Curriculum Vitæ. David John Benson. 17 March 1993

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1 Curriculum Vitæ David John Benson 17 March 1993 Work address: Mathematical Institute, St. Giles, Oxford OX1 3LB, England. Tel: Home address: 14 Aldrich Rd., Oxford OX2 7ST, England. Tel: Date and Place of Birth: 3rd December 1955, Farnborough, Hampshire, England. Education and Employment Dates Place Degree/Position 1986 University of Oxford University Lecturer and fellow of Wolfson College Northwestern University, Evanston, IL, USA Yale University, New Haven, CT, USA Assistant Professor (NSF grant held) Gibbs Instructor (NSF grant held) Århus Universitet, Danmark Royal Society exchange fellow Trinity College, Cambridge Ph. D. in Mathematics Trinity College, Cambridge Part III Mathematics, honours with distinction Trinity College, Cambridge First class honours in Mathematics tripos Entrance Scholar Senior Scholar Yeats Essay Prize Maidstone Grammar School 8 O-levels, 3 A/S-levels (Mathematics A1, Physics A, Chemistry A1) Sabbatical leave: September to December 1990, invited visitor for the program on Representations of Finite Groups, MSRI, Berkeley, California, USA. September to December 1991, School of Mathematics, University of Minnesota, Minneapolis, USA. 1

2 Ph. D. Thesis Supervisor: Prof.J.G.Thompson. Subject: Finite group theory. Title: The Simple Group J 4. Date Received: 12th June, Current Research Interests Cohomology of groups, modular representation theory, algebraic topology, invariant theory for finite groups. Publications [1] The Simple Group J 4. Ph. D. Thesis, Cambridge [2] The Loewy structure of the projective indecomposable modules for A 8 in characteristic 2. Comm. in Algebra 11 (13) (1983), MR84k:20006a, ZB [3] The Loewy structure of the projective indecomposable modules for A 9 in characteristic 2. Comm. in Algebra 11 (13) (1983), MR84k:20006b, ZB [4] (with R. A. Parker.) The Green ring of a finite group. J. Algebra 87 (1984), MR86a:16013, ZB [5] Lambda and psi operations on Green rings. J. Algebra 87 (1984), MR86g:16013, ZB [6] Modular representation theory: New trends and methods. Springer Lecture Notes in Mathematics 1081, Springer-Verlag, Berlin/New York (231 pp.) MR86g:20013, ZB [7] (with J. H. Conway.) Diagrams for modular lattices. J. Pure & Appl. Algebra 37 (1985), MR87b:06018, ZB [8] Brauer trees for 12M 22. J. Algebra 95 (1985), MR87d:20030, ZB [9] Some recent trends in modular representation theory. Proceedings of the group theory year at Rutgers, C.U.P. 1985, MR86j:20003, ZB

3 [10] (with W. Feit and R. Howe.) Finite linear groups, the Commodore 64, Euler and Sylvester. Amer. Math. Monthly 93 (9) (1986), MR87m:05011, ZB [11] Modules for finite groups: representation rings, quivers and varieties. (Three lectures given at the Fourth International Conference on the Representations of Algebras) Representation Theory II, Groups and Orders, Ottawa Springer Lecture Notes in Mathematics 1178, Springer-Verlag, Berlin/New York MR87m:20031 [12] (with J. F. Carlson.) Nilpotent elements in the Green ring. J. Algebra 104 (1986), MR88b:20021, ZB [13] Representation rings of finite groups. (Two lectures given at the 1985 Durham Symposium on the Representations of Algebras) Representations of Algebras, Durham London Math. Soc. Lecture Note Series 116, C.U.P., MR88i:16037, ZB [14] (with J. F. Carlson.) Diagrammatic methods for modular representations and cohomology. Comm. in Algebra 15 (1&2) (1987), MR87m:20032, ZB [15] (with N. Habegger.) Varieties for modules and a problem of Steenrod. J. Pure & Appl. Algebra 44 (1987), MR88e:55020, ZB [16] (with J. F. Carlson.) Complexity and multiple complexes. Math. Zeit. 195 (1987), MR88e:20050, ZB [17] Some remarks on the decomposition numbers for the symmetric groups. The Arcata Conference on Representations of Finite Groups, Proc. Symp. Pure Math. Amer. Math. Soc. 47, Part 1 (1987), MR89b:20037, ZB [18] Spin modules for symmetric groups. J. London Math. Soc. (2) 38 (1988), MR89k:20020, ZB [19] Projective modules for the group of the twenty-seven lines on a cubic surface. Comm. in Algebra 17 (5) (1989), MR90d:20022, ZB

