Type: Seminar

Distribution: World

Expiry: 29 Oct 2015

Calendar2: 6 Aug 2015 1400-1600

CalLoc2: Carslaw 830

Calendar3: 13 Aug 2015 1400-1600

CalLoc3: Carslaw 830

Calendar4: 20 Aug 2015 1400-1600

CalLoc4: Carslaw 830

Calendar5: 27 Aug 2015 1400-1600

CalLoc5: Carslaw 830

Calendar6: 3 Sep 2015 1400-1600

CalLoc6: Carslaw 830

Calendar7: 10 Sep 2015 1400-1600

CalLoc7: Carslaw 830

Calendar8: 17 Sep 2015 1400-1600

CalLoc8: Carslaw 830

Calendar9: 24 Sep 2015 1400-1600

CalLoc9: Carslaw 830

Calendar10: 8 Oct 2015 1400-1600

CalLoc10: Carslaw 830

Calendar11: 15 Oct 2015 1400-1600

CalLoc11: Carslaw 830

Calendar12: 22 Oct 2015 1400-1600

CalLoc12: Carslaw 830

Calendar13: 29 Oct 2015 1400-1600

CalLoc13: Carslaw 830

Auth: anthonyh@como.maths.usyd.edu.au

Professor Shrawan Kumar from the University of North Carolina, an expert in a wide range of areas in geometry and representation theory, will be visiting the School in Semester 2 as part of his sabbatical. He is going to give a lecture course on the proof of the Verlinde dimension formula, in which he will explain the relevant ingredients from algebraic geometry, representation theory, topology and mathematical physics. The course should be suitable for staff, PG students and possibly ambitious Honours students. He will give 2 hours of lectures each week of the semester, on Thursdays from 2pm in Carslaw 830. Details below. ----------------------------------------------------------------------------- Time: Thursday 2-4pm, each teaching week of Semester 2 Place: Carslaw 830 Lecturer: Shrawan Kumar (University of North Carolina) Title: Verlinde Dimension Formula for the Space of Conformal Blocks and the Moduli of G-bundles Outline: Classical theta functions can be interpreted in geometric terms as global sections of a certain determinant line bundle on the moduli space of line bundles of degree g-1 on a smooth projective curve C of genus g. This notion has a natural generalization where one replaces line bundles on C by G-bundles on C, for a simply-connected complex semisimple algebraic group G, giving rise to the space of generalized theta functions. This space is also identified with the space of conformal blocks arising in Conformal Field Theory, which is by definition the space of coinvariants in integrable highest weight modules of affine Kac-Moody Lie algebras. E. Verlinde conjectured a remarkable formula to calculate the dimension of the space of generalized theta functions. Various works, notably by Tsuchiya-Ueno-Yamada, Kumar-Narasimhan-Ramanathan, Faltings, Beauville-Laszlo, Sorger and Teleman culminated into a proof of the Verlinde formula. The main aim of this course will be to give a complete and self-contained proof of this formula derived from the Propagation of Vacua and the Factorization Theorem among others. The proof requires techniques from algebraic geometry, geometric invariant theory, representation theory of affine Kac-Moody Lie algebras, topology, and Lie algebra cohomology. Some basic knowledge of algebraic geometry and representation theory of semisimple Lie algebras will be helpful; but not required. I will develop the course from scratch recalling results from different areas as we need them. There is no text book available which is suitable for our course. We will mainly rely on my own notes on the subject. This course should be suitable for graduate students, postdocs and faculty members interested in the interaction between algebraic geometry, representation theory, topology and mathematical physics.