12 Categorical Quantum Mechanics—Dr Heunen and Dr Vicary—16Lectures HT
Prerequisites Ideal foundations for this course are given by the Michaelmas term course “Cate-gories, Proofs and Processes”, and the Hilary term course “Quantum Computer Science”. Studentswho have not taken these courses will need to be familiar with basic topics from category theory andlinear algebra, including categories, functors, natural transformations, vector spaces, Hilbert spacesand the tensor product.
Students wishing to do their dissertation with the Quantum Group are expected to sit this course,as well as the two mentioned above.
Overview Category theory gives a powerful mathematical framework for working with quantumtheory, and provides a high-level computer science perspective with which to understand it. Thiscourse gives an overview of some of the recent research in this exciting field, much of which wascarried out here at the Department of Computer Science. The focus is on reformulating quantum-mechanical concepts in category-theoretical terms, and applying this approach to quantum foun-dations and quantum information. The categorical formalism has a pictorial representation whichmakes deductions intuitive, and this will form a major part of the course.
While we concentrate on the applications to quantum theory, category theory forms an enormouslyimportant part of the modern mathematical landscape, and the tools we introduce in this coursehave close relationships to other area of mathematics, including representation theory, quantumfield theory and knot theory. They also have relevance to more applied disciplines, such as program-ming language semantics and computational linguistics. Further topics may be investigated if timepermits.
Synopsis This syllabus gives a suggestion of the topics which might be covered, and may not berigidly followed.
• Symmetric monoidal categories
• Graphical calculus
• Duals for morphisms
• Duals for objects
• Copying and deleting
• Frobenius algebras and classical structures
• Modelling quantum protocols
• Categories of completely positive maps
• Complementary observables
• Axiomatizing entangled states
• Automation
• Advanced topics
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Reading list Materials [1] to [5] cover topics relevant to the course in an introductory way; papers[6] to [11] are more advanced and provide good further reading.
• Samson Abramsky and Nikos Tzevelekos (2010) Introduction to categories and categoricallogic. In: New Structures for Physics, B. Coecke (ed), pages 3-94. Lecture Notes in Physics813. Springer-Verlag.
• John C. Baez and M. Stay (2010) Physics, topology, logic and computation: a Rosetta Stone.In: New Structures for Physics, B. Coecke (ed), pages 95-172. Lecture Notes in Physics 813.Springer-Verlag.
• Bob Coecke (2010) Quantum picturalism. Contemporary Physics 51, 59-83.
• Bob Coecke and Eric O. Paquette (2010) Categories for the practicing physicist. In: NewStructures for Physics, B. Coecke (ed), pages 173-286. Lecture Notes in Physics 813, Springer-Verlag.
• Peter Selinger (2011) A survey of graphical languages for monoidal categories. In: New Struc-tures for Physics, B. Coecke (ed.), pages 289-356. Lecture Notes in Physics 813, Springer-Verlag.
• Samson Abramsky and Bob Coecke (2004) A categorical semantics of quantum protocols. In:Proceedings of 19th IEEE conference on Logic in Computer Science, pages 415?425. IEEEPress. Revised version (2009): Categorical quantum mechanics. In: Handbook of QuantumLogic and Quantum Structures, K. Engesser, D. M. Gabbay and D. Lehmann (eds), pages261?323. Elsevier.
• Samson Abramsky, No-Cloning in Categorical Quantum Mechanics. In Semantic Techniquesin Quantum Computation, ed. S. Gay and I. Mackie, pages 1–28, Cambridge University Press2010.
• Krysztof Bar, Lucas Dixon, Ross Duncan, Benjamin Frot, Alex Merry, Aleks Kissinger andMatvey Soloviev (2011), “Quantomatic? software. http://sites.google.com/site/quantomatic/
• Bob Coecke and Ross Duncan (2011) Interacting quantum observables: categorical algebraand diagrammatics. New Journal of Physics 13, 043016.
• Peter Selinger (2007) Dagger compact closed categories and completely positive maps. Elec-tronic Notes in Theoretical Computer Science 170, pages 139-163. http://www.mscs.dal.ca/ selinger/papers.html
• Jamie Vicary (2008) A categorical framework for the quantum harmonic oscillator. Interna-tional Journal of Theoretical Physics 47, 3408-3447.
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14.13 Combinatorics — Prof. Scott — 16MT
Recommended Prerequisites Part B Graph Theory is helpful, but not required.
Overview An important branch of discrete mathematics concerns properties of collections of sub-sets of a finite set. There are many beautiful and fundamental results, and there are still many basicopen questions. The aim of the course is to introduce this very active area of mathematics, withmany connections to other fields.
Learning Outcomes The student will have developed an appreciation of the combinatorics offinite sets.
Synopsis Chains and antichains. Sperner’s Lemma. LYM inequality. Dilworth’s Theorem.
Shadows. Kruskal-Katona Theorem.
Intersecting families. Erdos-Ko-Rado Theorem. Cross-intersecting families.
VC-dimension. Sauer-Shelah Theorem.
t-intersecting families. Fisher’s Inequality. Frankl-Wilson Theorem. Application to Borsuk’s Con-jecture.
Combinatorial Nullstellensatz.
Reading
1. Bela Bollobas, Combinatorics, CUP, 1986.
2. Stasys Jukna, Extremal Combinatorics, Springer, 2007
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14.14 Computational Algebraic Topology — Prof Tillmann & Prof Abram-sky 14HT
Prerequisites The course will provide a self-contained, rapid introduction to (co)homology theoryfor simplicial sets. However, some familiarity with concepts from topology and homological algebrawill be of help. It should be noted that MFoCS students may offer both this course and the SectionA course Algebraic Topology in Michaelmas Term for examination. Those offering only this courseand with no familiarity with these topics would be well advised to at least sit in on the Section Acourse Algebraic Topology.
