Celebratio Mathematica

Michael F. Atiyah

Papers on index theory (1963–1984)

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M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors on com­pact man­i­folds,” Bull. Amer. Math. Soc. 69 : 3 (1963), pp. 422–​433. MR 157392 Zbl 0118.​31203 article

M. F. Atiyah and R. Bott: “The in­dex prob­lem for man­i­folds with bound­ary,” pp. 175–​186 in Dif­fer­en­tial Ana­lys­is: Pa­pers presen­ted at the in­ter­na­tion­al col­loqui­um (Bom­bay, 7–14 Janu­ary 1964). Tata In­sti­tute of Fun­da­ment­al Re­search Stud­ies in Math­em­at­ics 2. Ox­ford Uni­versity Press (Lon­don), 1964. MR 0185606 Zbl 0163.​34603 incollection

M. F. Atiyah: “The in­dex of el­lipt­ic op­er­at­ors on com­pact man­i­folds” in Sémin­aire Bourbaki. 15e an­née: 1962/63. Sémin­aire Bourbaki. Secrétari­at Math­ématique (Par­is), 1964. Ex­posé no. 253. Re­print of art­icle in Bull. Am. Math. Soc. 69 (1963). See also Sémin­aire Bourbaki 8 (1995). Zbl 0124.​31102 incollection

M. F. Atiyah: “The in­dex the­or­em for man­i­folds with bound­ary,” pp. 337–​351 in Sem­in­ar on the Atiyah–Sing­er in­dex the­or­em. Edi­ted by R. S. Pal­ais. An­nals of Math­em­at­ics Stud­ies 57. Prin­ceton Uni­versity Press, 1965. Ap­pendix I. Atiyah’s sole con­tri­bu­tion to Sem­in­ar on the Atiyah–Sing­er in­dex the­or­em (1965). Re­pub­lished in Atiyah’s Col­lec­ted works, vol. 3. See also sim­il­arly-titled art­icle in Dif­fer­en­tial ana­lys­is (1965). incollection

M. F. Atiyah and R. Bott: “A Lef­schetz fixed point for­mula for el­lipt­ic dif­fer­en­tial op­er­at­ors,” Bull. Am. Math. Soc. 72 : 2 (1966), pp. 245–​250. MR 0190950 Zbl 0151.​31801 article

M. F. Atiyah and R. Bott: “A Lef­schetz fixed point for­mula for el­lipt­ic com­plexes, I,” Ann. Math. (2) 86 : 2 (1967), pp. 374–​407. MR 0212836 Zbl 0161.​43201 article

M. F. Atiyah: “A Lef­schetz fixed-point for­mula for el­lipt­ic dif­fer­en­tial op­er­at­ors,” pp. 38–​39 in Sim­posio in­ternazionale di geo­metria al­gebrica (Rome, 30 Septem­ber–5 Oc­to­ber 1965), published as Rend. Mat. Ap­pl. V : 25. Issue edi­ted by G. Castel­nuovo. Cre­monese (Rome), 1967. See also Bull. Amer. Math. Soc. 72:2 (1966). Zbl 0149.​41201 incollection

M. F. Atiyah: “Al­geb­ra­ic to­po­logy and el­lipt­ic op­er­at­ors,” Comm. Pure Ap­pl. Math. 20 (1967), pp. 237–​249. MR 0211418 Zbl 0145.​43804 article

M. F. Atiyah and G. B. Segal: “The in­dex of el­lipt­ic op­er­at­ors, II,” Us­pehi Mat. Nauk 23 : 6 (144) (1968), pp. 135–​149. Rus­si­an trans­la­tion of art­icle in Ann. Math. 87:3 (1968). MR 0236953 article

M. F. Atiyah and G. B. Segal: “The in­dex of el­lipt­ic op­er­at­ors, II,” Ann. Math. (2) 87 : 3 (1968), pp. 531–​545. Rus­si­an trans­la­tion pub­lished in Us­pehi Mat. Nauk 23:6(144) (1968). MR 0236951 Zbl 0164.​24201 article

M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, I,” Ann. Math. (2) 87 : 3 (May 1968), pp. 484–​530. A Rus­si­an trans­la­tion was pub­lished in Us­pehi Mat. Nauk 23:5(143). MR 236950 Zbl 0164.​24001 article

M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, III,” Ann. Math. (2) 87 : 3 (May 1968), pp. 546–​604. A Rus­si­an trans­la­tion was pub­lished in Us­pehi Mat. Nauk 24:1(145). MR 236952 Zbl 0164.​24301 article

