America n Mathematica l Societ y

Colloquiu m Publication s Volum e 51

Chira l Algebra s

Alexande r Beilinso n Vladimi r Drinfel d

America n Mathematica l Societ y Providence , Rhod e Islan d

http://dx.doi.org/10.1090/coll/051

Editorial B o a r d

Susan J . Friedlander , Chai r Yuri Mani n

Peter Sarna k

2000 Mathematics Subject Classification. Primar y 17Bxx ; Secondar y 14Fxx .

For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages/col l -51

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Beilinson, Alexander , 1957 -Chiral algebra s / Alexande r Beilinson , Vladimi r Drinfeld .

p. cm . — (Colloquiu m publications , ISS N 0065-925 8 ; v. 51 ) Includes bibliographica l reference s an d index . ISBN 0-8218-3528- 9 (alk . paper ) 1. Geometry , Algebraic . 2 . Conformal geometry . 3 . Field theor y (Physics ) 4 . Mathematica l

physics. I . Drinfeld , V . G . II . Title . III . Colloquiu m publication s (America n Mathematica l Society) ; v. 51 .

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Contents

Introduction 1

Chapter 1 . Axiomati c Pattern s 1 1

1.1. Pseudo-tenso r categorie s 1 1

1.2. Complement s 1 8

1.3. Compoun d tenso r categorie s 2 4

1.4. Rudiment s o f compound geometr y 3 2

Chapter 2 . Geometr y o f D-scheme s 5 3

2.1. D-modules : Recollection s an d notatio n 5 3

2.2. Th e compoun d tenso r structur e 6 7

2.3. Dx-scheme s 7 9

2.4. Th e space s o f horizonta l section s 8 9

2.5. Lie * algebras an d algebroid s 9 5

2.6. Coisso n algebra s 11 3

2.7. Th e Tat e extensio n 11 7

2.8. Tat e structure s an d characteristi c classe s 12 9

2.9. Th e Harish-Chandr a settin g an d th e settin g o f c-stacks 14 3

Chapter 3 . Loca l Theory : Chira l Basic s 15 7

3.1. Chira l operation s 15 7

3.2. Relatio n t o "classical " operation s 16 3

3.3. Chira l algebra s an d module s 16 4

3.4. Factorizatio n 17 2

3.5. Operato r produc t expansion s 19 4

3.6. Fro m chira l algebra s t o associativ e algebra s 20 0

3.7. Fro m Lie * algebra s t o chira l algebra s 21 2

3.8. BRST , alia s semi-infinite , homolog y 22 7

3.9. Chira l differentia l operator s 23 8

3.10. Lattic e chira l algebra s an d chira l monoid s 25 7

CONTENTS

Chapter 4 . Globa l Theory : Chira l Homolog y 27 5

4.1. Th e cookwar e 27 5

4.2. Th e constructio n an d first propertie s 29 6

4.3. Th e B V structur e an d product s 31 3

4.4. Correlator s an d coinvariant s 32 3

4.5. Rigidit y an d flat projectiv e connection s 33 0

4.6. Th e cas e o f commutative Dxalgebra s 34 1

4.7. Chira l homolog y o f the d e Rham-Ch e valley algebra s 34 8

4.8. Chira l homolog y o f chira l envelope s 35 3

4.9. Chira l homolog y o f lattic e chira l algebra s 35 7

Bibliography 36 3

Index an d Notatio n

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Index an d Notatio n

action o f a Lie * algebr a o n a commutative * algebr a 1.4. 9 action o f a Lie * algebr a o n a chira l algebr a 3.3. 3 action o f a Li e algebroid o n a chira l algebr a 4.5.4 , twiste d 4.5. 6 action o f a pseudo-tenso r categor y o n a categor y 1.2.1 1 action o f a tenso r categor y o n a pseudo-tensor categor y 1.1.6(v ) admissible comple x o f sheaves o n X s 4.2. 1 admissible D-comple x o n X s 4.2. 6 algebraic 1)x -space 2.3. 1 annihilator 3.3. 7 augmentation functor , non-degenerat e 1.2.5 , reliable 1.4. 7 augmentation functo r i n compoun d settin g 1.3.10 , in I>-modul e settin g 2.2. 7 augmented compoun d tenso r category , functo r 1.3.1 6 augmented opera d 1.2. 4 augmented pseudo-tenso r categor y 1.2. 4 augmented pseudo-tenso r functor , uni t 1.2. 8

