Examiners’ Report: Final Honour School of Mathematics Part ... · Examiners’ Report: Final...

Examiners’ Report: Final Honour School of Mathematics Part

C Trinity Term 2010

October 21, 2010

Part I

A. STATISTICS

• Numbers and percentages in each class.

See Table 1, page 1.

When drawing comparison with historic data it should be noted that from 2009 clas-sification for Part C was based on Part C alone.

• Numbers of vivas and effects of vivas on classes of result.

As in previous years there were no vivas conducted for the FHS of Mathematics Part C.

• Marking of scripts.

The whole unit dissertations and half unit dissertations were double marked. Theremaining scripts were all single marked according to a pre-agreed marking scheme

Table 1: Numbers in each class

Number Percentages %2010 (2009) (2008) (2007) (2006) 2010 (2009) (2008) (2007) (2006)

I 49 (48) (44) (38) (52) 46.23 (50.53) (46.3) (45.8) (58.4)II.1 37 (30) (45) (35) (31) 34.91 (31.58) (47.4) (42.2) (34.8)II.2 15 (13) (6) (9) (6) 14.15 (13.68) (6.3) (10.8) (6.7)III 5 (3) (0) (1) (0) 4.72 (3.16) (0) (1.2) (0)P 0 (0) (0) (0) (0) 0 (0) (0) (0) (0)F 0 (1) (0) (0) (0) 0 (1.05) (0) (0) (0)Total 106 (95) (95) (83) (89) 100 (100) (100) (100) (100)

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which was strictly adhered to. For details of the extensive checking process, see PartII, Section A.

• Numbers taking each paper.

See Table 5 on page 7.

B. New examining methods and procedures

In the assessment of dissertations, it was newly introduced this year that each of the twoassessors, as well as proposing a mark, would also include an error bar around the mark;these were taken into account in the reconciliation process. For timed examinations, therewere were no changes this year (either in the number of questions available or the timeavailable). There had been several changes in 2009 (see: 2009 report), and it was felt thatthis should be a year of consolidation.

We adopted two suggestions from the previous year’s external examiners: the distributionof “BY HAND ONLY” folders which were used for transferring examinations between staff,and a message was sent to all assessors, requesting them to look at the paper afresh aftersome time had lapsed in order to iron out any misprints that might have occurred.

C. Changes in examining methods and procedures currently under discus-sion or contemplated for the future

It is being contemplated that in 2011 every half unit will be separately assessed, with aseparate USM for each.

D. Notice of examination conventions for candidates

The first notice to candidates was issued on 24th February 2010 and the second notice onthe 3rd May.

These can be found at

http://www.maths.ox.ac.uk/current-students/undergraduates/examinations/notices-part-c-candidates,and contain details of the examinations and assessments. The course handbook containsthe full examination conventions and all candidates are issued with this at Induction intheir first year. All notices and examination conventions are on-line athttp://www.maths.ox.ac.uk/notices/undergrad.

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http://www.maths.ox.ac.uk/current-students/undergraduates/examinations/notices-part-c-candidates

http://www.maths.ox.ac.uk/notices/undergrad

Part II

A. General Comments on the Examination

The examiners would like to thank in particular Helen Lowe, Waldemar Schlackow andCharlotte Rigdon for their commitment and dedication in running the examinations sys-tems; without their tireless efforts examiners would bear a much heavier load and we areindebted to them. We would also like to thank Sandy Patel and Margaret Sloper for alltheir work during the busy exam period. We also thank the assessors for their promptsetting of questions and for the care in checking their own and the other half unit. All theassessors and the internal examiners would like to thank the external examiners ProfessorsJames Vickers and Peter Giblin for their prompt and careful reading of the draft papersand insightful comments throughout the year.

Finally we thank the Physics Department for the prompt return of scripts for C7.4 (aheadof the deadlines Physics works to).

Timetable

The examinations began on Monday 31st May and finished on Saturday 12th June.

Medical certificates and other special circumstances

The examiners were presented with medical notes for three candidates. There were minormisprints in three papers which did not require any changes to the marking scheme.

Setting and checking of papers and marks processing

As is our usual practice, the questions were initially set by the course lecturer, with thelecturer of the other half of the course and the Subject Panel Convenor involved as checkersbefore the first drafts of the questions were presented to the examiners. The course lecturersalso acted as assessors, marking the questions on their course(s).

The internal examiners met in early January to consider the questions on Michaelmas termcourses, and changes and corrections were agreed with the lecturers. The revised questionswere then sent to the external examiners. Feedback from external examiners was givento examiners and the relevant assessor for each paper who responded to internal examinerfor their next meeting. Internal examiners met a second time to consider the externalexaminers’ comments and the assessor responses making further changes as necessary beforefinalising the questions. The same cycle was repeated towards the end of Hilary term forthe Hilary term courses, although the schedule here was much tighter. The Camera ReadyCopy was prepared for Michaelmas term courses and each assessor invited to sign this offso as to reduce the workload in early April when deadlines are much tighter. Following thepreparation of the Camera Ready Copy for HT courses, each assessor signed off their paperin time for submission to Examination schools in week 1 of Trinity term.

This year all examination scripts were collected from the Mathematical Institute ratherthan Examination Schools, which all worked smoothly.

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A team of graduate checkers, under the supervision of Helen Lowe, sorted all the scriptsfor each paper of this examination, carefully cross checking against the marks scheme tospot any unmarked questions or part of questions, addition errors or wrongly recordedmarks. Also sub-totals for each part were checked against the marks scheme, noting correctaddition. In this way a number of errors were corrected, each change signed by one of theexaminers who were present throughout the process. A check-sum is also carried out toensure that marks entered into the database are correctly read and transposed from themarks sheets.

Determination of University Standardised Marks

This year the Mathematics Teaching Committee issued each examination board with broadguidelines on the proportion of candidates that might be expected in each class. This wasbased on the average in each class over the last four years, together with recent historic datafor Part C, the MPLS Divisional averages, and the distribution of classifications achievedby the same group of students at Part B. It should be noted that at Part C there are asignificantly higher percentage of first class classifications awarded. This is primarily due tothe fact that it is, on average, the better students who proceed to Part C. The classificationsawarded at Part C broadly reflect the overall distribution of classifications which had beenachieved the previous year by the same students.

Table 2 on page 5 gives the final positions of the corners of the piecewise linear maps usedto determine USMs from raw marks. For each paper, P1, P2, P3 are the (possibly adjusted)positions of the corners above, which together with the end points (100, 100) and (0, 0)determine the piecewise linear map raw → USM. The entries N1, N2, N3 give the numberof incoming firsts, II.1s, and II.2s and below respectively from Part B for that paper, whichare used by the algorithm to determine the positions of P1, P2, P3. Where a half-paper isnot shown, it inherits the map from its whole paper parent. Where half papers are shown,the raw marks are out of 50, but have been scaled to marks out of 100.

In the Corners Table, Table 2 on page 5 the following key is used :

• † denotes the use of no corners or corners have been inserted by hand;

• ‡ denotes the removal of one corner.

