Dr Amit Kulshreshta
Dr Amit Kulshreshta is currently an Associate Professor and Head in the Department of Mathematical Sciences at Indian Institute of Science Education and Research Mohali. Prior to that, he was a visiting scientist at Indian Statistical Institute Bangalore/Delhi, and a postdoctoral researcher at Universite catholique de Louvain, Belgium. He did his PhD from Tata Institute of Fundamental Research, Mumbai in the broad area of algebra.
His research interests include central simple algebras, quadratic forms, linear algebraic groups, word maps on groups.
Title of the talk: Extracting Roots in Groups and Matrices
Abstract: Let $G$ be a group and $a \in G$. Solutions of the equation $X^n = a$ are called n-th roots of $a$. In this talk, the groups of our interest are matrix groups. More specifically, in the case of $SL_2(F)$, the special linear group of $2 \times 2$ matrices over a field $F$, we shall see how the problem of extracting roots can be connected to solutions of equations involving Fibonacci-like polynomials. As an interesting consequence of our computations, we shall derive that if $p$ is a prime and $F$ is a finite field of $q$ (odd) elements then every element in $SL_2(F)$ is a product of two p-th powers, except for the case $p = q = 3$. This is a joint work with Anupam Singh.