Batalin-Vilkovisky (BV ) algebra s 4.1. 6 BV extensio n o f a Li e algebroi d 4.1. 9 BV quantizatio n o f a n od d Poisso n algebr a 4.1. 6 BV structur e o n th e chira l chai n comple x 4.3. 1 BRST reduction , charge , differential : classica l 1.4.23 , quantu m 3.8.9 , 3.8.2 0 BRST propert y 1.4.24 , 3.8.10 , 3.8.2 1

calculus o f variations 2.3.2 0 cdo 3.9. 5 centralizer 3.3. 7 center o f a chira l algebr a 3.3. 7 central chira l module s 3.3. 7 Chern classe s ch^ 2.8.1 0 Che valley-Cousin comple x o f a chira l algebr a 3.4.11 , relativ e versio n 4.4. 9 Chevalley comple x o f a Lie * algebra , inne r 1.4.5 , 1.4.1 0 chiral actio n o f a Lie * algebr a 3.7.16 , o f a chira l Li e algebroi d 3.9.2 4 chiral actio n o f a chira l monoi d 3.10.1 7 chiral algebra s 3.3.3 , commutative 3.3.3 , non-unital 3.3.2 , universa l 3.3.1 4 chiral algebr a freel y generate d b y (N,P) 3.4.14 , chiral i?£^/-algebra s 3.9. 4 chiral enveloping algebr a o f a Lie* algebra 3.7.1 , o f a chiral Li e algebroid 3.9.1 1 chiral extensio n o f a Lie * algebroid 3.9.6 , rigidifie d 3.9. 8 chiral homolog y 4.2.1 1 chiral lattic e algebra s 3.10. 1

369

370 INDEX AN D NOTATIO N

chiral Li e algebroid s 3.9. 6 chiral module s 3.3. 4 chiral L-module s 3.7.1 6 chiral i?£^/-module s 3.9. 1 chiral monoi d 3.10.1 7 chiral operation s 3.1.1 , fo r (g , K)-modules 3.1.1 6 chiral ^-operation s 3.3. 4 chiral L c/l-operations 3.7.1 6 chiral produc t 3.3. 2 chiral pseudo-tenso r structur e 3.1. 2 Clifford algebra , coisso n 1.4.21 , chira l 3.8.6 , linea r algebr a versio n 3.8.1 7 coisson algebra s 1.4.18 , modules 1.4.20 , D-module settin g 2.6.1 , ellipti c 2.6. 6 commutative" algebras , module s 1.4. 6 commutative D^-algebra s 2.3. 1 complementary quotient s 1.3. 1 compound opera d 1.3.1 8 compound pseudo-tenso r categor y 1.3.7 , augmented 1.3.1 0 compound pseudo-tenso r functo r 1.3. 9 compound tenso r category , functo r 1.3.1 4 compound tenso r produc t map s 1.3.12 , binary 1.3.1 3 connections fo r Lie * algebroids 1.4.1 7 connections o n chira l homolog y 4. 5 Contou-Carrere symbo l 3.10.1 3 convenient Dj-algebr a 4.6. 1 convenient ^-module s 4.6. 3 coordinate syste m o n a T>x -scheme 2.3.1 7 c operation s 1.4.27 , i n D-modul e settin g 3.2. 5 correlators 4.4. 1 cotangent comple x 2.3.15 , 4.1.5 , 4.6. 6 cotorsor 3.4.16 , 3.10.1 2 Cousin D-comple x 4.2. 9 Cousin filtratio n 4.2.1 , 4.2.1 9 Cousin spectra l sequenc e 4.2.3 , 4.2.11 , 4.2.1 9 c-stack, c-ran k 2.9.1 0

X>x-algebras 2.3. 1 D-algebra poin t 2.3. 1 £)-modules: lef t an d righ t 2.1.1 , functorialit y 2.1.2 , induce d 2.1. 8 D-modules: quasi-induce d 2.1.11 , maxima l constan t quotien t 2.1.1 2 D-modules: topolog y a t a poin t x 2.1.13 , universal 2.9. 9 D-modules o n #(X) , lef t 3.4. 2 D-modules o n X s , righ t 3.4.1 0 de Rham-Chevalle y comple x o f a Lie * algebroid , inne r 1.4.1 4 de Rha m comple x o f a D-modul e 2.1. 7 de Rha m homolog y 2.1.12 , cohomolog y 2.1. 7 Dx-scheme 2.3.1 , formall y smooth , smooth , ver y smoot h 2.3.1 5 Dx-scheme: th e globa l spac e o f horizonta l section s 2.4. 1 Dx-scheme: th e loca l spac e o f horizonta l section s 2.4. 8 Dolbeault algebra s 4.1. 3