M. F. Atiyah and R. Bott: “A Lef­schetz fixed point for­mula for el­lipt­ic com­plexes, II: Ap­plic­a­tions,” Ann. Math. (2) 88 : 3 (November 1968), pp. 451–​491. MR 0232406 Zbl 0167.​21703 article

M. F. At’ja and I. M. Zinger: “The in­dex of el­lipt­ic op­er­at­ors, I,” Us­pehi Mat. Nauk 23 : 5(143) (1968), pp. 99–​142. Rus­si­an trans­la­tion of an art­icle pub­lished in Ann. Math. (2) 87:3 (1968). MR 232402 article

M. F. Atiyah: “Glob­al as­pects of the the­ory of el­lipt­ic dif­fer­en­tial op­er­at­ors,” pp. 57–​64 in Pro­ceed­ings of the In­ter­na­tion­al Con­gress of Math­em­at­ics (Mo­scow, 16–26 Au­gust 1966). Mir (Mo­scow), 1968. MR 0233378 Zbl 0204.​41902 incollection

M. F. Atiyah: “Hy­per­bol­ic dif­fer­en­tial equa­tions and al­geb­ra­ic geo­metry (after Pet­rowsky),” pp. 87–​99 in Sémin­aire Bourbaki 1966/1967. W. A. Ben­jamin (New York and Am­s­ter­dam), 1968. Ex­posé no. 319. Re­pub­lished in 1995. Zbl 0201.​12501 incollection

M. F. Atiyah and I. M. Sing­er: “In­dex the­ory for skew-ad­joint Fred­holm op­er­at­ors,” Inst. Hautes Études Sci. Publ. Math. 37 (1969), pp. 5–​26. MR 285033 Zbl 0194.​55503 article

M. F. Atiyah: “The sig­na­ture of fibre-bundles,” pp. 73–​84 in Glob­al ana­lys­is: Pa­pers in hon­or of K. Kodaira. Edi­ted by S. Iy­anaga and D. C. Spen­cer. Prin­ceton Math­em­at­ic­al Series 29. Uni­versity of Tokyo Press, 1969. MR 0254864 Zbl 0193.​52302 incollection

M. F. At’ja and I. M. Zinger: “The in­dex of el­lipt­ic op­er­at­ors, III,” Us­pehi Mat. Nauk 24 : 1(145) (1969), pp. 127–​182. The Eng­lish ori­gin­al was pub­lished in Ann. Math. (2) 87:3 (1968). MR 256417 article

M. F. Atiyah: “To­po­logy of el­lipt­ic op­er­at­ors,” pp. 101–​119 in Glob­al ana­lys­is (Berke­ley, CA, 1–26 Ju­ly 1968). Edi­ted by S.-S. Chern and S. Smale. Pro­ceed­ings of Sym­po­sia in Pure Math­em­at­ics 16. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1970. MR 0264700 Zbl 0207.​22601 incollection

M. Atiyah and F. Hirzebruch: “Spin-man­i­folds and group ac­tions,” pp. 18–​28 in Es­says on to­po­logy and re­lated top­ics: Mé­m­oires dédiés à Georges de Rham. Edi­ted by A. Hae­fli­ger and R. Narasim­han. Spring­er (New York), 1970. MR 0278334 Zbl 0193.​52401 incollection

M. F. Atiyah: “Glob­al the­ory of el­lipt­ic op­er­at­ors,” pp. 21–​30 in Func­tion­al ana­lys­is and re­lated top­ics (Tokyo, 1969). Edi­ted by S. T. Kur­oda. Uni­versity of Tokyo Press, 1970. MR 0266247 Zbl 0193.​43601 incollection

M. F. Atiyah: “Res­ol­u­tion of sin­gu­lar­it­ies and di­vi­sion of dis­tri­bu­tions,” Comm. Pure Ap­pl. Math. 23 : 2 (1970), pp. 145–​150. MR 0256156 Zbl 0188.​19405 article

M. F. Atiyah, R. Bott, and L. Gård­ing: “La­cunas for hy­per­bol­ic dif­fer­en­tial op­er­at­ors with con­stant coef­fi­cients, I,” Acta Math. 124 : 1 (July 1970), pp. 109–​189. A Rus­si­an trans­la­tion was pub­lished in Usp. Mat. Nauk 26:2(158). MR 0470499 Zbl 0191.​11203 article

M. F. Atiyah: “Riemann sur­faces and spin struc­tures,” Ann. Sci. École Norm. Sup. (4) 4 : 1 (1971), pp. 47–​62. MR 0286136 Zbl 0212.​56402 article