Overview Ideas and tools from algebraic topology have become more and more important incomputational and applied areas of mathematics. This course will provide at the masters level anintroduction to the main concepts of (co)homology theory, and explore areas of applications in dataanalysis and in foundations of quantum mechanics and quantum information.
Learning outcomes Students should gain a working knowledge of homology and cohomology ofsimplicial sets and sheaves, and improve their geometric intuition. Furthermore, they should gainan awareness of a variety of application in rather different, research active fields of applications withan emphasis on data analysis and contextuality.
Synopsis The course has two parts. The first part will introduce students to the basic conceptsand results of (co)homology, including sheaf cohomology. In the second part applied topics are in-troduced and explored.
Core: Homology and cohomology of chain complexes. Algorithmic computation of boundary maps(with a view of the classification theorm for finitely generated modules over a PID). Chain homotopy.Snake Lemma. Simplicial complexes. Other complexes (Delaunay, Cech). Mayer-Vietoris sequence.Poincare duality. Alexander duality. Acyclic carriers. Discrete Morse theory. (4 lectures)
Topic A: Persistent homology: barcodes and stability, applications todata analysis, generalisations.(4 lectures)
Topic B:Sheaf cohomology and applications to quantum non-locality and contextuality.Sheaf-theoreticrepresentation of quantum non-locality and contextuality asobstructions to global sections. Coho-mological characterizations and proofs of contextuality.(6 lectures)
Learning support: There will be two problem sheets and classes covering the core material. Inaddition there may be reading which students will be expected to present.
Reading List H. Edelsbrunner and J.L. Harer, Computational Topology -An Introduction,AMS(2010).
See also, U. Tillmann, Lecture notes for CAT 2012, in http://people.maths.ox.ac.uk/tillmann/CAT.html
Topic A:
G. Carlsson, Topology and data, Bulletin A.M.S.46 (2009), 255-308.
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H. Edelsbrunner, J.L. Harer, Persistent homology: A survey, Contemporary Mathematics 452A.M.S. (2008), 257-282.
S. Weinberger, What is ... Persistent Homology?, Notices A.M.S. 58 (2011), 36-39.
P. Bubenik, J. Scott, Categorification of Persistent Homology, Discrete Comput. Geom. (2014),600–627.
Topic B:
S. Abramsky and Adam Brandenburger, The Sheaf-Theoretic Structure Of Non-Locality and Con-textuality. In New Journal of Physics, 13(2011), 113036, 2011.
S. Abramsky and L. Hardy, Logical Bell Inequalities, Phys. Rev. A 85, 062114 (2012).
S. Abramsky, S. Mansfield and R. Soares Barbosa, The Cohomology of Non-Locality and Contex-tuality, in Proceedings of Quantum Physics and Logic 2011, Electronic Proceedings in TheoreticalComputer Science, vol. 95, pages 1–15, 2012.
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14.15 Computational Learning Theory — Prof. Worrell — 16MT
Prerequisites Students should have experience of reading and writing mathematical proofs. Fami-larity with calculus, probability theory, and linear algebra (to the level of the undergraduate Com-puter Science degree) is essential.
Overview Machine learning studies automatic methods for identifying patterns in complex dataand for making accurate predictions based on past observations. From predicting which movies a cus-tomer will like to assigning credit ratings, systems that learn are becoming increasingly widespreadand effective. Computational learning theory aims to develop rigourous mathematical foundationsfor machine learning, in order to provide guarantees about the behaviour of learning algorithms, toidentify common methods underlying effective learning procedures, and to understand the inherentdifficulty of learning problems. To address such issues we will bring together notions from probabilitytheory, optimisation, online algorithms, game theory, and combinatorics.
Learning Outcomes On completing this course, students should:
• understand key models of supervised and unsupervised learning and be able to formulatespecific learning problems in these models;
• understand a variety of learning algorithms and recognize when they are applicable.
Synopsis
• Introduction, PAC model [2 Lectures]
• Sample complexity, VC-dimension, the growth function [2 Lectures]
• Online learning, mistake bounds, the Perceptron and Winnow algorithms [2 lectures]
• Learning from expert advice, regret bounds, weighted majority and follow-the-leader [3 lec-tures]
• Weak learning, adaptive boosting, margin bounds [2 Lectures]
• Support Vector Machines and kernels [3 Lectures]
• Principal components analysis [1 Lecture]
• Johnson-Lindenstrauss lemma [1 lecture]
Syllabus PAC learning: Sample complexity, VC-dimension Online learning: mistake bounds,the Perceptron and Winnow algorithms Learning from expert advice: Deterministic & random-ized weighted majority, follow the leader Weak learning and boosting Support vector machines,kernels Principal components analysis Johnson-Lindenstrauss Lemma
Reading List Primary Text
• Mehryar Mohri, Foundations of Machine Learning, MIT Press, 2012.
Secondary Texts
• Michael Kearns and Umesh Vazirani. An Introduction to Computational Learning Theory,MIT Press, 1995.
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15 Computational Number Theory — Prof R Heath-Brown— Reading course TT
Prerequisites Despite the sophistication of this course the only pre-requisites are parts of a stan-dard elementary number theory course: Euclid’s algorithm, Quadratic residues, The law of reci-procity for Legendre and Jacobi symbols, Fermat’s theorem, primitive roots.
Aims and Synopsis This course aims to describe the algorithms used for efficient practical com-putations in number theory. It is based on recent research papers, along with parts of the text byCohen.
The course covers: The Euclidean Algorithm, computation of powers and square roots moduloprimes; the arithmetic of elliptic curves over finite fields; lattices and the LLL reduction algorithm;factorization algorithms.
Reading Henri Cohen, A course in computational algebraic number theory, Springer-Verlag (1993).