4 [20] Cyclic homology and path algebra resolutions. Math. Proc. Camb. Phil. Soc. 106 (1989), MR90f:18010, ZB [21] (with J. F. Carlson and G. R. Robinson.) On the vanishing of group cohomology. J. Algebra 131 (1990), MR91c:20073, ZB [22] (with L. Evens.) Isomorphisms in finite group cohomology. Comm. in Algebra 18 (1990), MR91h:20080 [23] Representations and Cohomology, Vol. I: Basic representation theory of finite groups and associative algebras. (xi pp.) Cambridge Studies in Advanced Mathematics 30, Cambridge University Press, MR92m:20005, ZB [24] Representations and Cohomology, Vol. II: Cohomology of groups and modules. (x pp.) Cambridge Studies in Advanced Mathematics 31, Cambridge University Press, [25] (with S. Martin.) Mod 2 cohomology of the unitary groups U 3 (2 n ). Comm. in Algebra 19 (1991), MR93a:20072 [26] (with V. Franjou.) Séries de composition de modules instables et injectivité dela cohomologie du groupe Z/2. Math. Zeitschrift 208 (1991), MR93b:55022, ZB [27] (with F. R. Cohen.) Mapping class groups of low genus and their cohomology. (iv pp.) Memoirs of the Amer. Math. Soc. 443, MR91g:57002, ZB732:57003 [28] (with M. Feshbach.) Stable splittings of classifying spaces of finite groups. Topology 31 (1992), [29] Resolutions and Poincaré duality for finite groups. In: Proceedings of the June 1990 Barcelona conference on Homotopy and Group Cohomology. Springer Lecture Notes in Mathematics 1509, Springer-Verlag, Berlin/New York [30] (with J. F. Carlson.) Projective resolutions and Poincaré duality complexes. To appear in Trans. Amer. Math. Soc. [31] (with J. F. Carlson.) The cohomology of extraspecial groups. Bull. London Math. Soc. 24 (1992), MR93b:

5 [32] (with J. F. Carlson.) Periodic modules with large period. Quarterly Journal of Mathematics 43 (1992), [33] (with J. F. Carlson.) Products in negative cohomology. J. Pure & Appl. Algebra 82 (1992), [34] The image of the transfer. To appear in Archiv der Mathematik, [35] (with M. Feshbach.) On the cohomology of split extensions. To appear in Proc. Amer. Math. Soc., [36] (with J. Greenlees.) The action of the Steenrod algebra on Tate cohomology. To appear in J. Pure & Appl. Algebra. [37] Poincaré series for the cohomology of compact Lie groups. Preprint, [38] (with J. F. Carlson.) Functional equations for Poincaré series in group cohomology. Submitted to the Journal of the London Math. Soc., [39] Polynomial Invariants of Finite Groups. (v pp.) To appear, L.M.S. Lecture Notes in Mathematics, C.U.P., [40] (with W. W. Crawley Boevey.) A ramification formula for Poincaré series, and a hyperplane formula for modular invariants. Preprint, [41] (with J. Wood.) Integral invariants and cohomology of BSpin(n). In preparation. [42] (with J. F. Carlson.) Cohomology of the double cover of the Mathieu group M 12. In preparation. [43] Simplicial and Cosimplicial Homotopy Theory. Printed lecture notes, Oxford, Book reviews [44] C. W. Curtis and I. Reiner. Methods of Representation Theory, with Applications to Finite Groups and Orders, Volume II. (Wiley & Sons Ltd., 1987) Bull. London Math. Soc. 20 (1988), Graduate Students John Fitzgerald June 87 Oct 90 Title of thesis: Weyl modules for groups of type B 2 and G 2. Laurence Barker Oct 87 Jun 89, Apr 90 Dec 91 Title of thesis: Blocks of endomorphism algebras of modules. Paul Day Oct 88 5