M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, V,” Ann. Math. (2) 93 : 1 (January 1971), pp. 139–​149. A Rus­si­an trans­la­tion was pub­lished in Us­pehi Mat. Nauk 27:4(166) (1972). MR 279834 article

M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, IV,” Ann. Math. (2) 93 : 1 (January 1971), pp. 119–​138. A Rus­si­an trans­la­tion was pub­lished in Us­pehi Mat. Nauk 27:4(166) (1972). MR 279833 Zbl 0212.​28603 article

M. F. At’ja, R. Bott, and L. Gord­ing: “La­cunas for hy­per­bol­ic dif­fer­en­tial op­er­at­ors with con­stant coef­fi­cients, I,” Usp. Mat. Nauk 26 : 2(158) (1971), pp. 25–​100. Rus­si­an trans­la­tion of an art­icle in Acta Math. 124:1 (1970). MR 0606062 Zbl 0208.​13201 article

M. F. At’ja and I. M. Zinger: “The in­dex of el­lipt­ic op­er­at­ors, IV,” Us­pehi Mat. Nauk 27 : 4(166) (1972), pp. 161–​178. Rus­si­an trans­la­tion of an art­icle pub­lished in Ann. Math. (2) 93:1 (1971). MR 385933 Zbl 0237.​58017 article

M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, V,” Usp. Mat. Nauk 27 : 4(166) (1972). Rus­si­an trans­la­tion of an art­icle pub­lished in Ann. Math. (2) 93:1 (1971). Zbl 0237.​58018 article

M. F. Atiyah, V. K. Pat­odi, and I. M. Sing­er: “Spec­tral asym­metry and Rieman­ni­an geo­metry,” Bull. Lon­don Math. Soc. 5 (July 1973), pp. 229–​234. MR 331443 Zbl 0268.​58010 article

M. Atiyah, R. Bott, and V. K. Pat­odi: “On the heat equa­tion and the in­dex the­or­em,” Matem­atika, Mosk­va 17 : 6 (1973), pp. 3–​48. Rus­si­an trans­la­tion of an art­icle in In­vent. Math. 19:4 (1973). Zbl 0364.​58016 article

M. Atiyah, R. Bott, and V. K. Pat­odi: “On the heat equa­tion and the in­dex the­or­em,” In­vent. Math. 19 : 4 (1973), pp. 279–​330. Ded­ic­ated to Sir Wil­li­am Hodge on his 70th birth­day. Er­rata were pub­lished in In­vent. Math. 28:3 (1975). A Rus­si­an trans­la­tion was pub­lished in Matem­atika 17:6 (1973). MR 0650828 Zbl 0257.​58008 article

M. F. Atiyah, R. Bott, and L. Gård­ing: “La­cunas for hy­per­bol­ic dif­fer­en­tial op­er­at­ors with con­stant coef­fi­cients, II,” Acta Math. 131 : 1 (December 1973), pp. 145–​206. A Rus­si­an trans­la­tion was pub­lished in Usp. Mat. Nauk 39:3(237). MR 0470500 Zbl 0266.​35045 article

M. F. Atiyah: The in­dex of el­lipt­ic op­er­at­ors, 1973. Notes dis­trib­uted at sev­enty-ninth an­nu­al meet­ing of the Amer­ic­an Math­em­at­ic­al So­ci­ety, Dal­las, TX, 25–28 Janu­ary 1973. Re­pub­lished in Atiyah’s Col­lec­ted works, vol. 3, pp. 475–498. Re­pub­lished in Fields Medal­lists’ lec­tures (1997). Zbl 0263.​58012 misc

M. F. Atiyah: El­lipt­ic op­er­at­ors and com­pact groups. Lec­ture Notes in Math­em­at­ics 401. Spring­er (Ber­lin), 1974. MR 0482866 Zbl 0297.​58009 book

M. F. Atiyah, V. K. Pat­odi, and I. M. Sing­er: “Spec­tral asym­metry and Rieman­ni­an geo­metry, II,” Math. Proc. Cam­bridge Philos. Soc. 78 : 3 (November 1975), pp. 405–​432. MR 397798 Zbl 0314.​58016 article

M. F. Atiyah, V. K. Pat­odi, and I. M. Sing­er: “Spec­tral asym­metry and Rieman­ni­an geo­metry, I,” Math. Proc. Cam­bridge Philos. Soc. 77 (1975), pp. 43–​69. MR 397797 Zbl 0297.​58008 article

M. F. Atiyah: “Ei­gen­val­ues and Rieman­ni­an geo­metry,” pp. 5–​9 in Man­i­folds (Tokyo, 1973). Edi­ted by A. Hat­tori. Uni­versity of Tokyo Press, 1975. MR 0372928 Zbl 0319.​53030 incollection