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Table 2: Position of corners of piecewise linear function

Paper P1 P2 P3 Additional corners N1 N2 N3

C1.1 (section 1) (86, 72) (44, 57) (16.3, 37) 1 3 0C1.1 (section 2) (89, 71) (38, 57) (7.9, 37) 1 3 0C1.2 (section 1) (61.6, 72) (34.6, 57) (15.9, 37) 2 3 0C1.2 (section 2) (86.8, 72) (34, 57) (9.6, 37) 2 3 0C2.1 (section 1) (46, 72) (41.4, 57) (19, 37) 3 0 0C2.1 (section 2) (53.4, 72) (49.4, 57) (22.7, 37) 3 0 0C2.2b (64.4, 72) (61.4, 57) (28.2, 37) 2 0 0C3.1a ‡ (64.4, 72) (12, 30) 3 0 1C3.2 (section 1) (56, 72) (35.4, 57) (16.3, 37) 3 0 0C3.2 (section 2)‡ (56.4, 72) (24.5, 37) 3 0 0C3.3 (section 1) (80, 72) (28, 57) (12.7, 37) 2 0 1C3.3 (section 2) ‡ (62.4, 72) (18, 31) 2 0 1C4.1 (section 1) (57.6, 72) (45.6, 57) (21, 37) 5 0 0C4.1 (section 2)‡ (60.4, 72) (26.4, 37) 5 0 0C5.1 (section 1) (65.6, 72) (38.6, 57) (17.7, 37) 4 2 0C5.1 (section 2) (72, 72) (42, 57) (19.3, 37) 4 2 0C5.2b (72, 72) (52, 57) (20, 37) 4 1 2C6.1a (75.6, 72) (33.6, 57) (15.4, 37) 7 5 3C6.2b (88, 70.5) (46, 57) (20, 37) 6 13 8C6.3 (section 1) (68.8, 72) (32.8, 57) (15, 37) (78, 83) 12 20 5C6.3 (section 2) (62, 72) (36, 57) (16.5, 37) 12 20 5C6.4a (64, 72) (36, 57) (16, 37) (44, 66) (32, 54) 11 20 1C7.1b (68, 72) (34, 57) (16, 37) 3 7 2C7.4 †C8.1 (section 1) (72, 72) (31, 57) (14.2, 37) (14, 36) (10, 30) 9 20 10C8.1 (section 2) (82, 72) (25.4, 57) (11.7, 37) 9 20 10C9.1 (section 1) (58, 72) (37.6, 57) (17.3, 37) 3 2 2C9.1 (section 2) (60, 72) (20, 50) (12, 37) (40, 66) 3 2 2C10.1 (section 1) (80, 70) (48, 57) (13.6, 37) 2 1 0C10.1 (section 2) (71.6, 72) (59.6, 57) (27.4, 37) 2 1 0C11.1 (section 1) (74, 72) (52, 57) (25.4, 37) 13 8 8C11.1 (section 2) (74, 72) (44, 57) (20.2, 37) (8, 20) 13 8 8C12.1 (section 1) (76, 72) (52, 57) (23.9, 37) 5 3 2C12.1 (section 2) (84, 72) (43.6, 57) (20, 37) 5 3 2C12.2 (section 1) ‡ (49.6, 57) (22.8, 37) 3 2 1C12.2 (section 2) (84, 72) (38, 57) (14.4, 37) 3 2 1MS1a ‡ (86, 74) (38.4, 37) 0 3 2MS2b (74, 72) (38, 57) (13.3, 37) 4 7 0MS3b (72, 70) (30, 57) (16, 37) 7 12 3MS5b‡ (64, 72) (25.5, 37) 0 1 3

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Table 3: Percentile table for overall USMs

Av USM Rank Candidates with this USM or above %94 1 1 0.9492 2 2 1.8991 3 4 3.7790 5 5 4.7289 6 6 5.6688 7 7 6.686 8 8 7.5585 9 9 8.4984 10 12 11.3283 13 13 12.2682 14 17 16.0481 18 18 16.9880 19 20 18.8779 21 23 21.778 24 27 25.4777 28 29 27.3676 30 31 29.2575 32 33 31.1374 34 34 32.0873 35 36 33.9672 37 37 34.9171 38 42 39.6270 43 49 46.2369 50 54 50.9468 55 62 58.4967 63 65 61.3266 66 66 62.2665 67 68 64.1564 69 71 66.9863 72 75 70.7562 76 81 76.4260 82 86 81.1359 87 87 82.0858 88 90 84.9157 91 92 86.7956 93 94 88.6855 95 96 90.5754 97 97 91.5152 98 100 94.3450 101 101 95.2849 102 103 97.1747 104 104 98.1140 105 106 100

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B. Equal opportunities issues and breakdown of the results by gender

Table 4: Breakdown of results by gender

Class Total Male FemaleNumber % Number % Number %

I 49 46.23 37 44.58 12 52.17II.1 37 34.91 28 33.73 9 39.13II.2 15 14.15 13 15.66 2 8.7III 5 4.72 5 6.02 0 0P 0 0 0 0 0 0F 0 0 0 0 0Total 106 100 83 100 23 100

Table 4, page 7 shows the performances of

candidates broken down by gender.

C. Detailed numbers on candidates’ performance in each part of the exam

Table 5: Numbers taking each paper

Paper Section Number of Avg StDev Avg StDevCandidates RAW RAW USM USM

C1.11 1 4C1.11 2 4C1.1a1 1 3C1.1b1 1 1C1.21 1 5C1.21 2 5C1.2a1 1 5C1.2b1 1 2C2.11 1 3C2.11 2 3C2.2b1 1 2C3.1a1 1 4C3.21 1 3C3.21 2 3C3.2b1 1 1C3.31 1 3C3.31 2 3C3.3a1 1 3C4.11 1 5C4.11 2 5

1Statistics are not reported for this paper as the number of candidates is five or fewer.

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Paper Section Number of Avg StDev Avg StDevCandidates RAW RAW USM USM

C4.1a1 1 4C5.1 1 6 32 5.97 72.17 8.21C5.1 2 6 34.5 8.41 73 12.43C5.1a 1 6 30.33 10.75 70.33 14.72C5.1b1 1 2C5.2b 1 7 34.86 14.81 73.71 23.91C6.1a 1 15 30.8 10.62 68.53 11.6C6.2b 1 26 34.92 10.41 67.62 13.84C6.3 1 37 27.43 7.01 66.92 7.79C6.3 2 37 27.3 7.7 67.57 11.36C6.3a 1 19 24.74 7.68 64 8.39C6.3b1 1 4C6.4a 1 31 25.42 8.91 65.81 11.85C7.1b 1 12 28.75 8.44 67.83 9.59C8.1 1 39 26.74 10.62 65.74 12.94C8.1 2 39 28.38 12.76 66.18 13.17C8.1a1 1 3C8.1b 1 19 30.68 12.67 67.16 14.92C9.1 1 7 26.86 11.11 68 16.49C9.1 2 7 22.14 14.39 62.29 19.88C9.1a 1 6 22.67 9.11 61.83 14.01C9.1b1 1 2C10.11 1 3C10.11 2 3C10.1a1 1 3C11.1 1 22 35.18 9.54 71.09 15.52C11.1 2 22 31.09 12.28 66.18 18.2C11.1a 1 27 33.41 6.31 69 10.57C12.1 1 9 36.89 7.3 72.44 11.22C12.1 2 9 36.78 10.26 71.33 13.51C12.1a 1 13 29.85 8.52 62.69 13C12.2 1 6 38 10.14 79.5 17.48C12.2 2 6 34.5 11.76 70.5 12.14C12.2a1 1 5C12.2b 2 7 35.57 12.55 71.43 14.13MS1a1 1 3MS2b 1 9 30.78 6.46 67.44 6.17MS3b 1 16 30.94 9.98 68.88 10.66MS5b1 1 1Half Unit CD1 4DissertationCD Dissertation 14 70.36 8.09OD Dissertation1 4

1Statistics are not reported for these papers as the number of candidates is five or fewer.