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15.1 Distributional Models of Meaning — Prof B Coecke— Readingcourse HT
Prerequisites This course will make heavy use of linear algebra, so students are expected to becomfortable with vectors and matrices. Some knowledge of discrete mathematics (sets, groups), ofcategory theory (categories, monoidal categories, functors), and of computational linguistics (phrase-structure grammar, parsing, semantics) is highly desirable, but can be acquired with supplementaryreading, and through available course notes for other courses in the department.
Overview Modelling the meaning of natural (as opposed to computer) languages is one of thehardest problems in artificial intelligence. Solving this problem has the potential to dramaticallyimprove the quality and impact of a wide range of text and language processing applications suchas text summarisation, search, machine translation, language generation, question answering, etc.
A host of different approaches to this problem have been devised throughout the years. One notableapproach is Formal Semantics, which treats natural languages as programming languages which‘compile’ to higher order logics. Another is Distributional Semantics, which models the meanings ofwords as points in high dimensional semantic spaces, determined by the contexts of occurrence.
Recent research has attacked the task or reconciling the strengths of both of these approaches toproduce compositional distributed (i.e. predominantly vector-based) models of meaning. This courseserves as an introduction to the theoretical end of this new and rapidly growing field. During it, wewill discuss algebraic approaches to reasoning about vectors and their compositionality, using toolsfrom Category Theory.
Learning Outcomes Students will be expected to:
• Strengthen background knowledge of category theory and their applications.
• Become familiar with the use of monoidal categories and other algebras (e.g. pregroup gram-mars) to represent and bring into interaction syntactic and semantic aspects of language.
• Have a reasonably complete grasp of core concepts present in the literature on distributionalmodels of semantics and their compositionality.
• Understand the logical properties of tensor-based models of semantics.
Synopsis QPL followed by a number refers to chapters from Quantum Physics and Linguistics.Chapter preprints will be made available on the course website.
Topics:
• Introduction to formal and distributional semantics (QPL 12)
• Overview of compositional distributional semantics (QPL 13)
• A diagrammatic language for processes (Coecke and Paquette 2009)
• Compact closed categories for composition (Coecke et al. 2010)
• Abstract algebra and compositional structure (QPL 1)
• Tensors and logic (Grefenstette 2013)
• Reasoning about function words with Frobenius algebras (Sadrzadeh et al. 2013)
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Reading
1. S. Clark, Type-driven syntax and semantics for composing meaning vectors (2013).
2. B. Coecke, An alternative gospel of structure: order, composition, processes, arXiv preprintarXiv:1307.4038, (2013).
3. B. Coecke and E. Paquette, Categories for the practising physicist, arXiv preprint or arXiv:0905.3010(2009).
4. B. Coecke, M. Sadrzadeh, and S. Clark Mathematical Foundations for a Compositional Dis-tributional Model of Meaning, (March, 2010).
5. E. Grefenstette, Towards a formal distribution semantics: Simulating logical calculi with ten-sors. arXiv preprint or arXiv:1304.5823 (2013).
6. C. Heunen, M.Sadrzadeh, and E. Grefenstette, editors. Quantum Physics and Linguistics: ACompositional, Diagrammatic Discourse. (Oxford University Press, 2013).
7. S. Pulman, Distributional Semantic Models. (Oxford University Press, 2013).
8. M. Sadrzadeh, S. Clark, and B. Coecke, The frobenius anatomy or word meanings i: subjectand object relative pronouns. (Journal of Logic and Computation, page ext044, 2013).
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15.2 Elliptic Curves — Prof. Kim — 16HT
Recommended Prerequisites It is helpful, but not essential, if students have already taken astandard introduction to algebraic curves and algebraic number theory. For those students who mayhave gaps in their background, I have placed the file “Preliminary Reading” permanently on theElliptic Curves webpage, which gives in detail (about 30 pages) the main prerequisite knowledge forthe course. Go first to: http://www.maths.ox.ac.uk/courses/material then click on “C3.7 EllipticCurves” and then click on the pdf file “Preliminary Reading”.
Overview Elliptic curves give the simplest examples of many of the most interesting phenomenawhich can occur in algebraic curves; they have an incredibly rich structure and have been the testingground for many developments in algebraic geometry whilst the theory is still full of deep unsolvedconjectures, some of which are amongst the oldest unsolved problems in mathematics. The coursewill concentrate on arithmetic aspects of elliptic curves defined over the rationals, with the study ofthe group of rational points, and explicit determination of the rank, being the primary focus. Usingelliptic curves over the rationals as an example, we will be able to introduce many of the basic toolsfor studying arithmetic properties of algebraic varieties.
Learning Outcomes On completing the course, students should be able to understand and useproperties of elliptic curves, such as the group law, the torsion group of rational points, and 2-isogenies between elliptic curves. They should be able to understand and apply the theory of fieldswith valuations, emphasising the p-adic numbers, and be able to prove and apply Hensel’s Lemmain problem solving. They should be able to understand the proof of the Mordell–Weil Theorem forthe case when an elliptic curve has a rational point of order 2, and compute ranks in such cases, forexamples where all homogeneous spaces for descent-via-2-isogeny satisfy the Hasse principle. Theyshould also be able to apply the elliptic curve method for the factorisation of integers.
Synopsis Non-singular cubics and the group law; Weierstrass equations.Elliptic curves over finite fields; Hasse estimate (stated without proof).p-adic fields (basic definitions and properties).1-dimensional formal groups (basic definitions and properties).Curves over p-adic fields and reduction mod p.Computation of torsion groups over Q; the Nagell–Lutz theorem.2-isogenies on elliptic curves defined over Q, with a Q-rational point of order 2.Weak Mordell–Weil Theorem for elliptic curves defined over Q, with a Q-rational point of order 2.Height functions on Abelian groups and basic properties.Heights of points on elliptic curves defined over Q; statement (without proof) that this gives a heightfunction on the Mordell–Weil group.Mordell–Weil Theorem for elliptic curves defined over Q, with a Q-rational point of order 2.Explicit computation of rank using descent via 2-isogeny.Public keys in cryptography; Pollard’s (p−1) method and the elliptic curve method of factorisation.