INDEX AN D NOTATIO N 371

Dolbeault £>#(x)-algebr a 4.2. 7 Dolbeault resolution s 4.1. 4 Dolbeault-style algebr a 4.1. 4 Dolbeault-style D#(x)-algebr a 4.2.1 6 duality 1.3.11 , 2.2.1 6 duality fo r d e Rha m cohomology , globa l 2.2.17 , loca l 2.7.1 0

elliptic morphism , Lie * algebroi d 2.5.22 , coisson algebr a 2.6. 6 enveloping algebr a o f a n operadi c algebr a 1.2.16 , of a Li e algebroi d 2.9. 2 enveloping B V algebr a o f a B V algebroi d 4.1.8 . 4.1. 9 enveloping chira l algebr a o f a chira l Lie * algebroid 3.9.1 1 Euler-Lagrange equation s 2.3.2 0

factorization algebra s 3.4.1 , D-modul e settin g 3.4.4 , truncate d 3.4.1 3 factorization algebras : canonica l D-modul e structure 3.4.7 , commutative 3.4.2 0 factorization algebr a freel y generate d b y (iV , P) 3.4.1 4 factorization ^-module s 3.4.1 8 factorization structur e 3.4. 4 filtration o n a chira l algebra , commutative , unita l 3.3.1 2 flabby comple x o f sheaves 4.2. 2 formal groupoi d 1.4.1 5 formally smooth/etal e morphis m o f Dx-scheme s 2.3.1 6 Fourier-Mukai trnasfor m 4.9. 9

(g, i^)-modules 2.9.7 , chira l structur e 3.1.1 6 (g,K)-structure 2.9. 8 Gelfand-Dikii coisso n algebr a 2.6. 8 Gelfand-Kazhdan structur e 2.9. 9 group actio n o n a chira l algebr a 3.4.1 7

hamiltonian reductio n 1.4.1 9 handsome complexe s o f !-sheave s o n X s 4.2. 2 Harish-Chandra pair , modul e 2.9. 7 Heisenberg Lie * algebra 2.5. 9 Heisenberg grou p 3.10.1 3 homotopically Ox - an d Dx-fla t complexe s 2.1. 1 homotopy uni t commutativ e algebr a 4.1.1 4 homotopy unita l commutativ e algebr a 4.1.14 , B V algebr a 4.1.1 5 Hopf chira l algebra s 3.4.1 6

ind-scheme 2.4. 1 induced module s 3.7.15 , 3.9.2 4 inner Horn , inne r P object s 1.2.1 , i n augmente d sens e 1.2. 7

jet schem e 2.3.2-2.3. 3

Kac-Moody extensio n 2.5. 9 Kashiwara's lemm a 2.1. 3 Knizhnik-Zamolodchikov (KZ ) equation s 4.4. 6

lattices, c - an d d - 2.7. 7 Lie* algebra s an d module s 1.4.4 , D-module settin g 2.5.3-2.5. 4 Lie algebroi d 2.9. 1

372 INDEX AN D NOTATIO N

Lie* algebroi d 1.4.11 , D-modul e settin g 2.5.16 , ellipti c 2.6. 6 Lie coalgebroid 1.4.1 4

matrix * algebra 1.4. 2 middle d e Rha m cohomolog y shea f h 2.1. 6 Miura torso r 2.8.1 7 module operad s 1.2.1 1 modules ove r operadi c algebra s 1.2.1 3 morphisms o f DG Dx-algebras : semi-free , elementar y 4.3. 7 multijet schem e 3.4.2 1 mutually commutin g morphism s o f chira l algebra s 3.4.1 5

n-coisson algebr a 1.4.1 8 n-Poisson algebr a 1.4.1 8 nice complexe s o f !-sheave s o n X s 4.2. 1 non-degenerate pair s (# , Vh) 2.5.2 3 normally ordere d tenso r produc t 3.6. 1