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The tables that follow give the question statistics for each paper for Mathematics candidates.

Question statistics are not reported for the following papers as they were taken by five orfewer mathematics candidates.

C1.1: Godel’s Incompleteness Theorems and Model Theory

C1.1a: Godel’s Incompleteness Theorems

C1.1b: Model Theory

C1.2: Analytic Topology and Axiomatic Set Theory

C1.2a: Analytic Topology

C1.2b: Axiomatic Set Theory

C2.1: Lie Algebras and Representation Theory of Symmetric Groups

C2.2b: Representation Theory of Semisimple Lie Algebras

C3.1a: Topology and Groups

C3.2: Lie Groups and Differentiable Manifolds

C3.2b: Differentiable Manifolds

C3.3: Abelian, Nilpotent & Solvable Groups and Groups, Trees & Hyperbolic Spaces

C3.3a: Abelian, Nilpotent & Solvable Groups

C4.1: Functional Analysis and Banach and C∗-Algebras

C4.1a: Functional Analysis

C5.1b: Fixed Point Methods for Nonlinear PDEs

C6.3b: Applied Complex Variables

Paper C8.1a: Mathematics and the Environment

C9.1b: Elliptic Curves

C10.1: Stochastic Differential Equations and Brownian Motion in Complex Analysis

C10.1a: Stochastic Differential Equations

C12.2a: Approximation Theory

MS1a: Graphical Models and Inference

MS5b: High-throughput Data Analysis

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Paper C5.1: Methods of Functional Analysis for PDEs and Fixed Point Methodsfor Nonlinear PDEs

Question Mean Mark Std Dev Number of AttemptsAll Used Used Unused

Q1 17.67 17.67 3.33 6 0Q2 14.4 14.4 3.21 5 0Q3 14 14 – 1 0Q4 15.5 15.5 5.97 4 0Q5 16.4 19.5 8.62 4 1Q6 16.75 16.75 3.77 4 0

Paper C5.1a: Methods of Functional Analysis for PDEs


Q1 17.17 17.17 4.07 6 0Q2 17.75 17.75 4.57 4 0Q3 8 8 – 1 0

Paper C5.2b: Calculus of Variations


Q1 16.57 16.57 7.98 7 0Q2 18.29 18.29 7.30 7 0Q3 4 – 4.24 0 2

Paper C6.1a: Solid Mechanics


Q1 14.93 14.93 6.26 14 0Q2 15.6 15.6 5.03 15 0Q3 19 19 – 1 0

Paper C6.2b: Elasticity and Plasticity


Q1 16.68 16.68 5.56 22 0Q2 20.64 20.64 4.74 22 0Q3 10.2 10.88 5.33 8 2

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Paper C6.3: Perturbation Methods and Applied Complex Variables


Q1 13.67 14.15 4.04 33 3Q2 14.41 14.77 4.58 26 1Q3 9.35 10.93 4.17 15 8Q4 11.37 11.37 4.40 27 0Q5 15.49 15.49 4.16 37 0Q6 12.25 13 5.01 10 2

Paper C6.3a: Perturbation Methods


Q1 12.33 12.71 4.087 17 1Q2 13.06 14.5 5.88 14 2Q3 5.45 7.29 3.33 7 4

Paper C6.4a: Topics in Fluid Mechanics


Q1 12.70 12.70 6.09 27 0Q2 9.18 9.7 4.24 10 1Q3 13.92 13.92 4.88 25 0

Paper C7.1b: Quantum Theory and Quantum Computers


Q1 13.92 15 3.65 10 2Q2 12.8 12.8 7.39 10 0Q3 16.75 16.75 1.26 4 0

Paper C8.1: Mathematics and the Environment and Mathematical Physiology


Q1 6.73 6.95 3.22 20 2Q2 15.37 15.37 5.11 38 0Q3 16 16.84 5.72 19 1Q4 11.26 11.26 6.88 34 0Q5 15.41 15.41 6.99 37 0Q6 22 22 2.94 7 0

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Paper C8.1b: Mathematical Physiology


Q1 12.74 12.74 8.02 19 0Q2 16.87 16.87 6.15 15 0Q3 22 22 1.41 4 0

Paper C9.1: Analytic Number Theory and Elliptic Curves


Q1 13.57 13.57 6.40 7 0Q2 11.67 11.67 3.67 6 0Q3 23 23 – 1 0Q4 10.29 10.29 5.97 7 0Q5 11.5 11.5 10.60 4 0Q6 12.33 12.33 8.96 3 0

Paper C9.1a: Analytic Number Theory


Q1 11.33 11.33 6.44 6 0Q2 10.6 10.6 2.61 5 0Q3 15 15 – 1 0

Paper C11.1: Graph Theory and Probabilistic Combinatorics


Q1 19.24 19.24 4.43 21 0Q2 12.13 12.14 6.62 7 1Q3 16.21 17.81 5.87 16 3Q4 17.05 17.05 6.30 22 0Q5 13.11 14 7.29 8 1Q6 13.47 14.07 6.77 14 1

Paper C11.1a: Graph Theory


Q1 17.72 18.13 4.30 24 1Q2 14.38 14.92 4.18 13 3Q3 15.2 16.06 3.68 17 3

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Paper C12.1: Numerical Linear Algebra and Continuous Optimization


Q1 16 16 3 9 0Q2 20.89 20.89 5.11 9 0Q3 – – – – –Q4 20.44 20.44 5.17 9 0Q5 15.33 15.33 6.15 6 0Q6 18.33 18.33 6.35 3 0

Paper C12.1a: Numerical Linear Algebra


Q1 14.23 14.23 4.15 13 0Q2 17.5 17.5 4.60 10 0Q3 7 9.33 4.74 3 2

Paper C12.2: Approximation Theory and Finite Element Methods


Q1 17.67 17.67 6.68 6 0Q2 21.33 21.33 3.21 3 0Q3 19.33 19.33 4.93 3 0Q4 17.6 17.6 6.27 5 0Q5 16.2 16.2 7.12 5 0Q6 19 19 4.24 2 0

Paper C12.2b: Finite Element MEthods for Partial Differential Equations


Q1 15.43 15.43 6.55 7 0Q2 22.4 22.4 3.29 5 0Q3 14.5 14.5 13.44 2 0

Paper MS2b: Stochastic Models in Mathematical Genetics


Q1 14.88 15 3.23 7 1Q2 15.6 15.6 5.46 5 0Q3 15.67 15.67 2.34 6 0

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Paper MS3b: Levy processes and Finance


Q1 13.72 14.7 6.81 10 1Q2 15.87 15.87 5.93 15 0Q3 13.78 15.71 5.97 7 2

D. Recommendations for Next Year’s Examiners and Teaching Committee

We strongly recommend that a prize is instituted for dissertations (unanimously endorsedby all examiners, including the external examiners).