Reading
1. J.W.S. Cassels, Lectures on Elliptic Curves, LMS Student Texts 24 (Cambridge UniversityPress, 1991).
2. N. Koblitz, A Course in Number Theory and Cryptography, Graduate Texts in Mathematics114 (Springer, 1987).
3. J.H. Silverman and J. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Math-ematics (Springer, 1992).
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4. J.H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106 (Springer,1986).
Further Reading
1. A. Knapp, Elliptic Curves, Mathematical Notes 40 (Princeton University Press, 1992).
2. G, Cornell, J.H. Silverman and G. Stevans (editors), Modular Forms and Fermat’s Last The-orem (Springer, 1997).
3. J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts inMathematics 151 (Springer, 1994).
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15.3 Finite Dimensional Normed Spaces — Dr Sanders — 16TT
Recommended Prerequisites The course will have a strongly analytic flavour and familiaritywith spectral theory would be very useful.
Overview It is well-known that any two norms on a finite dimensional space are equivalent, soone might think that there is not much more to say about these spaces. However, the constant ofequivalence will, in general, grow as the dimension of the space grows. This is the first of a range ofquantitative situations which show that things are subtler than they at first appear, and this courseis about their study.
We have two external motivations. This first is embeddings in computer science. A typical applica-tion of embeddings is to the sparsest cut problem: given a graph G we want to find a partition ofthe vertex set into two sets S and S such that e(S; S) / |S||S| is minimal- the sparsest cut of thegraph. This problem is known to be NP hard, but techniques based on metric embeddings can giveus good approximations.
The second motivation comes from harmonic analysis and additive combinatorics. Here the problemis more philosophical. We shall take the view that the purpose of a proof is to explain why somethingis true, not so much to show that it is true. In this light there are various results in harmonic analysiswhere the real explanation comes from some naturally arising normed space.
Synopsis Our approach will be highly quantitative and we shall cover (finite dimensional versionsof) a range of topics including basis constants, Banach spaces without bases, Johnson’s uniquenessof norm theorem, absolutely summing operators, tensor products, embeddings, the Banach-Alaoglutheorem, the Dvoretzsky-Rogers theorem, Khintchine’s inequality and Nazarov’s inequality.
Some adjustments as a result of timing are likely but the avour of the results will be very much asabove.
Reading There is no one standard reference from which the course material will be taken, al-though the following all include aspects of the course.
• (AK06) F. Albiac and N. J. Kalton. Topics in Banach space theory, volume 233 of Graduate
• Texts in Mathematics. Springer, New York, 2006.
• (Gre04) B. J. Green. Spectral structure of sets of integers. In Fourier analysis and convex-ity,Appl. Numer. Harmon. Anal., pages 83-96. Birkhauser Boston, Boston, MA, 2004.
• (MS86) V. D. Milman and G. Schechtman. Asymptotic theory of finite-dimensional normedspaces, volume 1200 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986.
• (Nao10) A Naor. Local theory of Banach spaces. 2010.
• (Naz97) F. L. Nazarov. The Bang solution of the coef cient problem. Algebra i Analiz,9(2):272-287, 1997.
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• (Woj91]) P. Wojtaszczyk. Banach spaces for analysts, volume 25 of Cambridge Studies inAdvanced Mathematics. Cambridge University Press, Cambridge, 1991.
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15.4 Machine Learning - Prof Nando de Freitas - 24HT
Recommended Prerequisites Machine Learning is a mathematical discipline, and students willbenefit from a good background in probability, linear algebra and calculus. Programming experienceis essential.
Overview Machine learning techniques enable us to automatically extract features from data soas to solve predictive tasks, such as speech recognition, object recognition, machine translation,question-answering, anomaly detection, medical diagnosis and prognosis, automatic algorithm con-figuration, personalisation, robot control, time series forecasting, and much more. Learning systemsadapt so that they can solve new tasks, related to previously encountered tasks, more efficiently.This course lays out the foundations of machine learning. It covers maximum likelihood estimation,empirical risk minimization, contrastive learning, and Bayesian inference. To this end, it reviewsoptimisation, probabilistic modelling, Monte Carlo integration, statistics and linear algebra.
The course highlights the use of nonparametric Bayesian techniques for decision making. Thisenables us to attack problems such as automatic A/B tests, interactive optimisation, multi-armedbandit problems and algorithm configuration.
The course covers the newly exciting fields of deep learning and big data. By drawing inspiration fromneuroscience and statistics, it introduces the basic background on neural networks, back propagation,Boltzmann machines, autoencoders and convolutional neural networks. Subsequently, it builds onthis background to illustrate how deep learning and big data are impacting our understanding ofintelligence and contributing to the practical design of intelligent machines.
Learning Outcomes On completion of the course students will be expected to:
Have a good understanding of the two numerical approaches to learning optimization and integrationand how they relate to maximum likelihood and the Bayesian approach. Have an understanding ofhow to choose a probabilistic model to describe a particular type of data. Know how to evaluatea learned model in practice. Understand the role of machine learning in massive scale automation.Have a good understanding of the problems that arise when dealing with very small and very bigdata sets, and how to solve them. Understand the mathematics necessary for constructing novelmachine learning solutions. Be able to design and implement various machine learning algorithmsin a wide range of real-world applications. Understand the background on deep learning and be ableto implement deep learning models to solve predictive tasks, such as classification.
Synopsis Mathematics of machine learning. Overview of supervised, unsupervised, multi-task,transfer, active and reinforcement learning techniques.