6 Subject: The May spectral sequence for group algebras. Jon Clark Oct 90 Subject: Cohomology of nilpotent groups of class two. Dan Brown Jun 91 Subject: Relative cohomology of finite groups. Katrina Hicks Apr 92 Subject: Representation theory. Teaching Experience Cambridge University (undergraduate supervisions, ) Group theory Representation theory Algebraic geometry Galois theory Number theory First year algebra and analysis Second year complex variables Yale University : (undergraduate) Calculus 112 (Differentiation) Calculus 115 (Integration) (graduate) Modular representation theory : (undergraduate) Early Concentration Mathematics for Freshmen (Topics included: ordinary differential equations, Fourier series and Fourier transforms, and some analytic number theory) (graduate) Ordinary representations of finite Chevalley groups Northwestern University : (undergraduate) B20 Calculus of several variables A11 Finite mathematics (graduate) E11 1 Integral and modular representation theory 6

7 : (undergraduate) C35 Number theory (graduate) D46 1 Algebraic topology: Homotopy theory D46 3 Topological K-theory University of Oxford : (graduate) Representation theory and cohomology of groups I, II : (3rd year undergraduate) Representation theory of finite groups (graduate) Representations of algebras : (graduate) Representation theory and cohomology of finite groups I, II, III : (graduate) Block theory Introduction to algebraic geometry : (graduate) Commutative algebra I: Intersection theory Commutative algebra II: Invariant theory : (3rd year undergraduate) Representation theory of finite groups (graduate) Classifying spaces of finite groups (seminar) Cohomology of groups seminar (2 terms) : (graduate) Simplicial and cosimplicial homotopy theory (2 terms) Invited lectures The simple group J 4. Conference, Oberwolfach, West Germany: Gruppentheorie, 27 April 3 May The Green ring of a finite group. Conference, Paris VII, France: Théorie des blocs, 7

8 June Diagrams for modular lattices. Conference, Oberwolfach, West Germany: Darstellungstheorie endlicher Gruppen, July, Modules for finite groups. Three lectures. Conference, Troisième Cycle Romand de Mathématique, Les Plans sur Bex, Switzerland: Représentations des groupes finis, March Modules for finite groups: representation rings, quivers and varieties. Three lectures. Fourth International Conference on the Representations of Algebras, Ottawa, Canada, August, Nilpotent elements in representation rings. Algebra seminar, University of Illinois at Urbana-Champagne, U.S.A., December Representation rings of finite groups. Conference, Northwestern University, Illinois, U.S.A: Cohomology of groups. March Representation rings of finite groups. Two lectures. Durham (U.K.) symposium: Representations of Algebras, July, Semidihedral groups and Bott periodicity. Conference, Oberwolfach, West Germany: Topologische Methoden in der Gruppentheorie, 1 7 June, Representation rings. Conference, Arcata, California, U.S.A: Representations of finite groups and related topics July, Spin modules for symmetric groups. Conference, Arcata, California, U.S.A: Representations of finite groups and related topics July, Specht modules and the cohomology of mapping class groups. Conference, Oberwolfach, West Germany: Darstellungstheorie endlicher Gruppen, 31 May 6 June Cyclic homology and block theory. Representation theory seminar, Manchester, U.K., March Astronomical clocks, continued fractions and the back wheels of a car. Invariant Society, Oxford, U.K., 24 May, Cyclic homology of path algebras. Conference, Marseille-Luminy, France: Homotopy Theory, July On the vanishing of group cohomology. Topology seminar, University of Kentucky, Lexington, U.S.A., September Three lectures: Specht modules and the cohomology of mapping class groups, On the vanishing of group cohomology, and How to build a sporadic group. University of Minneapolis, Minnesota, U.S.A., September Group theory seminar: The Evens norm map and Steenrod operations in group cohomology, and K-theory seminar (Karoubi): Stable splittings of classifying spaces of finite groups. Paris, France, March