E. Comments on sections and on individual questions

The following comments were submitted by the assessors.

C1.1a: Godel’s Incompleteness Theorems

Thirteen candidates in TT 2010 took C1.1/C1.1a Godel’s Incompleteness Theorems, 7 fromMathematics, 5 from Mathematics and Philosophy, and 1 from Mathematics and ComputerScience. The average mark per question overall (on each candidate’s two highest marks)was 17.2; the average mark for the 7 Mathematics candidates was 13.8 and the average markfor the 6 candidates from Mathematics and Philosophy and Mathematics and ComputerScience was 21.2. One candidate had a total of 10 marks for his/her highest two questions;all other candidates scored at least 14 on one of their two questions, five candidates scored 20or above on both their highest two questions, and one candidate scored 25 on two questions.Excluding the candidate whose total of marks was 10, the overall average mark per questionwas 18.2, and the average mark for the remaining 6 Mathematics candidates was 15.25.

Excluding weakest third answers (of which there were 4), question 1 received 13 answers(i.e. answered by all candidates), question 2 received 8 answers, and question 3 received5 answers. There were three perfect answers to question 1, and two perfect answersand one nearly perfect answer to question 3. The highest mark on question 2 was 20,and no candidate gave a correct answer to question 2(d) (if for each sentence X, PA `(Pr∗PA(pXq) ⊃ Pr∗PA(pPr∗PA(pXq)q)), then with 2(c),Pr∗PA(v1) would be a provabilitypredicate for PA, but then by the Second Incompleteness Theorem, for every sentence X,PA 0∼ Pr∗PA(pXq), contradicted by 2(b)). There were two misprints in question 2—in thethird line of 2(c), Pr∗S(pXq) should have been Pr∗PA(pXq), and in the second line of 2(d),(Pr∗(pXq) ⊃ Pr∗(pPr∗(pXq)q)) should have been (Pr∗PA(pXq) ⊃ Pr∗PA(pPr∗PA(pXq)q)),but there was no indication in candidates’ answers to these questions that they had anydifficulty reading them as what was intended.

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C1.1b: Model Theory

9 students took the exam, with all but one choosing question 4 and 6 (resp. 1 and 3 forthe half-unit). It seems that 5 of these students are very strong (with more than 41 pointsin total). Two students got 34 and 36 points (2i), the other two got 28 (2.ii) and a poor 17(3). Each of question questions 4 and 6 received 25 marks at least one, the only attempt atquestion 5 (resp. 2) got 3 marks. I don’t think that this question was particularly difficult,it had just a bit more text to it, and covered some of the rather late material in the course(though by no means throughout).

C1.2a: Analytic Topology

Overall the students found the non-bookwork parts harder than I expected, whereas thebookwork was mostly done well.

Of note is that every part of every question (with one tiny exception in Question 2) wascompletely answered by some student, so everything was doable. The highest overall rawmark was 38/50 achieved 3 times.

Comments on individual questions:

Question 1: All but one student attempted this question. The bookwork was done well ingeneral, with 3 common problems: forgetting to include a lower separation axiom in thedefinition of T3.5; not being able to prove that the embedding was open; characterizing theproduct as ‘the coarsest topology such that the projections are all continuous’. As the lastof these could be cast (with some difficulty) into a characterization of the product in termsof maps into the product and projections I gave 1 out of 2 marks for it.

Two students did not define Sierpinski space and hence did not prove that the maps theygave were continuous; 3 students gave (different) universal spaces for non-T0 spaces, butmost students did not even attempt this part of gave the indiscrete space as an example(which it is not).

In the last part, most students showed that the compact set C was normal, but then appliedTietze’s extension theorem to X; others only required a T3.5 space in their statement ofTietze’s theorem.

Overall there were 3 marks at 18 or above with one student gaining full marks. Most othermarks were in the 14-16 mark range.

Question 2: All students attempted this question. Again, the bookwork was mostly welldone; there was one mistake done by all students (who got that far): they obtained a mapwhich was the identity on a dense subset of a topological space and simply said that ittherefore had to be the identity everywhere without mentioning that the space was Haus-dorff. I have checked in my lecture notes and I did remark that Y is Hausdorff there (butof course might not have drawn the attention to it during the lectures).

The second half proved surprisingly hard with only very few students gaining significantmarks in either direction of the equivalence: 1 student got the backwards direction with6/6, 2 others the forwards direction with 6/6. These 3 just obtained a first class mark(18/25) whereas most others only got 12 or 13 marks with three very weak answers (2,4,8marks).

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Question 3: Only 1 student attempted this question and obtained full marks. I foundthis surprising because it is the least fiddly of the questions but of course requires a goodhigh-level understanding and some skill in set-algebra.

C1.2b: Axiomatic Set Theory

13 students sat the exam. The number of candidates is lower than previous years, as is thenumber of those who attended lectures.

All but one students demonstrated sufficient understanding of the material. Four studentsscored more than 20 marks in two questions (only two out of 27 last year).

Question 4. This was attempted by most of the candidates, the marks are between 5 and24, rather evenly spread. So right difficulty level.

Question 5. As popular as question 4. On cardinal arithmetic, including Koenig’s Lemma.The proof of the Lemma itself wasn’t a major problem, although some made serious tech-nical mistakes in the proof. More candidates this year than previous years demonstratedreasonable skills in direct cardinal arithmetic calculations. Marks are quite good, slightlybetter than in question 4.

Question 6. Epsilon-induction and the axiom of foundation. Not difficult but somewhat un-expected type of question for many. Hence - the least popular question but three candidatesdid it quite well, 20, 22 and 24 marks.

C2.1a: Lie Algebras

The examination was taken by seven students. It was clear from lectures, classes and theexamination that this group of students was substantially stronger than groups that tookthe course in previous years. In devising questions and allocating credit to parts of questionsI took this into account; there were parts (with just a few marks) to distinguish the reallyable students from the merely First Class. More than half of the solutions to questions wereFirst Class.

Q. 1. Attempted by all seven candidates; marks 9–25, average 19.

Only one candidate could not do the bookwork from early in the course. The exercises inpart (c) discriminated effectively the candidates who could manipulate the material in acreative way.

Q. 2. Three attempts, marks 3, 13, 23.

Most candidates recognised that this question was less routine than the others. (The can-didate who scored 3 attempted all three questions, so that mark can be disregarded.) Thechallenge in the example after the mildly disguised bookwork is to avoid heavy matrixcomputations; one candidate succeeded excellently.