Reading
1. Kevin P. Murphy. Machine Learning: A Probabilistic Perspective, MIT Press 2012.
2. Christopher M. Bishop.Pattern Recognition and Machine Learning, Springer 2007.
3. T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning. Springer2011.
4. S. Haykin. Neural networks and learning machines. Pearson 2008.
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15.5 Networks — Prof. Porter — 16HT
Recommended Prerequisites None [in particular, C5.3 (Statistical Mechanics) is not¯
required],though some intuition from modules like C5.3, the Part B graph theory course, and probabilitycourses (at the level that everybody has to take anyway) can be useful. However, everything is self-contained, and none of these courses are required. Some computational experience is also helpful,and ideas from linear algebra will certainly be helpful.
Overview This course aims to provide an introduction to network science, which can be used tostudy complex systems of interacting agents. Networks are interesting both mathematically andcomputationally, and they are pervasive in physics, biology, sociology, information science, andmyriad other fields. The study of networks is one of the “rising stars” of scientific endeavors, andnetworks have become among the most important subjects for applied mathematicians to study.Most of the topics to be considered are active modern research areas.
Learning Outcomes Students will have developed a sound knowledge and appreciation of someof the tools, concepts, and computations used in the study of networks. The study of networks ispredominantly a modern subject, so the students will also be expected to develop the ability to readand understand current (2015) research papers in the field.
Synopsis 1. Introduction and Basic Concepts (1-2 lectures): nodes, edges, adjacencies, weightednetworks, unweighted networks, degree and strength, degree distribution, other types of networks
2. Small Worlds (2 lectures): clustering coefficients, paths and geodesic paths, Watts-Strogatznetworks [focus is on modelling and heuristic calculations]
3. Toy Models of Network Formation (2 lectures): preferential attachment, generalizations of pref-erential attachment, network optimization
4. Additional Summary Statistics and Other Useful Concepts (2 lectures): modularity and assorta-tivity, degree-degree correlations, centrality measures, communicability, reciprocity and structuralbalance
5. Random Graphs (2 lectures): Erdos-Renyi graphs, configuration model, random graphs with clus-tering, other models of random graphs or hypergraphs; application of generating-function methods[focus is on modelling and heuristic calculations; material in this section forms an important basisfor sections 6 and 7]
6. Community Structure and Mesoscopic Structure (2 lectures): linkage clustering, optimization ofmodularity and other quality functions, overlapping communities, other methods and generalizations
7. Dynamics on (and of) Networks (3-4 lectures): general ideas, models of biological and socialcontagions, percolation, voter and opinion models, temporal networks, other topics
8. Additional Topics (0-2 lectures): games on networks, exponential random graphs, network infer-ence, other topics of special interest to students [depending on how much room there is and interestof current students]
Reading (most important are [2] and [3]):
1. A. Barrat et al, Dynamical Processes on Complex Networks, Cambridge University Press, 2008
2. M. E. J. Newman, Networks: An Introduction, Oxford University Press, 2010 [also, Newman’s2003 review article in SIAM Review for ”older” topics]
3. M. A. Porter, A Terse Introduction to Networks, Springer, in preparation
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4. Various papers and review articles (see the Math C6.2b blog at http://networksoxford.blogspot.co.ukfor examples). The instructor will indicate a small number of specific review articles that shouldbe read along with the notes, and other helpful (but optional) articles will also be indicated.”
5. Other networks books are also useful. (I will point interested students to them if they ask, butI have listed enough things here. They can also look in the references in the textbook I amwriting.)
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15.6 Probabilistic Combinatorics — Prof. McDiarmid — 16HT
Recommended Prerequisites Part B Graph Theory and Part A Probability. C8.3 Combina-torics is not as essential prerequisite for this course, though it is a natural companion for it.
Overview Probabilistic combinatorics is a very active field of mathematics, with connections toother areas such as computer science and statistical physics. Probabilistic methods are essential forthe study of random discrete structures and for the analysis of algorithms, but they can also providea powerful and beautiful approach for answering deterministic questions. The aim of this course isto introduce some fundamental probabilistic tools and present a few applications.
Learning Outcomes The student will have developed an appreciation of probabilistic methodsin discrete mathematics.
Synopsis First-moment method, with applications to Ramsey numbers, and to graphs of highgirth and high chromatic number.Second-moment method, threshold functions for random graphs.Lovasz Local Lemma, with applications to two-colourings of hypergraphs, and to Ramsey numbers.Chernoff bounds, concentration of measure, Janson’s inequality.Branching processes and the phase transition in random graphs.Clique and chromatic numbers of random graphs.
Reading
1. N. Alon and J.H. Spencer, The Probabilistic Method (third edition, Wiley, 2008).
Further Reading
1. B. Bollobas, Random Graphs (second edition, Cambridge University Press, 2001).
2. M. Habib, C. McDiarmid, J. Ramirez-Alfonsin, B. Reed, ed., Probabilistic Methods for Algo-rithmic Discrete Mathematics (Springer, 1998).
3. S. Janson, T. Luczak and A. Rucinski, Random Graphs (John Wiley and Sons, 2000).
4. M. Mitzenmacher and E. Upfal, Probability and Computing : Randomized Algorithms andProbabilistic Analysis (Cambridge University Press, New York (NY), 2005).
5. M. Molloy and B. Reed, Graph Colouring and the Probabilistic Method (Springer, 2002).
6. R. Motwani and P. Raghavan, Randomized Algorithms (Cambridge University Press, 1995).
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16 Theory of Data and Knowledge Bases— Prof T Lukasiewicz— 16 lectures HT
Recommended prerequisites Basic knowledge of automata theory (e.g., finite state automata),databases (e.g., the relational data model, relational algebra, SQL), and complexity theory (e.g.,NP-completeness) will be assumed. This can be gained from the undergraduate courses Modelsof Computation, Databases and Computational Complexity, or the MSc courses Introduction toFoundations of Computer Science and Databases. Students who have not studied these coursesshould talk to the lecturer.