O-modules o n Jl(X) 3.4. 2 odd coisso n algebr a 1.4.1 8 odd Poisso n algebr a 1.4.1 8 ope algebra , associative , commutativ e 3.5. 9 oper 2.6. 8 operad 1.1.4 , of Jacob i typ e 3.7. 1 operadic algebr a 1.1.6(iii) , augmented 1.2. 8 operator produc t expansio n 3.5. 8

perfect B V algebr a 4.1.1 8 perfect commutativ e D G algebr a 4.1.1 7 perfect complexe s 4.1.1 6 Poincare-Birkhoff-Witt theore m fo r Lie * algebra s 3.7.14 , twiste d cas e 3.7.2 0 PBW theore m fo r usua l Li e algebroid s 2.9.2 , 3.9.1 2 PBW theore m fo r chira l Li e algebroid s 3.9.1 1 polydifferential * operations 1.4.8 , D-modul e settin g 2.3.1 2 pre-factorization algebr a 3.4.1 4 projective connection s 2.5.1 0 pseudo-tensor categor y 1.1. 1 pseudo-tensor category : A- , £;- , additive , abelia n 1.1. 7 pseudo-tensor functors , adjointnes s 1.1. 5 pseudo-tensor /c-categor y 1.1. 7 pseudo-tensor produc t 1.1. 3 pseudo-tensor structur e 1.1. 2 pseudo-tensor subcategory , ful l 1.1.6(i )

quantization o f a coisso n algebr a 3.3.11 , mo d t 2 3.9.1 0 quasi-factorization algebr a 3.4.1 4

Ran's spac e 3.4. 1 reasonable topologica l algebr a 2.4. 8 represent able pseudo-tenso r structur e 1.1. 3 resolutions o f commutative D^-algebra s 4.3. 7 rigidification o f a Lie * algebroid 1.4.13 , o f a Lie algebroi d 2.9. 1

INDEX AN D NOTATIO N 373

rigidified 5-extensio n 3.9. 8

yi-structure 1.2.1 1 * algebra s 1.4.1 , D-modul e settin g 2.5. 1 * pairing , non-degenerat e 1.4. 2 * operations fo r D-module s 2.2.3 , induce d cas e 2.2.4(i ) Schouten-Nijenhuis bracke t 1.4.1 8 semi-free Dx-algebra s 4.3.7 , 4.6. 1 semi-free module s 4.1.5 , 4.6. 3 smooth I)x-algebra s 2.3.1 5 special V xi-module 3.1. 6 stress-energy tenso r 3.7.2 5 Sugawara's constructio n 3.7.2 5 super comple x 1.1.1 6 super convention s 1.1.1 6

/-topology 2.1.1 7 El-topologies: E x- 2.1.13 , E% ie 2.5.12 , E* L 2.5.18 , E 8/ 2.7.1 1 --topologies: E c

xois 2.6.3 , E% s 3.6.4

tangent Lie * algebroid 1.4.16 , for a D-scheme : Remar k (ii ) i n 2.3.12 , 2.3.1 5 Tate extension : D-modul e settin g 2.7.2 , 2.7.3 , o n a D^-schem e 2.7. 6 Tate extension : linea r algebr a settin g 2.7.8 , chiral approac h 3.8. 5 Tate extensio n o f a Lie * algebra 2.8.15 , 3.8. 7 Tate structur e o n a vecto r D-bundl e 2.8. 1 Tate vecto r space , compact , discret e 2.7. 7 tensor produc t o f chira l algebra s 3.4.1 5 tensor produc t o f pseudo-tensor categorie s 1.1. 9 topological associativ e algebr a 3.6. 1 topological commutativ e algebr a 2.4. 1 topological Li e algebroid 2.5.1 8 transversal quotient s 1.3. 1 twists o f chira l algebra s 3.4.1 7

unit objec t i n a compound tenso r categor y 1.3.16 , strong 1.3.1 7

vector D-bundle s 2.1.5 , on a Dx-schem e 2.3.1 0 vector D^-schem e 2.3.1 9 vertex operato r 3.5.1 4 very smoot h T>x -algebras 2.3.1 5 Virasoro extensio n 2.5.1 0 Virasoro vecto r 3.7.2 5

W-algebras 3.8.1 6 Weyl algebra , chira l an d coisso n 3.8.1 , linea r algebr a versio n 3.8.1 7 Wick algebr a 3.6.11 , globa l 3.6.2 0