Q. 3. Five attempts; marks 10–25, average 19.

Variations on themes from lectures; easy for most members of this talented class.

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C2.1b: Representation Theory of Symmetric Groups

There were a small number of candidates who did the questions for this course. Most ofthe takers of the course produced excellent solutions.

C2.2b: Representation Theory of Semisimple Lie Algebras

Qu1. There was a mistake in the question. One student attempted this question. Thecandidate easily spotted the mistake and knew how to fix it. This caused the candidate nodifficulties with the rest of the question. The candidate’s answer did not give full details.

Qu2. Two students attempted this question. One gave a complete answer the other asketchy one.

Qu3. One student attempted this. The student gave a full answer.

The questions were of appropriate difficulty.

C3.1a: Topology and Groups

Summary of individual questions:

Question 1

Eight out of the eleven candidates attempted this question and it seemed to differentiatethem well. The range of marks was as low as 3 from a candidate who was only able to givethe most basic definitions and nothing more, to a candidate who scored 22, which representsfull marks apart from the last part of the question at which point it looks likely that he/sheran out of time.

The main differentiating parts of this question at the upper end were the last two parts. Inparticular the penultimate part saw a range of marks from 0/10 to 10/10 with several scoresin between, showing the candidates’ varying ability to argue with precision and detail.

With reference to borderlines, my feeling is that this question was set at the correct standard.The two candidates who scored 3 and 4 clearly failed this question. I would rate thecandidate who scored 9 as a low IIii on this question. I think the candidates who scored14 and 15 points were a the medium to low end to IIi territory for this question, and thecandidate who scored 22, whilst only 2 points more was a clear first who would probablyhave gained full marks had he/she completed the question.

Question 2

All eleven of the candidates attempted this question and it was generally well done. Nine ofthe candidates scored 19 or more, which possibly suggests that the question was somewhaton the easy side and something a little more challenging should have been added at theend. Nevertheless I was pleased to see that candidates seemed to have a good grasp of thegroup theoretic concepts in this question.

The candidate who scored 3 on this question is for me a clear fail on this question. Thecandidate who scored 8 was for me a class III on this question. I find the rest of thecandidates to be either at IIi standard or first class standard for this question. I wouldprobably place the cut off at around 21 or 22 because the question was probably a little

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easy.

Question 3

This question was attempted by three candidates, who scored 22, 23 and 24. In contrast toquestion 2 which I now think was probably a little easy, I maintain that this question wasat roughly the correct standard, and certainly not too easy. The candidates who attemptedit were clearly proficient which material right to the end of the course, and they presentedtheir answers in a fashion which was pleasing to read. I rate all three candidates’ solutionsas first class on this question.

Overall summary:

There are two general issues regarding this paper which I would like to draw to the attentionof the examiners.

First, both questions 1 and 3 required knowledge of covering spaces, so any candidates whodid not revise this part of the course were destined to have problems. I think this was thecase for at least one or two candidates.

Second, whilst I now think that question 2 was a little easy, I do not think that the paper asa whole was too easy because of the more difficult questions 1 and 3. Further compensatingfor the easiness of question 2, I think the paper was perhaps on the long side, as evidencedby the fact that a clearly very able candidate was unable to finish.

C3.2a: Lie Groups

There were only few attempts to the exam. Only one candidate did very well. It is probablyfair to say that this year’s exam was somewhat harder and longer than in previous years.

Question 1: (3 attempts)

The first part should have been straight forward with a similar question having appearedon a previous exam. Unfortunately only one of the candidates remembered that the Liealgebra associated to SO(3) is the same as R3 with the cross product and was able to usethat to give a complete proof. In part (b), all but one of the answers were confused, andthe last part was not correctly solved by any of the candidates.

Question 2: (4 attempts)

The first three parts were generally well-done. For part (d), only one of the candidatesspotted that the decomposition follows directly from Schur’s Lemma, as intended in thisquestion. Another reproduced the books and lecture notes’ approach to this decompositionwhich received full marks. There were a couple of reasonable attempts at the last part.

Question 3: (1 attempt)

In retrospect quite a tough question. But the only attempt at this question showed alreadyweakness in the first and quite straight forward part of the question.

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C3.2b: Differentiable Manifolds

Of the questions, the middle one was not attempted, the first elicited somewhat disap-pointing solutions, the third, the bookwork question, was answered satisfactorily on thewhole.

C3.3a: Abelian, Nilpotent and Solvable Groups

11 students took the exam, 5 of them as half unit. There was a wide spread of marks 14/50to 44/44. Most of the candidates did quite well and I think were quite strong. Despite thewide range of marks I don’t think there was a problem with the paper. My feeling is thatthe students that did poorly hadn’t assimilated basic parts of bookwork.

Q1 was on basic algebraic properties of torsion free nilpotent groups. The last two parts ofthe question were trickier but still 4 students got nearly perfect marks on this question.

Q2 was on solvable groups and growth. The last part was quite challenging as they had toinfer the algebraic structure of a group from its growth type. Candidates did generally didwell in the algebraic part of the question but only one managed to get full nearly marks.

Q3 was the least popular question as was more geometric in nature. Only two candidatesattempted this and got reasonable marks. I don’t think the question was too hard butprobably the mixture of algebra and geometry intimidated the candidates.

C3.3b: Groups, Trees and Hyperbolic Spaces

6 students took the exam, 5 of them as half unit. There was a widespread of marks 6/50 to39/50. Most of the candidates were very strong and I was pleased with their performance.Among the students that took this as half unit most did better in the questions coming fromC3.3a. However the one candidate that took this as half unit did quite well so I think thequestions were reasonable. Probably the reason most candidates concentrated on the otherhalf during the exam was that the questions were more familiar and algebraic in nature.

Q1 This was a basic question about free groups. Students did well in parts a and c buthad difficulties with part b as they tried to solve this by word calculations instead of usingsubgroup theorems.

Q2 This was a quite long question on amalgamated products. Students had difficulties inthe parts that required to use the action on the tree to understand elements with infinitelymany roots.

Q3 This question was attempted by only 2 students. It was on thelast part of the coursedealing with quasi-isometries and Stallings theorem. Nobody did the last part, which, Ithink now, was probably too hard.

C4.1a: Functional Analysis

The half-paper was taken by 10 candidates, whose marks ranged from 24 to 47 out of 50.All three questions attracted a reasonable number of attempts, with a good success rate.There are no technical issues that I wish to report on.

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My impression is that the questions were by no means easy and that the five scores of over40 are not just first-class, but good (to very good) first-class.

C4.1b: Banach and C∗-Algebras

General comments

Only six candidates sat the paper. This represents a considerable decline over recent years,probably caused by the absence of a proper analysis course in the second year, and anunpopular Banach space course in the third year. The standard was high, the average markper question being approximately twenty.

Individual questions

(Q4) Only two candidates attempted this question and neither was able to manage tocomplete the final part of the question.

(Q5) This was the most popular question and the marks achieved were high. Marks werelost mainly because of omissions of some parts of the proofs.