Overview The lecture series provides an understanding of the logical foundations of databasequery languages and knowledge representation formalisms, the expressive power of such languages/formalisms,and the complexity of query answering and reasoning with such languages/formalisms.
Synopsis We will try to cover the following topics:
1. A brief introduction to databases, finite model theory and descriptive complexity.
2. Conjunctive queries: Complexity and optimization.
3. SQL and First Order Queries: Complexity, limits of expressive power
4. Reasoning with propositional Horn theories.
5. Datalog and fixed-point queries: Complexity and expressive power.
6. Querying semi-structured data.
7. Elements of non-monotonic reasoning and closed-world reasoning.
8. Problems of data integration and data exchange.
9. Advanced material (time permitting)
It is roughly planned to spend two hours for each of the topics 1-8, and to insert additional material(9) dynamically where appropriate.
Reading Material Books:
• Abiteboul, S., Hull, R. and Vianu, V. Foundations of Databases. Addison-Wesley, 1995.
• Leonid Libkin, Elements of Finite Model Theory, Springer, 2004 (selected chapters)
A list of papers covering various relevant topics will be supplied in due course.
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.1 Safety Information
These notes give some information about the safety arrangements at the Mathematical Institute.For further information, please see http://www.maths.ox.ac.uk/notices/safety.
.2 Action in Case of Emergency
To summon the FIRE BRIGADE, AMBULANCE SERVICE and/or POLICE, DIAL9-999. Note that 9-999 can be dialled from any internal University telephone extension, even if itis otherwise barred from making outside calls.
For SERIOUS ACCIDENTS or FIRES on University premises, immediately after arranging forthe emergency servises, telephone again the Security Services (89999).
Remember that unless there is a continuing risk to others or to property, the law requires that incases of serious accidents or fires the scene must remain undisturbed until it is examined by theHealth and Safety Executive, the University Safety Office and Trade Union safety representatives.Some types of serious accident must be reported immediately. In those cases, the Safety Office isresponsible for contacting the Health and Safety Executive.
.3 Statement of Safety Policy
It is the policy of the University to ensure that all members of the University and its staff are awareof their individual responsibility to exercise care in relation to themselves and those who work withthem. To this end individuals are enjoined to:
1. familiarise themselves with University Safety Policy and any departmental safety requirements(http://www.admin.ox.ac.uk/safety/hs-mgement-policy/);
2. take reasonable care that all procedures used are safely carried out, and seek expert advice inany case of doubt;
3. warn of any special or newly identified hazards in present procedures or risks in new proceduresabout to be introduced;
4. report accidents or incidents promptly;
5. familiarise themselves with fire and emergency drills (including the location of emergencytelephones) and escape routes.
.4 Statement of Health and Safety Organisation
Please see http://www.maths.ox.ac.uk/notices/safety/statement for a statement of Health and SafetyOrganisation
.5 Departmental Health and Safety Committee
The membership and responsibilities of the Safety Committee can be found athttp://www.maths.ox.ac.uk/notices/committees/safety.
.6 Code of Practice-Harassment
The University code of practice relating to harassment can be found via the following link:http://www.admin.ox.ac.uk/eop/harassmentadvice/
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.7 Smoking
Smoking is not permitted anywhere in the Andrew Wiles Building. Smoking on the Radcliffe Infir-mary sight is permitted in designated smoking areas only.
.8 Electricity
All electrical equipment (including personal property) must be tested for safety beforeit is used in the Department of Computer Science buildings. Equipment must not be dis-mantled. If equipment is faulty, do not attempt to repair it—please email [email protected] Donot tamper with electrical supply equipment. Do not unplug equipment without express permission.Please report any problems to the Laboratory’s technicians.
.9 Equipment rooms - Department of Computer Science
Electrical power in the various equipment rooms (including the Teaching Laboratory in the ThomBuilding) can be cut by an ‘emergency stop’. In the Thom Building, this is a white break-glassunit; in the Wolfson Building, it is a red button (either just inside or just outside the door to eachequipment room); it is normally clearly labelled with a green ‘Emergency stop’ sign. Please notethat it will need the support staff to restart circuits.
.10 Lighting - Department of Computer Science
Do not switch off any corridor lighting at any time. Please report any faulty corridor or staircaselighting to the technical staff. Please advise the Administrator if there are any other areas whichare poorly lit.
.11 Other Safety Information
A blue ring binder containing a Statement of the Department of Computer Science’s Safety Organi-sation, a Statement of the University Safety Policy, and a collection of University and DepartmentalGuidance Notes, is available in the Library (room 240) and in the Common Room (room 103).
The University Safety Office has a library of safety publications and other material at 10 ParksRoad.
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A University’s Policy on Plagiarism
A.1 What is plagiarism?
Plagiarism is the copying or paraphrasing of other people’s work or ideas into your own work withoutfull acknowledgement. All published and unpublished material, whether in manuscript, printed orelectronic form, is covered under this definition. Collusion is another form of plagiarism involving theunauthorised collaboration of students (or others) in a piece of work. Cases of suspected plagiarism inassessed work are investigated under the disciplinary regulations concerning conduct in examinations.Intentional or reckless plagiarism may incur severe penalties, including failure of your degree orexpulsion from the university.
A.2 Why does plagiarism matter?
It would be wrong to describe plagiarism as only a minor form of cheating, or as merely a matterof academic etiquette. On the contrary, it is important to understand that plagiarism is a breachof academic integrity. It is a principle of intellectual honesty that all members of the academiccommunity should acknowledge their debt to the originators of the ideas, words, and data which formthe basis for their own work. Passing off another’s work as your own is not only poor scholarship, butalso means that you have failed to complete the learning process. Deliberate plagiarism is unethicaland can have serious consequences for your future career; it also undermines the standards of yourinstitution and of the degrees it issues.