9 The Evens norm map and Steenrod operations in group cohomology. Representation theory seminar, Universität Essen, West Germany, April Two lectures: Poincaré duality and cohomology of finite groups, and Stable splittings of classifying spaces of finite groups. Université degenève, Switzerland, April Projective resolutions and complexes. Conference, Arcata, California, U.S.A: Representations of finite and related algebraic groups July, Projective resolutions of modules for finite groups. Århus Universitet, October 1989 and Københaven Universitet, Denmark, October The classifying space of a finite group. Colloquium, Birmingham University, U.K., November Projective resolutions of modules for finite groups. ETH Zürich, Switzerland, January Representations and cohomology of finite groups. British Mathematics Colloquium, Norwich, U.K., April Resolutions and Poincaré duality for finite groups. Barcelona conference on homotopy theory, Spain, 6 12 June Mapping class groups of surfaces of low genus and their cohomology. University of Virginia at Charlottesville, U.S.A., joint Algebra/Topology seminar, 11 September Colloquium: Reflection groups, braids and mapping class groups, and seminar: Projective resolutions of modules for finite groups. University of Illinois at Urbana-Champaign, Illinois, U.S.A., Oct On the Becker Gottlieb transfer. Topology seminar, University of Minnesota, Minneapolis, U.S.A., 24 Oct Projective resolutions of modules for finite groups. Colloquium, University of Rochester, New York, U.S.A., 31 Oct Extraspecial groups and modules with large period. Northeast Topology Meeting, Albany, New York, U.S.A., 2 Nov Reflection groups, braids and mapping class groups. Seminar, University of Georgia, Athens GA, U.S.A., 8 Nov Two lectures: Projective resolutions of modules for finite groups, and Extraspecial groups and modules with large period. Conference on representations of finite groups, MSRI Berkeley, California, U.S.A., Dec Reflection groups, braids and mapping class groups. Colloquium, Liverpool University, U.K., 22 Feb Why are most products zero in negative degree cohomology? Algebra seminar, Cambridge University, U.K., 4 March Products in negative cohomology. Representation theory meeting, Warwick University, 9

10 U.K., 6 June The image of the transfer. Seminar, Universität Göttingen, Germany, 7 August Why are most products zero in negative degree cohomology? Conference on classifying spaces of compact Lie groups, Göttingen, 12 Sept The image of the transfer. Group theory seminar, University of Chicago, U.S.A., 24 Oct Why are most products zero in negative degree cohomology? Midwest Group Theory Meeting, University of Chicago, U.S.A., 26 Oct The transfer map in group cohomology. Colloquium, University of Virginia at Charlottesville, U.S.A., 12 Dec The image of the transfer. Algebra Seminar, Yale University, New Haven CT, U.S.A., 1 April Why are most products zero in negative Tate cohomology? Algebra Seminar, University of Virginia at Charlottesville, U.S.A., 14 April Colloquium, University of Virginia at Char- Polynomial invariants of finite groups. lottesville, U.S.A., 16 April The index of a module for a finite group. Invited talk at the summer meeting of the Canadian Math. Society, Toronto, Canada, 12 June Projective resolutions and Poincaré duality for finite and compact Lie groups. Seminar, University of Rochester, New York, U.S.A., 17 June Poincaré duality in the cohomology of the classifying spaces of finite and compact Lie groups. Invited lecture at the Amer. Math. Soc Summer Research Conference on Cohomology, Representations and Actions of Finite Groups, South Hadley, Massachusetts, U.S.A., 25 June Four lectures on the cohomology of finite groups, University of Virginia at Charlottesville, U.S.A., September The transfer map in group cohomology. Colloquium, Bristol University, U.K., 8 December On the cohomology of finite groups. Colloquium, Sheffield University, U.K., 24 February Musical Activities Playing the flute since 1975 Singing (bass) since 1987 Playing the piano since