(Q6) This question was set on the final part of the course and it is gratifying to see that itreceived several attempts, all of which indicated that candidates had understood almost allaspects of the course.

C5.1a: Methods of Functional Analysis for PDEs

14 candidates attempted 1 or more questions on the paper.

Overall the performance was good, and it was clear that the candidates preferred questions1 and 2. Only very few attempted question 3, and only one with an amount of success. Ialso believe that question 3 was the hardest on the paper, so it was somewhat expected.

The marking scheme suggested on the model solutions for the paper worked well, and Iadhered to it throughout. I therefore recommend that the usual limits are used for theclassification.

C5.1b: Fixed Point Methods for Nonlinear PDEs

Question 1: This question was only attempted by four candidates. Some could not re-member the proof of the second part of (a) even though this was bookwork. Part (b) wasgenerally done well, as was the first part of (c). The main problem was to show a suitablebound for iterates of the operator K. A rough bound was not sufficient.

Question 2: This was the most popular question, not surprisingly since parts (a) and (b)were bookwork. Four out of eight candidates scored full marks.

Question 3: This was perhaps the most difficult question. Several students had difficultiescomputing the directional derivative of the area functional even though this was done inlectures. Several students had also problems to show that the area functional is not coercive.Part (c) was mostly done well. The only problem was to recognize that in (ii) the functionalis not well-defined if n > 2.

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C5.2b: Calculus of Variations

Question 1 was attempted by all candidates. Bookwork was not too complicated except thesecond part of (a). Not all candidates could recognize that it is a particular case of the firstquestion in Problem Sheet 3. Part (b) was a routine application of necessary and sufficientconditions for local minima with a small perturbation in (i). In the second variation, oneshould first get rid of the independent variable and then apply the theory or Poincare ’sinequality.

Question 2 was also attempted by all candidates. Part (a) was standard bookwork. Part (b)was well done by several candidates but not all of them exploited the connection betweenthe functional and its second variation. This would give the immediate answer to question(i).

Question 3 was on application of Tonelli’s existence theorem and traditionally was attemptedby very few candidates.

C6.1a: Solid Mechanics

Comments on the questions:

1. Every candidate except one attempted Q.1 and Q.2. This is probably due to the factthat Q.3 is new material that did not appear in previous exams and, moreover, mostof it was taught in the final 4 lectures of the course.

2. The bulk of the questions were possibly a little too easy; with sufficiently conscientiousstudy of the course material one ought to easily obtain 18 to 19 points on each question.As a result the marks are concentrated slightly towards the upper and lower ends.

3. The last parts of the questions seemed of appropriate difficulty; only few candidatesattempted them successfully. The last part of Q.1 was probably a little bit easier thanthose of Q.2 and Q.3 and one should therefore put the first-class limit for Q.1 slightlyhigher.

4. The“difficult” part of Q.3 gave 6 points as opposed to 5 in the other questions; hence19 points is an essentially perfect script up to that point.

C6.2b: Elasticity and Plasticity

A total of 27 students were registered for the course, of which one was on the Mathematicsand Statistics course. The general performance was good and the students had clearlyfollowed the lectures and the problem sheets.

There were a total of three questions - the first question was on the basics of linear elasticity,the second question focussed on travelling-wave solutions of the Navier equations and thethird question focussed on linear plate theory coupled with calculus of variations. Mostof the students attempted questions 1 and 2. Question 1 was generally well-attemptedalthough a lot of students struggled with the concept of a second-rank tensor and had

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problems in deriving an identity from first principles. The second question was very well-attempted. Some students could not recall the definition of dispersive waves and otherscould not correctly locate the points where the functions, tan and cot, blow up.

Question 3 was generally not well-attempted. The students were not comfortable with thevariational nature of the question and had significant difficulty in dealing with the boundaryintegrals. It is worth pointing out that some students have done extremely well and thecourse seems to have been a successful one.

C6.3a: Perturbation Methods

1. 56 candidates attempted problem 1, so this seems to have at least been perceived aseasier (or, at least, a problem in which one can get the first, say, 10 points without muchissue). I would have preferred a roughly equal split in attempts between the 3 questions.

2. Somewhat fewer people seemed to do #3 than did #2, and many people who did #3seemed to do it as their 3rd problem (so most of them didn’t have time to give it a seriousattempt).

(Subject to my faulty memory but...) I think my exam this year was harder to get a reallyhigh score than last year, but it was easier for the weaker students to get a few points thanlast year. I didn’t actually view the problem as harder to get the final points than in pastyears, but I think what might have happened is because some of the parts were a bit long,the very top students didn’t have time to get the couple of extra points that they mighthave gotten whereas the other students wouldn’t have gotten those anyway.

Comments on the individual problems:

1. Parts (i) and (ii) should have been easy for anybody who knows anything, but somestudents still managed to mess things up a bit. (They still got most of the points, asthose parts were intended for the weaker students to get some points.) Almost all studentsmastered dominant balance, which I had stressed massively. Part (iii) wasn’t hard, but thestudents did need to know boundary layer theory well (and many students didn’t realizethat both possible BLs were actual BLs — in particular, some tried to apply a criterion from2nd order linear equations that is specifically not true for 3rd order linear equations andwhich was in fact the reason I decided to make this question about a 3rd order problem inthe first place; I wanted to see who memorized that criterion and who actually understoodwhere those results come from!). In retrospect, this part of the problem was a bit too long,and this caused some issues at the top end of the scores, as some of the 18s or 19s were forpeople who probably would have been able to get a few more points had they not run outof time and others were for people who would have been at 18 or 19 even with more time.I would say that there was a distinction between 18 and above versus 16 and below on thisproblem. (I think that a single student, or perhaps two of them, got a 17.) Some peopleweren’t efficient on parts (i) and (ii), and that gave them less time to do part (iii). Anamazing number of students (4th year Oxford students!) seem not to understand the needfor constants of integration. This is depressing. (Many students also couldn’t handle thechain rule or a simple integrating factor DE that arose along the way, which they shouldhave already mastered in Mods.) In general, I think that this turned out to be a goodquestion except that it was a bit too long. In (ii), some students also spent time thinkingabout whether there were interior layers even though there was no need. (However, given

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that the calculation in (ii) was exactly the same and that part (iii) would see the lack ofneed for an interior layer, no time had to be wasted on this if a student actually thoughtabout it. Some of them saw this and just wrote a sentence indicating this scaling in case itwas relevant, and then they correctly moved on.) The highest score was 21.

