Academic proposal for AIS in Representation Theory
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1 Academic proposal for AIS in Representation Theory Name of the school, venue, and dates AIS in Representation Theory ISI Bangalore Centre June 2010 Names of organizers K. N. Raghavan N. S. Narasimha Sastry Matscience, Chennai ISI, Bangalore Names of resource persons Bhaskar Bagchi (ISI, Bangalore) Shripad Garge (IIT, Bombay) Upendra Kulkarni (ISI, Bangalore) Amit Kulshrestha (IISER, Mohali) K. N. Raghavan (Matscience, Chennai) N. S. Narasimha Sastry (ISI, Bangalore) Anupam Kumar Singh (IISER, Pune) Maneesh Thakur (ISI, Delhi) 1
2 A brief description of the school for publicity The representation theory of finite groups has been an area of intense research ever since its birth in the late nineteenth century. The theory is an important tool in several branches of mathematics, e.g., geometry, number theory, and combinatorics. Moreover, the unsolved mysteries and conjectures intrinsic to the theory, like the conjectures of Lusztig, Alperin, and Dade, are expected to keep it an active area of research. This AIS is aimed at those pursuing (or intending to pursue) research in any related area of mathematics (interpreted in a broad sense). The basic concepts and results of both the ordinary and the modular theory (particularly the latter) will be introduced. A basic knowledge of mathematics (especially of algebra) as taught in the foundational schools will be assumed but no more. Those working with or expecting to work with groups of any kind whatsoever (finite, Lie, algebraic, p- adic, quantum,... ) are encouraged to apply. Students with the pre-requisite background but still undecided about their area of research are also encouraged to apply. Participants apart, post-docs and young faculty working in the area will be invited to participate (to be in charge of tutorials or just to attend). 2
3 Syllabus Speaker No. of lectures Syllabus I (BB) 3 Linear representations of groups; modules over group rings; indecomposable and irreducible representations; Schur s lemma and complete reducibility; constructing new representations from given ones: tensor, symmetric and exterior products; duals. Induced representations; Frobenius reciprocity; Mackey s criterion for irreducibility. II (SG) 3 Radicals of modules and rings; Artin-Wedderburn structure theory of semi-simple rings and modules over them; applications to group rings of finite groups. III (BB) 3 Ordinary theory: characters; orthogonality and other properties of irreducible characters. Burnside s p a q b theorem. IV (UK) 2 Artin-Brauer characterization of a character; Clifford theory. V (AKS) 3 See (1) and (3) of note below VI (AK) 3 See (2) and (4) of note below. VII (NSNS) 3 p-adic numbers and p-adic integers; p-modular system for a group; lifting of idempotents, projective and free modules; VIII (KNR) 5 Cartan-Brauer triangle and its properties; Brauer characters and their orthogonality. IX (KNR) 5 Vertices and sources; trivial intersection, Green correspondence. X (NSNS) 7 Blocks and defect groups; Brauer correspondence; Brauer s theorems and Alperin s conjecture. Resource persons Bhaskar Bagchi (ISI, Bangalore) (BB), Shripad Garge (IIT, Bombay) (SG), Upendra Kulkarni (ISI, Bangalore) (UK), Amit Kulshrestha (IISER, Mohali) (AK), K. N. Raghavan (Matscience, Chennai) (KNR), N. S. Narasimha Sastry (ISI, Bangalore) (NSNS), Anupam Kumar Singh (IISER, Pune) (AS), Maneesh Thakur (ISI, Delhi) (MT) Note (1) Quick review of: Noetherian and Artinian rings and modules; existence of composition series of modules. Projective, injective, relatively projective and relatively injective modules Dornhoff 39 and 51 (2) I-adic topologies of rings and modules, completeness of modules relative to I-adic topologies Dornhoff 42, Theorems and See Feit Chap I 9, Theorems 9.3, 9.10, 9.11 (3) Local rings, unique decomposition theorem, Krull-Schmidt theorem. 3
4 Feit, Chap. I 10, 11 Dornhoff 43. (4) Finite extensions of complete local domains Dornhoff 49 Theorems 49.4, See also Feit, Chap I, 18 4
5 Timetable Day Date Lecture 1 Lecture 2 Tutorial 1 Tutorial to nd I V I V 2 3rd I V I V 3 4th I V I V 4 5th III II III II 5 7th III II III II 6 8th III II III II 7 9th III VI III VI 8 10th IV VI IV VI 9 11th IV VI II VI 10 12th VII VII (Tutorial) th VII VIII VII VIII 12 15th VII VIII VII VIII 13 16th X VIII X VIII 14 17th X VIII X VIII 15 18th X VIII X VIII 16 19th IX IX (Tutorial) st X IX X IX 18 22nd X IX X IX 19 23rd X IX X IX 20 24th X IX X IX The Unity of Mathematics lectures will be scheduled during on some three or four days of the School. 5
6 Contact details of organizers K. N. Raghavan N. S. Narasimha Sastry Institute of Mathematical Sciences Division of Theoretical Statistics and Mathematics C. I. T. Campus Indian Statistical Institute Taramani 8th Mile, Mysore Road, R. V. College Post Chennai Bangalore (44) (Off) (80) (Off) (44) (Res) (80) (Res) (Mobile) 6
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