A Symposium in Mathematics 2017


Overview Bhaskaracharya Pratishthana is organising a Symposium in Mathematics for Mathematics students and teachers on the occasion of the Birth Day of Founder Director Professor Shreeram S. Abhyankar, a world famous Mathematician. Prof. Abhyankar was born on 22nd July 1930 in Ujjain.
Venue Bhaskaracharya Pratishthana
Date Saturday, 22nd July 2017


Sr. Name Affiliation
1 Prof. Balwant Singh Centre for Basic Sciences, Mumbai
2 Prof. R. V. Gurjar IIT Bombay
3 Prof. S. M. Bhatwadekar Bhaskaracharya Pratishthana


Sr. Time Speaker Title ABSTRACT:

03:00 PM
04:00 PM

Prof. Balwant Singh The Jacobian Conjecture

The Jacoboian Conjecture in dimension two states that if f = f (X, Y ) and g = g(X, Y ) are polynomials in two variables X and Y with real coefficients (or cofficients in any field k of characteristic zero) such that their Jacobian

J(f, g) = (∂f /∂X)(∂g/∂Y ) − (∂f /∂Y )(∂g/∂X)

is a nonzero constant then X and Y can be expressed as polynomials in f and g. In terms of rings, the conclusion can be stated as the equality k[f, g] = k[X, Y ].

There is a similar conjecture in dimension n.

The conjecture was formulated in 1939, and it is still open even for n = 2. The efforts of several mathematicians over the past decades have yielded only partial results.

We shall describe some of these partial results.

2 04:20 PM
05:20 PM
Prof. R. V. Gurjar Singularities In this lecture we will discuss how singular points occur naturally when dealing with zeros of polynomials. We will look at some powerful tools developed by mathematicians to study them. Several beautiful results proved using these methods will be mentioned.
3 05.35 PM
06.35 PM
Prof. S. M. Bhatwadekar Epimorphism problems in Affine Geometry

One of most significant contributions of (late) Professor Shriram Abhyankar is the following famous Abhyankar-Moh/Suzuki Epimorphism Theorem:

Theorem. Let k be a field of characteristic zero. Let F ∈ k[X, Y ] be such that .k[X, Y ]/(F ) ≈ k[T ]. Then F is a variable in k[X, Y ] i.e. ∃ G ∈ k[X, Y ] such that k[X, Y ] = k[F, G].

This result led to many important questions in affine geometry. To cite one (known as Abhyankar-Sathaye Epimorhism Problem):
Question. Let k be an algebraically closed field of characteristic zero. Let I be an ideal in the polynomial ring k[X1 , X2 , · · · , Xn] such that k[X1 , X2 , · · · , Xn]/I is a polynomial ring in m variables over k. Is it true that there exist F1 , · · · , Fn−m ∈ I and G1, · · · , Gm ∈ k[X1 , X2 , · · · , Xn ] such that

k[X1 , X2 , · · · , Xn ] = k[F1, · · · , Fn−m, G1 , · · · , Gm]?

In my talk, I will address this (and some other related questions) and report on the progress made so far.


1 Please click here for the registration form
2 List of selected participants will be announced later