0-datum 3.10. 3

Nota t ion

{A) 2.4.1 , 4.2.1 6 Aa

xs 2.4.8 , 3.6.2 , 3.6.4 , 3.6.1 3

374 INDEX AND NOTATIO N

Aas 3.6.1 4 ALie 3.3. 3 A(P) 3.4.1 7 A™ 3.6.11 AW(X) 3.6.2 0 B(M) 3.4.1 8 SV, £V U, £ V 4.1. 6 C(A) 3.4.1 1 C(B,i4) 4.4. 9 C^pC, 4 ) 4.2.1 1 C'cfc(X,i4)a, C ch(X,A)VQ 4.2.1 2 Ccf t(X,A,{M5}), C ch(X,A{Ms})g>Q, e c h ( X , A , { M j ) y Q 4.2.1 9 Gch(X,A,M)TQ, e d ( X , i , M , ) M 4.2.1 9 Cch(X,B,yl)3-Q 4.4. 9 SA{X) 3.3. 3 GApO* 3.4.1 6 e ^ O M , e r

fi(,c,-), e R (£ ) 2.9. 1 e^"( i? ) 2.9. 1 CM(Ji(X)) 4.2. 6 e^'Ci?) 1.4.1 8 Corny 1.4.6 , 4.1. 6 Gomuv{X) 2.3. 1 AlS) 3.4.1 0 "DDR 3.9.1 8 £>M(ft(.X)) 4.2. 6 £,nd*(V) 2.2.15 , 2.5.6(a ) £zp(.X) 3.4. 1 gl(V) 2.5.6(a ) 0/(F)b 2.7.4 , 2.7. 8 Hch(X, A) 4.2.11 HDR(X,M) 2.1. 7 HDR(X,M) 2.1.1 2 WoC 4.1.7 3Z 2.3. 2 L t 1.1.1 6 L* 1.4.16 , 2.9. 1 Ml, N r 2.1. 1 M c o n s t 2.1.1 2 Mx 2.8.1 7 M r (X) , M'(X) , M(X ) 2.1. 1 M(X)*' 2.2. 6 M(X)0 '1 3.1.2 , M(X)cl 3.2. 5 M(X,i?*) 2.3. 5 M(X,^) ,M(X,A) c f t 3.3. 4 M(X 8 ) 3.4.1 0 M^ttpQ ) 3.4.2 O/ 3.5. 8 7>ch(A,L) 3.9. 6

INDEX AND NOTATIO N 37 5

7ch(L) 3.9. 6 3>d(£) 2.8. 2 7(F) 3.4.1 6 Q(I) 1.3. 1 Q{I,m) 1.4.2 7 #(X) 3.4. 1 S 1.1. 1 § 1.2. 4 $chv(X) 2.3. 1 Spec(i?,£} 2.5.1 8 Spf Q 2.4. 1 Taie(V), Tate(y ) 2.8. 1 UW 3.1. 1 t/[J/7l 3.4. 4 f/(L) 3.7. 1 V(a) 3.6.1 4 Z(A) 3.3. 7 A, 2.2.2 , 3.1. 4 (8)*, ®c/l 3.4.1 0 ® 3.6. 1 »£s 2.4.8 , 3.6. 4 Ex 2.1.1 3 ~£ i e 2.5.1 2 Z* 2.4.1 1 3 ^ 2.5.1 9 ]_ch = X ch 4 2.16 .