2. Many students couldn’t get the basic definition in (i), which is depressing but unsurprising(and this was true even for students who could do calculations reasonably well... when itcomes to precise definitions, however...). I gave partial credit for definitions that had someof the correct ideas but had some problems. More students than expected got confused in(ii), as they made the problem harder for themselves than it actually was. Some studentslost points for not showing sufficient working or deriving the result using an assumption(—xt— ¡ 1) to use a step that they never needed to use for the derivation (and whichbecomes iffy anyway if they’re not really careful with scaling when they take t -¿ infinity).Several students showed rigorously that the terms being dropped were indeed of smallerorder than the ones being kept, which is very good though I wasn’t actually expecting that,so in retrospect I should either have expected it or explicitly stated that it wasn’t required.[Part (ii) was meant as a way for even the weaker students to get some points.] Part (iv)of this problem was the most difficult part of the entire exam. Students didn’t catch thesubtleties that were in it, and because there were multiple ways to go about doing it, Ineeded to adjust how to get partial marks on the problem from the way I wrote in themarking scheme (and this corresponded rather closely to an alert that the checker gave me;I made a remark about this alert in the solution sheet). If a student had a vague idea ofhow to deal with things (and showed some correct manipulations), I gave them 3/10 (or1/10 if there was really only a small hint). If they also got one of the scalings right, I gavethem 5/10. If they got both scalings right and generally knew what they were doing butdidn’t get all of the subtleties, then they got 8/10. (Only one student got 10/10, and thatstudent had seen the problem before in a book that I recommended the students use tohelp with revisions.) Part (iii) was easy for many students, as was intended. [Because, asthe checker noted, there were different ways to organize one’s solution to (iv), the precisesubdivisions in my solutions need not be exactly the ones in the student write-ups, andthat is indeed what happened. This is where this subdivision of points comes from.] Aswith problem 1, the top dividing line between students seems to be at 18 and above versus16 and below. One person had evidently studied parts (iii) and (iv) of the problem fromthe book from which I obtained it (this was a book on the reading list that I repeatedlyencouraged the students was a good place to study for revisions) and got a 24/25 (even withonly a 9/25 on the other problem attempted). Overall, I think this was an effective problemeven though some of what the students did surprised me. Part (iv) served its purpose ofseeing who really knew their stuff, aside from 1 person who might have gotten lucky (or wassimply paying attention to certain homework problem solutions when I encouraged that theintegral in part (iii) would be a good one to practice!).

3. It was easy to get at least 6 points in part (i), though missing some points up happeningbecause students messed up things they should have mastered in Mods (i.e., finding ’par-ticular solutions’ using the method of undetermined coefficients). Some people lost pointsunnecessarily in part (i) because they didn’t bother following directions (the expressions foru0, u1, and u2 were explicitly asked for ... many students seemed not to quite understandthe phrasing the question, as they thought that u2 wasn’t required when it explicitly wasrequired ... only a couple of students seemed to understand that I wanted through the termwith the first secularity, which implies that I wanted u2 instead of that being the first term

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that was dropped, which is was what most students thought — I have read this questionover and over again, and I still think my phrasing was clear!). The part of problem #3 thatcaused the most problems for students was discerning the scaling at the beginning of (ii)even with the strong hint from going through the calculation in (i), and even people who gotexactly the right answer for when the secularity first appears in part (i) [which was almosteverybody] often flubbed this part of (ii) [despite the strong hint from what they saw in (i)],and this made it hard to get things right with subsequent calculations—significant pointscould be awarded with propagation errors after that point [for both (ii) and (iii), whichcould also be affected by propagation errors], but some calculations simply won’t work outright if there is an issue there. (And most students who went down the wrong path eitherdidn’t notice the issues that showed up or pretended they weren’t there, even though self-correction was distinctly possible here, though of course time constraints can curtail that.)It was very common for weaker students to get 8-12 points, stronger ones got in the midto upper teens, and the highest score was 20. Somewhat fewer students did problem #3than did the other two, so I wonder if they didn’t like my explicitly asking about scaling atthe beginning of (ii) instead of giving it to them? (This scaling arises specifically from thequadratic term, and in some sense makes this problem on the harder end of multiple scalesproblems that can reasonably show up on an exam.) Similar to other problems, the divisionbetween the best students versus the others seems to be 17 and above versus everybodyelse. Note that many of the people who got low scores on #3 concentrated on the othertwo problems and presumably were stronger students who just didn’t have time to do #3properly. The highest score was 20, and some of the calculations might have made theproblem a bit too long.

On this problem especially (but it was also sometimes true for #1 and #2), I am continuallyamazed by many people not bothering to answer questions that I explicitly asked. Isfollowing instructions really that hard? It’s a shame when points are needlessly lost in thisfashion (and a highly nontrivial number of 4th-year students seem to be doing this...).

C6.3b: Applied Complex Variables

No errors on the paper.

I was disappointed with the standard of the answers this year. I must

have set quite challenging questions (making them think a bit), but even the

bookwork was not well done in general. There were only a handful of good

answers, and lots of marks around the 8-13 level.

C6.4a: Topics in Fluid Mechanics

One small (obvious) typo Q2 which did not affect any answers. One

student was confused by x being a vertical coordinate in Q1, but this was

clarified by the invigilator. No other errors on the paper.

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C7.1b: Quantum Theory and Quantum Computers

As remarked last year a course which falls evenly into two halves is not easily tested inthree questions, and most people concentrated on the questions covering the first half of thecourse. Most answers indicated a good understanding of the bookwork. Overall there wasa reasonable spread of marks, but the calculations proved more demanding than expected,and minor slips meant that there were fewer high scores than usual.

The integral part of question 1 proved a little too challenging, and only a couple of candidatesgot the answer. (It involved an integration by parts, but candidates were clearly expectinga simplification along the way.)

In question 2 the problem was that many either wrote down the wrong integral (forgettingthat it was a volume integral and not one- or two-dimensional), or got the radial part ofthe Laplacean wrong. (A few candidates were extremely confused about the derivation ofthe variational method, and attempted to obtain it from the Heisenberg picture!)

Those who tackled the quantum computing question usually made a detour and proved theSchmidt decomposition by the method used in lectures, rather than using earlier part andmethod indicated. Correct proofs by any method got full credit, but the detour made thefinal part harder, and there were no completely satisfactory solutions.

C7.2b: General Relativity I

Only 6 maths candidates answered questions on this course but with candidates in Physicsand Philosophy and Physics included there were enough attempts to get a reasonableoverview. One question was noticably less popular, and was perhaps the longest. Therewere some high scores on all questions, so that all the questions were do-able, but theaverages were low and I think the questions would have benefitted from being shorter.

C7.4: Theoretical Physics I

See Physics Part C examiners report.

C8.1a: Mathematics and the Environment

Question 1

22 4th year Oxford mathematics undergraduates had little idea how to do a relativelystraightforward question. Not a single one could do the elementary arithmetic necessary tocompute a certain dimensionless number, the closest value being three orders of magnitudeaway. My M.Sc. students in Limerick are better than this.

Question 2

I was responsible for Question 2 on C8.1. Overall I felt the students did quite well thoughI was a little surprised by the number who made some basic mistakes with the non-dimensionalisation in 2(ii). There did appear to be a clear distinction between the majorityof students who knew the material well enough as opposed to those who had a deeperunderstanding that performed very well.

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C8.1b: Mathematical Physiology

Question 4:

Candidates struggled with this question both conceptually and algebraically even thoughit was covered in the problems classes. Several students were unable to apply the law ofmass action correctly in (i) and were unable to derive the production rate in (iv) using arecursion relation. Only a few were able to carry out the quasi-steady state reduction in(vi).

