Prof. Abhyankar Birthday Symposium (2023)

Bhaskaracharya Pratishthana, Pune has organized

‘Abhyankar Birthday Symposium’

on the occasion of birthday of Prof. Shreeram S. Abhyankar (1930-2012).

The symposium will be held

on Saturday, 22nd July 2023 at Bhaskaracharya Pratishthana, Pune.

Venue: Audio-visual Hall, Bhaskaracharya Pratishthana Pune.

Online Mode: https://us06web.zoom.us/j/83696584949?pwd=WjFMWmlEd1JoNmpmd3k3MngydGN6UT09
 

Meeting ID: 836 9658 4949 Passcode: 496555

Find your local number: https://us06web.zoom.us/u/kbS915ieMc

Speakers and Topics

Title: Exact integrals via symplectic geometry

Abstract:
I will give an account of the formalism of Duistermaat-Heckman and its application in a particular case, which in turn motivates an elegant way of computing integrals over a polytope attached to any trivalent graph.

Here is the context, which I will probably only briefly touch upon: Given a genus-$g$ Riemann surface $ \Sigma $, the moduli space of rank two vector bundles with trivial determinant is, by the Narasimhan-Seshadri Theorem, in bijection with the space of (equivalence classes of) representations in $ SU(2) $ of the fundamental group of the surface.

In the latter avatar, the space $ \cM_g $ has a symplectic structure and a corresponding finite measure, the Liouville measure. Normalised to total mass one this gives a probability measure. There is a natural class of real-valued functions $ W_C: \cM_g \to [0,1] $, parameterised by isotopy classes of loops, $ C \subset \Sigma $. These are called Wilson loop functions by physicists and Goldman functions by mathematicians. With applications in mind, I present a simple scheme to compute joint distributions of these functions for families of loops. This is possible because of the miracle of symplectic geometry called the Duistermaat-Heckman formalism (whose applicability in this context is due to L. Jeffrey and J. Weitsman) and a continuous analogue of the Verlinde algebra. The Verlinde algebra is well-known to physicists, representation theorists and algebraic geometers in the context of "generalised theta functions".
This is based on the preprint: https://arxiv.org/abs/2206.07455

Chair: Prof. S. M. Bhatwadekar ,Bhaskaracharya Pratishthana, Pune, India.

Title: Topological and Group-theoretic Background of "Historical Ramblings"

Abstract: 
Abhyankar wrote his now-famous paper "Historical Ramblings in Algebraic Geometry and Related Algebra" in American Mathematical Monthly in 1976. I shall first comment on this paper and try to provide a topological and group-theoretic background to understand this paper. At the end, I shall try to give a proof of two specific results, one on graphs, and the other on surfaces.

Chair: Prof. Vinayak Joshi, Savitribai Phule Pune University, Pune, India.

Title: A conjecture on D algebras
Abstract: The goal of this talk is to int roduce D algebras, formulate a conjecture about these algebras, and prove this conjecture in several cases.

 

Chair: Prof. Neena Gupta, Indian Statistical Institute Kolkata, India. (Online)