11 Choral Concerts Brahms: A German Requiem. Oxford Harmonic Society, Sun 26 Nov 1989, Oxford Town Hall, with the Oxford Symphony Orchestra, conducted by Brian Trowell. J. S. Bach: Mass in B minor. Oxford Harmonic Society, Sat 17 March 1990, Sheldonian Theatre, Oxford, with the Guildhall Chamber Orchestra, conducted by Brian Trowell. Beethoven: Choral Symphony. Oxford Harmonic Society, Thu 3 June 1990, Deddington Church, Oxfordshire, with the Oxford Symphony Orchestra, conducted by Brian Trowell. Gilbert and Sullivan: Trial by Jury, and Elgar part songs: Deep in my soul (Op. 53 No. 2), Owls (Op. 53 No. 4), There is sweet music (Op. 53 No. 1) and O wild west wind (Op. 53 No. 3). Oxford Harmonic Society, Wed 23 June 1990, Holywell music room, Oxford, conducted by Brian Trowell. J. Haydn: St. Nicholas Mass, and Bruckner: two motets, Ave Maria and Ecce sacerdos. Worcester/Somerville Choral Society, Oxford, Sat. 2 Mar 1991, St. Aloysius church, Oxford. E. Elgar: The Dream of Gerontius. Oxford Harmonic Society, Sat. 23 Mar 1991, Sheldonian Theatre, Oxford, with the Oxford Symphony Orchestra, conducted by Robert Secret. Charpentier: Te Deum. Oxford Classical Chorus, Sat 8 June 1991, Keble College Chapel, Oxford, with the Oxford Classical Orchestra, directed by Neil Farrow and Hilary Norris. Berlioz: Requiem. Oxford Harmonic Society, Wed 12 June 1991, Sheldonian Theatre, Oxford, with the Oxford Symphony Orchestra, conducted by Robert Secret. Mozart: Mass in B flat K. 427, Regina Cœli K. 127, Litaniæ K University of Minnesota Chamber Singers, director Thomas Lancaster, with the Concert Choir and Chamber Orchestra, conducted by Murray Sidlin. Mon Nov. 18, 1991, Calvary Lutheran Church, Golden Valley, Minnesota; Tue Nov. 19, College of St. Benedict, St. Joseph, Minnesota (this performance was recorded and played on National Public Radio on Sun Dec. 8); Tue Dec. 3, Northrup Auditorium, University of Minnesota. Vivaldi: Gloria, Doyle: Non Nobis Domine, Balfour Gardiner: Evening Hymn, Tchaikovsky: The Crown of Roses. Kodály Choir, Sat 28 Nov. 1992, Merton College Chapel, Oxford, conducted by Anna Smith. Mendelssohn: Elijah. Oxford Harmonic Society, Sun 14 March 1993, Oxford Town Hall, with the Oxford Symphony Orchestra, conducted by Robert Secret. Operatic Choral Rôles Mozart: Idomeneo. Oxford University Opera Club, 1 4 May 1991 (four fully staged performances with period instruments), Playhouse, Oxford, conducted by Jonathan Williams and produced by Alison Brown. 11

12 Tchaikovsky: Eugene Onegin. Oxford University Opera Club, 6 9 May 1992 (four fully staged performances), Playhouse, Oxford, conducted by Duncan Hinnells and produced by Rosie Johnston. 12