Special lecture course: Kumar -- Verlinde dimension formula

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.

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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.