Question 5:

The majority of candidates performed well on this question, although the phase planeanalysis was rather sloppy.

Question 6:

Although a relatively small number of students tackled this question, the answers tendedto be very good.

C9.1a: Analytic Number Theory

It is hard to judge the difficulty of the paper as the marks range from extremely low tovery high. However, based on the students’ performance I think the paper is overall fairlystandard. Four out of 16 students got over 35, among them one did extremely well. Twostudents were very poor.They could not even do some standard homework and bookworkproblems.

C9.1b: Elliptic Curves

This paper was sat by just 12 candidates. There was a good spread of marks, if somewhaton the low side. Each question had a decent number of attempts.

Q1. There were no really good treatments of the variant of Hensel’s Lemma; and nosuccessful treatments for the final part.

Q2. The bookwork was well done. Each of the examples at the end saw successful answers.

Q3. Again the bookwork was well done, but only one candidate reached the end (almost!).

C10.1a: Stochastic Differential Equations

No report submitted.

C10.1b: Brownian Motion in Complex Analysis

Of the four candidates choosing to take this paper all attempted Questions 4 and 6. Ingeneral, the candidates seemed to find the questions a little more challenging than I wasexpecting but nonetheless there were some very good answers, especially to Question 6.The attempts at Question 4 were however a little disappointing. Several of the candidatesgave convoluted, but ultimately correct, answers in response to the standard calculation in

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4(a) and I think this may have deprived them of time for second half of the question. Allcandidates had some difficulty identifying the image of the different parts of the infinite halfstrip S under the conformal map f. This was surprising because, even though this particularexample was unseen, examples of this type are rehearsed almost to the point of routine onthe problem sheets.

C11.1a: Graph Theory

The bookwork parts of the exam were for the most part well done. The unseen questionswere found difficult by many candidates, although there were some extremely good solutions.

C11.1b Probabilistic Combinatorics

Question 4 was generally well done, though many candidates didn’t give quite enough detailat the end. Also, several candidates made the calculations harder for themselves by notusing part (b) to do part (c). This question was attempted by almost all candidates andwas with hindsight a little too straightforward.

Question 5 had a wide spread of marks. For the last part, all that was required was showingawareness of the result for the non-bipartite case, and a very brief indication why it is thesame in the bipartite case.

Those who attempted the whole of question 6 generally answered it reasonably well, exceptthat only one candidate answered the last part correctly. (Others came close.) Some verypartial attempts were also submitted.

C12.1a : Numerical Linear Algebra and Approximation

This exam seems to have been a fair test. Most candidates attempted the 1st question(on projections and the SVD) though many had some difficulties identifying the orthogonalprojector onto the Null space. A significant proportion also attempted the 2nd question (onGauss Elimination and orthogonal similarity transforms) with several very good attempts.The 3rd question (on sparse matrices and Conjugate Gradients) was less popular but therewere a few good attempts.

C12.1b : Continuous Optimisation

Most of the students answered questions 4 and 5, with only a couple trying question 6. Mygeneral impression is that most students did reasonably well with question 4, but struggledmore with question 5.

Those few students that tried question 6 seemed to do relatively well.

Generally, the students seemed to be rushed for questions 5 and 6; presumably they wererunning out of time and rushed. In fact, a couple of students only wrote down a handful oflines for their second attempted question. With that said, there were also several studentsthat did not seem to have any issue with time, since it took me a fortnight to read theirsolutions!

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I do not think there was any question that essentially everyone missed.

C12.2a Approximation of Functions

Question 1 Unseen application of Peano Kernel theory and bookwork part on cubic splineapproximation.

10 attempts, average 14.7

Peano Kernel material either very well done or not at all/badly.

Cubic spline approximation well understood. The two part format of question resulted inroughly half with high marks and half with good marks for second part only.

Question 2 L2 approximation question, large theory part with unseen application.


The large theory part was well done and most candidates scored well on that part, theapplication was more difficult resulting in only two attempts over 21 marks.

Question 3 Proof of de La Vallee Poussin theorem and convergence of a one point exchangealgorithm, linear best approximation problem.


Proof of de La Vallee Poussin theorem uniformly well done, proof of convergence of exchangealgorithm patchy, linear best approximation example well done.

C12.2b: Finite Element Methods for Partial Differential Equations

The exam appeared a fair test. Most candidates chose to attempt the 1st question (onweak form and minimisation form) and the 2nd question (on error analysis and Aubin-Nitsche dualityfor P2 elements), though the 3rd question (on practical manipulation for Q1element) also had several attempts. There were several high scores on each question, butalso so blatant occurrences where candidates had simply not read the questions.

Statistic Half-Units

Reports of the following courses may be found in the Mathematics and Statistics examinersreport.

MS1a : Graphical Models and Inference

MS2a: Bioinformatics and Computational Biology

MS2b: Stochastic Models in Mathematical Genetics

MS3b: Levy Processes and Finance

MS5b: High-throughput Data Analysis

MS6a: The Analysis of Biological Networks

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Computer Science Half-Units

Reports on the following courses may be found in the Mathematics and Computer Scienceexaminers report.

Quantum Computer Science

Categories, Proofs and Processes

Automata, Logic and Games

F. Comments on performance of identifiable individuals

Removed from the public version.

G. Names of members of the Board of Examiners

• Examiners:

Dr A DancerProf. P ChruscielDr A CurnockProf. EV Flynn (Chair)Prof. P GiblinDr M PorterProf. J VickersProf. J Wilson

Dr Curnock and Prof. Chrusciel were originally examiners and Dr Dancer took overduring the course of the year.

• Assessors for Papers C1.1–C12.2

Prof. P BressloffDr H BuiDr T CassProf. P ChruscielDr A CowardProf. S ChapmanProf. C DrutuDr M EdwardsDr K ErdmannDr A FowlerDr H GrambergDr K HannabussProf. R HaydonProf. R Heath-Brown

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Prof. N HitchinDr D IssacsonDr J KoenigsmannDr K Kremnizer Dr J KristensenProf. T LyonsDr A MajumdarProf. B NiethammerDr C OrtnerDr P PapazoglouDr S PeppinProf. O RiordanDr D Robinson Prof. G SanderProf. A ScottProf. G SereginDr I SobeyDr R SuabedissenProf. U TillmannProf. P TodProf. L N TrefethenDr WathenProf. B Zilber

• Assessors for dissertations

Dr P BeeleyDr C BrewardProf. M BridsonDr H BuiProf. K BurrageProf. J ChapmanDr X de la OssaPorf. M du SautoyDr R ErbanDr E GaffneyProf. D GavaghanDr K HannabussDr R HauserDr P HowellProf. S HowisonDr H JinDr G KochDr A LauderDr A MuenchDr P NeumannDr J OblojDr H OckendonProf. J Ockendon

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Dr J OliverDr P PapazoglouDr A PapachristodoulouDr S PeppinDr Z QianProf. G SereginProf. A ScottDr I SobeyDr P TarresDr B WardhaughDr A WathenDr R Whittaker

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