A Symposium in Mathematics On the Birthday of Late Professor Shreeram S. Abhyankar

The link for the lectures is:  https://meet.google.com/wzi-bhee-zfu

Please join by 3.15 pm on Thursday, 22nd July 2021.

Time

Speaker

Topic and Abstract

3.30 pm

to 4.30 pm

IST

Prof. Steven Spallone

IISER, Pune

 Title: A Chinese Remainder Theorem for Partitions.

Abstract: Given a natural number $t$ and a partition $\lambda$, there is a notion of a "remainder of $\lambda$ upon division by $t$”, called the $t$-core of $\lambda$. It results from successively removing $t$-hooks from the diagram of $\lambda$. The theory of $t$-hooks is widely used in symmetric group representation theory and in studying the partition function. I will present an analogue of the Chinese Remainder Theorem in this context, jointly discovered with IISER Pune BS-MS alumna K. Seethalakshmi.

 

 

4.45 pm

to 5.45 pm

IST

Prof. Steven D. Cutcosky

University of Missouri

Title: Towards resolution of singularities in positive characteristic.

 

Abstract:  Most of what is known about resolution of singularities in positive characteristic is due to Professor Abhyankar, who proved the existence of a resolution of singularities   in dimension less than or equal to three in the 1950s and 1960s. Resolution of singularities in higher dimensions is still open.

 

We will begin by discussing what a singularity is, leading to Zariski's definitive definition.

We will then discuss the techniques of Zariski for resolution in characteristic zero, explain  how his algorithm will fail in positive characteristic if there is ``defect'', and discuss Abhyankar's methods which allowed him to get around this for surfaces and threefolds. We will also discuss why it is difficult to extend these results to higher dimensions.

 

6 pm

to 7 pm

IST

Prof. Avinash Sathaye

University of Kentucky

 

Title: Revisiting the Jacobian Problem

 

Abstract: Given polynomials $f,g \in k[x,y]$ where $k$ is an algebraically closed field of characteristic $0$.

The classical formulation of the problem is:

 

We want to show that if $J = J_{x,y}(f,g)$ is a nonzero constant, where $\deg(f)=n$ and $\deg(g) =m$,

then one of $n,m$ divides the other.

 

We try to investigate the problem:

 

Given $f,g$ of degrees $n<m$, what are the possible degrees of $J$, when we assume that $f$ is monic of degree $n$ in $y$.

 

To facilitate the discussion we use the $f$-adic expansion $g = A_0f^r+ \cdots + A_{r-1}f + A_r$ with the extra assumption that $r>1$ and $A_i$ are polynomials in $f$ of degree less than $n$. We study the sequence of jacobians of $A_i$ with $f$ for a fixed $f$ and any such $g$. Conditions on the sequence of the jacobians seem to give help to show that the last jacobian and hence $J$  itself cannot be a nonzero constant, i.e. the minimum degree is bigger than $0$.

 

I will outline the process and the arguments.