M. F. Atiyah: “Clas­sic­al groups and clas­sic­al dif­fer­en­tial op­er­at­ors on man­i­folds,” pp. 5–​48 in Dif­fer­en­tial op­er­at­ors on man­i­folds (Var­enna, Italy, 24 Au­gust–2 Septem­ber 1975). Edi­ted by E. Vesentini. Cre­monese (Rome), 1975. MR 0650830 incollection

M. Atiyah, R. Bott, and V. K. Pat­odi: “Er­rata to: ‘On the heat equa­tion and the in­dex the­or­em’,” In­vent. Math. 28 : 3 (1975), pp. 277–​280. Er­rata for art­icle in In­vent. Math. 19:4 (1973). MR 0650829 Zbl 0301.​58018 article

M. F. Atiyah and E. Rees: “Vec­tor bundles on pro­ject­ive 3-space,” In­vent. Math. 35 : 1 (1976), pp. 131–​153. MR 0419852 article

M. F. Atiyah, V. K. Pat­odi, and I. M. Sing­er: “Spec­tral asym­metry and Rieman­ni­an geo­metry, III,” Math. Proc. Cam­bridge Philos. Soc. 79 : 1 (1976), pp. 71–​99. MR 397799 Zbl 0325.​58015 article

M. F. Atiyah: “El­lipt­ic op­er­at­ors, dis­crete groups and von Neu­mann al­geb­ras,” pp. 43–​72 in Col­loque “Ana­lyse et To­po­lo­gie” en l’hon­neur de Henri Cartan (Or­say, 1974). As­térisque 32–​33. So­ciété Math­ématique de France (Par­is), 1976. MR 0420729 Zbl 0323.​58015 incollection

M. Atiyah and W. Schmid: “A geo­met­ric con­struc­tion of the dis­crete series for semisimple Lie groups,” In­vent. Math. 42 : 1 (1977), pp. 1–​62. Re­pub­lished in Har­mon­ic ana­lys­is and rep­res­ent­a­tions of semisimple Lie groups (1980). Er­rata pub­lished in In­vent. Math. 54:2 (1979). MR 0463358 Zbl 0373.​22001 article

M. Atiyah and W. Schmid: “Er­rat­um: ‘A geo­met­ric con­struc­tion of the dis­crete series for semisimple Lie groups’,” In­vent. Math. 54 : 2 (1979), pp. 189–​192. Er­rat­um for art­icle in In­vent. Math. 42:1 (1977). See also Har­mon­ic ana­lys­is and rep­res­ent­a­tions of semisimple Lie groups (1980). MR 550183 article

M. Atiyah and W. Schmid: “A geo­met­ric con­struc­tion of the dis­crete series for semisimple Lie groups,” pp. 317–​378 in Har­mon­ic ana­lys­is and rep­res­ent­a­tions of semisimple Lie groups (Liège, 5–17 Septem­ber 1977). Edi­ted by J. A. Wolf, M. Ca­hen, and M. de Wilde. Math­em­at­ic­al Phys­ics and Ap­plied Math­em­at­ics 5. D. Re­idel (Dordrecht), 1980. Re­pub­lished from In­vent. Math. 42:1 (1977). Zbl 0466.​22012 incollection

M. Atiyah and W. Schmid: “Er­rat­um to the art­icle ‘A geo­met­ric con­struc­tion of the dis­crete series for semisimple Lie groups’” in Har­mon­ic ana­lys­is and rep­res­ent­a­tions of semisimple Lie groups (Liège, 5–17 Septem­ber 1977). Edi­ted by J. A. Wolf, M. Ca­hen, and M. de Wilde. Math­em­at­ic­al Phys­ics and Ap­plied Math­em­at­ics 5. D. Re­idel (Dordrecht), 1980. Er­rat­um for art­icle in same volume. Re­pub­lished from In­vent. Math. 54:2 (1979). Zbl 0466.​22013 incollection

M. F. Atiyah, H. Don­nelly, and I. M. Sing­er: “Geo­metry and ana­lys­is of Shim­izu \( L \)-func­tions,” Proc. Nat. Acad. Sci. U.S.A. 79 : 18 (September 1982), pp. 5751. MR 674920 Zbl 0503.​12007 article

M. F. Atiyah, H. Don­nelly, and I. M. Sing­er: “Eta in­vari­ants, sig­na­ture de­fects of cusps, and val­ues of \( L \)-func­tions,” Ann. Math. (2) 118 : 1 (July 1983), pp. 131–​177. An ad­dendum to this art­icle was pub­lished in Ann. Math. (2) 119:3 (1984). MR 707164 Zbl 0531.​58048 article