A.3 What forms can plagiarism take?
• Verbatim quotation of other people’s intellectual work without clear acknowledgement. Quo-tations must always be identified as such by the use of either quotation marks or indentation,with adequate citation. It must always be apparent to the reader which parts are your ownindependent work and where you have drawn on someone else’s ideas and language.
• Paraphrasing the work of others by altering a few words and changing their order, or byclosely following the structure of their argument, is plagiarism because you are deriving yourwords and ideas from their work without giving due acknowledgement. Even if you include areference to the original author in your own text you are still creating a misleading impressionthat the paraphrased wording is entirely your own. It is better to write a brief summary ofthe author’s overall argument in your own words than to paraphrase particular sections of hisor her writing. This will ensure you have a genuine grasp of the argument and will avoid thedifficulty of paraphrasing without plagiarising. You must also properly attribute all materialyou derive from lectures.
• Cutting and pasting from the Internet. Information derived from the Internet must be ade-quately referenced and included in the bibliography. It is important to evaluate carefully allmaterial found on the Internet, as it is less likely to have been through the same process ofscholarly peer review as published sources.
• Collusion. This can involve unauthorised collaboration between students, failure to attributeassistance received, or failure to follow precisely regulations on group work projects. It is yourresponsibility to ensure that you are entirely clear about the extent of collaboration permitted,and which parts of the work must be your own.
• Inaccurate citation. It is important to cite correctly, according to the conventions of yourdiscipline. Additionally, you should not include anything in a footnote or bibliography thatyou have not actually consulted. If you cannot gain access to a primary source you must makeit clear in your citation that your knowledge of the work has been derived from a secondary
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text (e.g. Bradshaw, D. Title of Book, discussed in Wilson, E., Title of Book (London, 2004),p. 189).
• Failure to acknowledge. You must clearly acknowledge all assistance which has contributedto the production of your work, such as advice from fellow students, laboratory technicians,and other external sources. This need not apply to the assistance provided by your tutor orsupervisor, nor to ordinary proofreading, but it is necessary to acknowledge other guidancewhich leads to substantive changes of content or approach.
• Professional agencies. You should neither make use of professional agencies in the productionof your work nor submit material which has been written for you. It is vital to your intellectualtraining and development that you should undertake the research process unaided.
• Autoplagiarism. You must not submit work for assessment which you have already submitted(partially or in full) to fulfil the requirements of another degree course or examination.
A.4 Not just printed text!
The necessity to reference applies not only to text, but also to other media, such as computer code,illustrations, graphs etc. It applies equally to published text drawn from books and journals, and tounpublished text, whether from lecture handouts, theses or other student’s essays. You must alsoattribute text or other resources downloaded from web sites.
All matters relating to plagiarism are taken very seriously and would lead to a Disciplinary matter.
For further information see The Proctors’ and Assessors’ Memorandum Essential Information forStudents Section 9, also available online athttp://www.admin.ox.ac.uk/proctors/info/pam/section9.shtml.
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B Electronic Resources for Mathematics
Oxlip — Oxford Libraries Information Portal http://www.bodley.ox.ac.uk/ox lip/
Databases
Resource name Subject coverage AccessMathSciNet Mathematical Reviews produced by the
AMSOxlip
Zentralblatt MATH Euro-pean Mathematical Society
Pure and Applied Mathematics and Historyof Mathematics
Internet
http://www.zentralblatt-math.org/zmath/en/advanced/INSPEC Physics, Engineering, Computing and Ap-
plied MathematicsOxlip
Web of Knowledge — Webof Science — Science Cita-tion Index
Science and Technology Oxlip
Electronic Journals Oxford University e-journals portalhttp://sfx7.exlibrisgr oup.com/oxford/az provides an extensive collection of e-journalspublished by the main societies and publishers: Association for Computing — ACMDigital Archive, American Mathematical Society, London Mathematical Society, IEEEElectronic Library — Computer Society (Digital Library), SIAM journals includingLOCUS archive, Cambridge University Press — Computer Science and Mathematics,Science Direct, Springer journals, Wiley InterScience, etc.
Resource name Subject coverage AccessDOAJ Directory of OpenAccess Journals
Full-text journals Oxlip
E-print and preprint serversArXiv.org
Full-text articles in physics, mathe-matics, computing uk.ArXiv.org athttp://xxx.soton.ac.uk/
OpenAccess
The Computing ResearchRepository (CoRR)
Online repository. Full-text articles.http://arxiv.org/corr/home
OpenAccess
The Mathematical InstituteE-prints Archive (OxfordUniversity)
Open access resource, full-text articleshttp://eprints.maths.ox.ac.uk/
OpenAccess
Oxford University ResearchArchive — ORA
Full-text articles, conference proceedings,theses, reports
Oxlip
ZETOC journals — BritishLibrary
Electronic Tables of Contents Bibliographicdetails (references only)
Oxlip
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Electronic Books
Resource name Subject coverage AccessEncyclopaedia of Mathe-matical Physics
e-book Oxlip
Handbook of MathematicalFunctions
e-book Oxlip
Handbook in Economicsseries (Science Direct)including: Handbook ofMathematical Economics;Handbook of Computa-tional Economics; Hand-book of the Economics ofFinance; more . . .