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Titles i n Thi s Serie s

51 Alexande r Beil inso n an d Vladimi r Drinfeld , Chira l algebras , 200 4

50 E . B . Dynkin , Diffusions , superdiffusion s an d partia l differentia l equations , 200 2

49 Vladimi r V . Chepyzho v an d Mar k I . Vishik , Attractor s fo r equation s o f mathematical physics , 200 2

48 Yoa v Benyamin i an d Jora m Lindenstrauss , Geometri c nonlinea r functiona l analysis ,

Volume 1 , 200 0

47 Yur i I . Manin , Probeniu s manifolds , quantu m cohomology , an d modul i spaces , 199 9

46 J . Bourgain , Globa l solution s o f nonlinea r Schrodinge r equations , 199 9

45 Nichola s M . Kat z an d Pete r Sarnak , Rando m matrices , Frobeniu s eigenvalues , an d monodromy, 199 9

44 Max-Alber t Knus , Alexande r Merkurjev , an d Marku s Rost , Th e boo k o f

involutions, 199 8

43 Lui s A . Caffarell i an d Xavie r Cabre , Full y nonlinea r ellipti c equations , 199 5

42 Victo r Guil lemi n an d Shlom o Sternberg , Variation s o n a them e b y Kepler , 199 0

41 Alfre d Tarsk i an d Steve n Givant , A formalizatio n o f se t theor y withou t variables , 198 7

40 R . H . Bing , Th e geometri c topolog y o f 3-manifolds , 198 3

39 N . Jacobson , Structur e an d representation s o f Jorda n algebras , 196 8

38 O . Ore , Theor y o f graphs , 196 2

37 N . Jacobson , Structur e o f rings , 195 6

36 W . H . Gottschal k an d G . A . Hedlund , Topologica l dynamics , 195 5

35 A . C . Schaeffe r an d D . C . Spencer , Coefficien t region s fo r Schlich t functions , 195 0

34 J . L . Walsh , Th e locatio n o f critica l point s o f analyti c an d harmoni c functions , 195 0

33 J . F . Rit t , Differentia l algebra , 195 0

32 R . L . Wilder , Topolog y o f manifolds , 194 9

31 E . Hill e an d R . S . Phil l ips , Functiona l analysi s an d semigroups , 195 7

30 T . Rado , Lengt h an d area , 194 8

29 A . Weil , Foundation s o f algebrai c geometry , 194 6

28 G . T . Whyburn , Analyti c topology , 194 2

27 S . Lefschetz , Algebrai c topology , 194 2

26 N . Levinson , Ga p an d densit y theorems , 194 0

25 Garret t Birkhoff , Lattic e theory , 194 0

24 A . A . Albert , Structur e o f algebras , 193 9

23 G . Szego , Orthogona l polynomials , 193 9

22 C . N . Moore , Summabl e serie s an d convergenc e factors , 193 8

21 J . M . Thomas , Differentia l systems , 193 7

20 J . L . Walsh , Interpolatio n an d approximatio n b y rationa l function s i n th e comple x

domain, 193 5

19 R . E . A . C . Pale y an d N . Wiener , Fourie r transform s i n th e comple x domain , 193 4

18 M . Morse , Th e calculu s o f variation s i n th e large , 193 4

17 J . M . Wedderburn , Lecture s o n matrices , 193 4

16 G . A . Bliss , Algebrai c functions , 193 3

15 M . H . Stone , Linea r transformation s i n Hilber t spac e an d thei r application s t o analysis ,

1932

14 J . F . Ri t t , Differentia l equation s fro m th e algebrai c standpoint , 193 2

13 R . L . Moore , Foundation s o f poin t se t theory , 193 2

12 S . Lefschetz , Topology , 193 0

11 D . Jackson , Th e theor y o f approximation , 193 0

TITLES I N THI S SERIE S

10 A . B . Coble , Algebrai c geometr y an d thet a functions , 192 9

9 G . D . Birkhoff , Dynamica l systems , 192 7

8 L . P . Eisenhart , Non-Riemannia n geometry , 192 7

7 E . T . Bell , Algebrai c arithmetic , 192 7

6 G . C . Evans , Th e logarithmi c potential , discontinuou s Dirichle t an d Neuman n problems , 1927

5.1 G . C . Evans , Functional s an d thei r applications ; selecte d topics , includin g integra l equations, 191 8

5.2 O . Veblen , Analysi s situs , 192 2

4 L . E . Dickson , O n invariant s an d th e theor y o f number s

W . F . Osgood , Topic s i n th e theor y o f function s o f severa l comple x variables , 191 4

3.1 G . A . Bliss , Fundamenta l existenc e theorems , 191 3

3.2 E . Kasner , Differential-geometri c aspect s o f dynamics , 191 3

2 E . H . Moore , Introductio n t o a for m o f genera l analysi s

M. Mason , Selecte d topic s i n th e theor y o f boundar y valu e problem s o f differentia l equations

E. J . Wilczynski , Projectiv e differentia l geometry , 191 0

1 H . S . W h i t e , Linea r system s o f curve s o n algebrai c surface s

F. S . Woods , Form s o n noneuclidea n spac e

E. B . Va n Vleck , Selecte d topic s i n th e theor y o f divergen t serie s an d o f continue d fractions, 190 5