M. F. Atiyah, H. Don­nelly, and I. M. Sing­er: “Sig­na­ture de­fects of cusps and val­ues of \( L \)-func­tions: The non­split case,” Ann. Math. (2) 119 : 3 (1984), pp. 635–​637. Ad­dendum to an art­icle pub­lished in Ann. Math. (2) 118:1 (1983). MR 744866 Zbl 0577.​58030 article

M. F. At’ya, R. Bott, and L. Gård­ing: “La­cunas for hy­per­bol­ic dif­fer­en­tial op­er­at­ors with con­stant coef­fi­cients, II,” Usp. Mat. Nauk 39 : 3(237) (1984), pp. 171–​224. Rus­si­an trans­la­tion of an art­icle in Acta Math. 131:1 (1973). MR 747794 Zbl 0568.​35058 article

M. F. Atiyah: “Char­ac­ters of semi-simple Lie groups,” pp. 489–​558 in Col­lec­ted works, vol. 4: In­dex the­ory 2. Ox­ford Sci­ence Pub­lic­a­tions. Clar­en­don Press and Ox­ford Uni­versity Press (Ox­ford, New York), 1988. Lec­tures giv­en at Math­em­at­ic­al In­sti­tute, Ox­ford, 1976. incollection

M. Atiyah: Col­lec­ted works, vol. 4: In­dex the­ory 2. Ox­ford Sci­ence Pub­lic­a­tions. The Clar­en­don Press and Ox­ford Uni­versity Press (Ox­ford and New York), 1988. MR 951895 book

M. Atiyah: Col­lec­ted works, vol. 3: In­dex the­ory 1. Ox­ford Sci­ence Pub­lic­a­tions. The Clar­en­don Press and Ox­ford Uni­versity Press (Ox­ford and New York), 1988. MR 951894 Zbl 0724.​53001 book

M. F. Atiyah: “The in­dex of el­lipt­ic op­er­at­ors,” pp. 475–​498 in Col­lec­ted works, vol. 3: In­dex the­ory 1. Ox­ford Sci­ence Pub­lic­a­tions. Ox­ford Uni­versity Press, 1988. Col­loqui­um Lec­tures (Dal­las, 1973), Amer­ic­an Math­em­at­ic­al So­ci­ety.

V. Vassiliev: “The math­em­at­ic­al leg­acy of ‘La­cunas for hy­per­bol­ic dif­fer­en­tial equa­tions’,” pp. xxiii–​xxviii in Raoul Bott: Col­lec­ted pa­pers, vol. 2: Dif­fer­en­tial op­er­at­ors. Edi­ted by R. D. MacPh­er­son. Con­tem­por­ary Math­em­aticians. Birkhäuser (Bo­ston, MA), 1994. Com­ment­ary on two-part art­icle pub­lished in Acta Math. 124 (1970) and Acta Math. 131 (1973). MR 1290364 incollection

M. F. Atiyah: “Hy­per­bol­ic dif­fer­en­tial equa­tions and al­geb­ra­ic geo­metry (after Pet­rowsky),” pp. 87–​99 in Sémin­aire Bourbaki 10: An­nées 1966/67–1967/68. So­ciété Math­ématique de France (Par­is), 1995. Ex­posé no. 319. Re­pub­lic­a­tion of 1968 ori­gin­al. MR 1610456 incollection

M. F. Atiyah: “The in­dex of el­lipt­ic op­er­at­ors on com­pact man­i­folds,” pp. 159–​169 in Sémin­aire Bourbaki, vol. 8, An­nées 1962–63, 1963–64. As­térisque. Sémin­aire Bourbaki. So­ciété Math­ématique de France (Par­is), 1995. Ex­posé no. 253. Re­print of art­icle in Bull. Am. Math. Soc. 69 (1963). See also Sémin­aire Bourbaki. 15e an­née: 1962/63 (1964). MR 1611539 incollection

M. F. Atiyah: “The in­dex of el­lipt­ic op­er­at­ors,” pp. 115–​127 in Fields Medal­lists’ lec­tures. Edi­ted by M. F. Atiyah and D. Iag­ol­nitzer. World Sci­entif­ic Series in 20th Cen­tury Math­em­at­ics 5. World Sci­entif­ic (River Edge, NJ), 1997. Re­pub­lic­a­tion of notes dis­trib­uted at AMS con­fer­ence (1973). MR 1622942 incollection