E-book series in Economics, Finance, Com-puter Science
Oxlip
Handbook of Statistics (Sci-ence Direct)
e-book Oxlip
Lecture Notes in Mathe-matics (Springer)
e-book series Oxforde-journals
Lecture Notes in ComputerScience (Springer)
e-book series Oxforde-journals
CREDO Reference Sciencedictionaries online
e-books Oxlip
Oxford Reference OnlineOxford University Press
e-books, English Dictionaries and ScienceDictionaries
Oxlip
Oxford Scholarship Online— OUP
e-books, Economics and Finance Collection Oxlip
Dissertations
Resource name Subject coverage AccessDissertation Abstracts On-line
Doctoral dissertations and Masters Theses Oxlip
Electronic Index to Theses Dissertations from UK universities OxlipOxford Doctoral Disserta-tions
Science, Technology and Medicine Print/RSL
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Conference Proceedings
Resource name Subject coverage AccessBritish Standards On-Line— BSOL
Full-text standards Oxlip
IIEE Conference Proceed-ings,
Engineering, Physics, Applied Sciences Internet
Standards http://www.ieee.org/web/publications/home/index.htmlIndex to Scientific andTechnical Proceedings
ISI Web of Knowledge — Science, Technol-ogy and Engineering
Oxlip
Lecture Notes in Mathe-matics
Conference Proceedings, e-books Oxforde-journals
General and reference resources and software
Resource name Subject coverage AccessEndNoteWeb/EndNote Reference management software (web ver-
sion)Oxlip
RefWorks Reference management software (web ver-sion)
Oxlip
RefWorks and EndNoteWeb are online research management, writing and collabo-ration tool, designed to help researchers easily gather, manage, store and share alltypes of information, as well as generate citations and bibliographies. Any bona fidemember of the University may freely create an account, though you must be withinthe Oxford Internet domain to do so.
Internet Gateways (quality assessed Internet Resources)Resource name Subject coverage AccessAmerican Mathematical So-ciety (AMS)
Directory of Mathematics Pre-print and e-Print Servershttp://www.mathontheweb.org/mathweb/mi-preprints.html
Internet
EMIS — The EuropeanMathematical InformationService
Pure and Applied Mathematics, Statistics.http://www.maths.soton.ac.uk /EMIS/European Mathematical Society — EMS
Internet
Intute — Internet resourcesfor Science, Engineeringand Technology
Quality assessed Internet Resources Mathe-maticshttp://www.intute.ac.uk/sciences/mathematics/
Oxlip/Internet
The London MathematicalSociety
LMS http://www.lms.ac.uk/ Internet
PhysMathCentral Peer reviewed e-journals (physics and maths)http://www.physmathcentral. com/
OpenAccess
The Royal Society Independent scientific academy of the UKhttp://royalsociety.org/
Internet
Society for Industrial andApplied Mathematics SIAM
Books, Journals, Conference Proceedingshttp://www.siam.org/
Internet
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Accessing electronic resources:Electronic resources can be accessed directly on the University of Oxford network.For remote access to databases, electronic reference works and e-book or e-journalspackages: Use OxLIP+ beta version (http://oxlip-plus.ouls.ox.ac. uk) and login withyour Oxford Single Sign-On (SSO) (http://www.oucs .ox.ac.uk/registration/oxford/).Alternatively, click directly on a resource name and it will prompt you for your SSOauthentication (which will override the padlock).
For remote access to individual e-journals:Use OU E-journals (http://ejournals.ouls.ox.ac. uk/). Users will be prompted tologin with their Oxford Single Sign-On (SSO) when clicking through from the list ofe-journal titles to the individual journal homepage.
For print resources check OLIS Library Catalogue http://www.lib.ox.ac.uk/olis/Please note that this List of resources is not exhaustive.
Your subject librarian can advise on other relevant resources for your research topic.Contact: Ljilja Ristic, Physical Sciences Librarian Subject Consultant, Radcliffe Sci-ence Library [email protected], Tel. (01865) 272816.
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C Applying for Computer Resources
For an account on the Mathematical Institute network, you should complete theapplication form included in your induction pack, and return it to Monica KundanFinlayson, who will provide your logon details. Nobody may use the resources ofthe Mathematical Institute without signing an appliction form, nor continue to do soonce their account has expired.
If you experience difficulties in using any of the machines or networks, please sendelectronic mail to [email protected].
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C.1 University Policy on Intellectual Property
The University of Oxford has in place arrangements governing the ownership andexploitation of intellectual property generated by students and researchers in thecourse of, or incidental to, their studies. These arrangements are set out in theUniversity’s Statutes 2013 under which the University claims ownership of certainforms of intellectual property which students may create. The main provisions in theStatutes can be found in the Regulations for the Administration of the University’sIntellectual Property Policy (http://www.admin.ox.ac.uk/statutes/regulations/182-052.shtml)
C.2 Regulations Relating to the Use of Information Technology Facilitiesand University Policy on Data Protection and Computer Misuse
Students must familiarise themselves with regulations relating to the use of infor-mation tecnology, data protection and computer misuse. Regulations and policiesrelating to the use of IT facilities can be found via the following links:
http://www.it.ox.ac.uk/legal/rules
http://www.admin.ox.ac.uk/councilsec/compliance/dataprotection
C.3 Equal Opportunities
The department subscribes to the University Equal Opportunities Statement,see http://www.admin.ox.ac.uk/eop/universityofoxfordequalitypolicy/.and the Race Equality Policy (http://www.admin.ox.ac.uk/eop/race/policy/
It also has its own Disability Statement:
“The Institute will do everything within its power to make available its teachingand other resources to students and others with disabilities to ensure that they arenot at a disadvantage. In some cases, this may require significant adjustments tothe building and to teaching methods. Those with disabilities are encouraged todiscuss their needs with the Academic Administrator [tel: 01865 273525, email [email protected]] at the earliest possible opportunity.
The Executive Committee is responsible for the department’s disability policy.
In addition, individuals may seek specialist advice and support from the UniversityDisability Office http://www.admin.ox.ac.uk/eop/disab/ or [email protected] tel: 01865 615203.
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