Shiv Prakash Patel (IIT, Delhi)

Title:  Multiplicity one theorems in representation theory

Speaker: Shiv Prakash Patel (IIT, Delhi)

Date: Tuesday, February 25, 2020

Time: 4:00 pm 

Venue: Seminar Room, School of Physical Sciences (SPS)

ABSTRACT

A representation $\pi$ of $G$ is called multiplicity free if the dimension of the vector space $Hom_{G}(\pi, \sigma)$ is at most 1 for all irreducible representation $\sigma$ of $G$. Let $H$ be a subgroup of $G$ and $\psi$ an irreducible representation of $H$. The triple $(G,H, \psi)$ is called Gelfand triple if the induced representation $Ind_{H}^{G} (\psi)$ of $G$ is multiplicity free. There is a geometric way to prove if some triple is Gelfand triple, which is called Gelfand's trick. Multiplicity free representations play an important role in representations theory and number theory, e.g. the use of Whittaker models. We will discuss Gelfand's trick and its use in a simple cases for Whittaker models of the representations of the group $GL_{n}(R)$ where $R$ is finite local ring.

Manish Mishra, IISER, Pune

Title: A generalization of the 3d distance theorem

Speaker: Manish Mishra, IISER Pune

Where: Seminar Room, School of Physical Sciences (SPS), C V Raman Marg, JNU
When: Tuesday, February 11, 2020, 4 PM

Abstract: 

Let P be a positive rational number. Call a function f : R → R to have finite gaps property mod P if the following holds: for any positive irrational α and positive integer M, when the values of f(mα), 1 ≤ m ≤ M, are inserted mod P into the interval [0,P) and arranged in increasing order, the number of distinct gaps between successive terms is bounded by a constant kf which depends only on f. In this note, we prove a generalization of the 3d distance theorem of Chung and Graham. As a consequence, we show that a piecewise linear map with rational slopes and having only finitely many non- differentiable points has finite gaps property mod P. We also show that if f is distance to the nearest integer function, then it has finite gaps property mod $1$ with $k_f ≤ 6$. This is a joint work with Amy Philip.