Talks Given: 



Title: On the duality involution on derived subgroups.


    Abstract: Let G be a reductive p-adic group with a duality involution. In this talk, we shall address the following

     question: When does the duality involution of G restrict to a duality involution of the derived subgroup of G? We shall

     discuss a few examples based on a work in progress. 




Title: An informal introduction to Langlands Program.


Abstract: The Langlands Program- often referred to as the ‘Grand Unified Theory of Mathematics’, predicts remarkable connections between two seemingly distant areas of Mathematics: Number Theory and Harmonic Analysis. These far-reaching ideas played a crucial role in one of the greatest mathematical achievements of the 20th century- the proof of Fermat’s Last Theorem.

In this talk, I shall offer an informal introduction to the Langlands Program and highlight some of its developments.


Title: Unitarily induced representations of real reductive groups (Three lectures).

Abstract: In this series of three talks, we describe the unitarily induced representations of connected linear reductive/semisimple Lie groups. Explicit constructions are given for the group SL(2,R). 


Title: Evolution of Mathematics: The Role of Pythagoras' Theorem. 


Title: Hecke algebras of GL(2) over p-adic division algebras.

Abstract: Let D be a division algebra defined over non-archimedean local field F of characteristic zero. We will address the following question: To what extent does the representation theory of GL(n,D) depend on the division algebra D (or the underlying field F)? Using Bushnell-Kutzko type theory and Bernstein decomposition of Hecke algebra H(GL(2,D)), we show that the category of smooth representations of GL(2,D) does not depend on D (or on the field F). This is joint work with Dr. Basudev Pattanayak.


     Title: Swan conductor and slope of Galois representations.

     Abstract: The ‘Swan conductor’ and ‘slope’ of a smooth, complex, semisimple local Galois representation are certain invariants of the representation under Local Langlands Correspondence (LLC). Though these two notions are related for irreducible representations, ‘slope’ is generally more challenging to study. In this talk, we will present the behaviour of Swan conductors under various functorial lifts. We will also describe the slope of adjoint representation with applications. This is a work in progress with Dr. Arindam Jana.


     

Title: The doubling construction for integral representations of L-functions.

Abstract: In this talk, we will present an overview of the doubling construction of an integral representation of L-functions. An explicit construction will be given for quasi-split special orthogonal groups.

     





Abstract: The Swan conductor of a smooth, complex, semisimple local Galois representation is an additive invariant of the representation. In this talk, we will describe the best possible upper bounds of the Swan conductor of the following functorial lifts: symmetric square, exterior square and Asai lift. This is part of a work in progress with Arindam Jana. 




Abstract: Euler was interested in the infinite sum and product of numbers. Along the lines of Euler, Ramanujan studied several arithmetic functions, including the tau function, which has several applications in elementary and modern mathematics. Ramanujan made a conjecture about tau function. In this talk, we will show how part of the conjecture can be proved using the notion of modular forms, a ‘fundamental operation’ in modern mathematics.





   Abstract: Nature is full of symmetry. In this talk, we will see how Mathematics can be used to study these symmetries.        Then, we will see how symmetries in mathematical objects (numbers, functions, sets, etc.) can be used to answer questions in mathematics. 



     Abstract: In this talk, we will discuss the depth-zero representations of reductive p-adic groups as compactly induced representations. First, we will classify all the depth-zero supercuspidal representations of general linear groups. Then, we will describe a possible generalization of this mechanism for studying smooth representations of reductive p-adic groups and the connection to the associated Hecke algebras. 



     Abstract: In this talk, we will present an overview of an explicit realization of the contragredient of irreducible smooth representations of p-adic groups, known as the duality theorems. In particular, we will present a proof of the duality theorem for p-adic general spin groups. 



     Abstract: The duality theorem for p-adic groups aims to realize the dual of irreducible admissible representations on the space of the given representation itself. In this talk, we will present a proof of the duality theorem for p-adic general spin groups by constructing a suitable duality involution on the group.



     Abstract: L-functions are ubiquitous in number theory. Langlands' conjecture on L-functions describes the expected properties of the L-functions associated with automorphic representations. Integral representations of L-functions play a central role in proving this conjecture. In this talk, we will describe a particular construction of integral representations, known as the 'Doubling Construction'. Towards the end, we will present recent progress in the doubling construction of integral representations for quasi-split special orthogonal groups.

 

     Abstract: In this talk, we will survey the contragredient of irreducible smooth representations of p-adic groups. In particular, we will present recent progress on the contragredient representations of general spin groups. This is a joint work with Santosh Nadimpalli.


     Abstract: In this talk, we will present an overview of the doubling construction for integral representations of L-functions. An explicit construction will be given for quasi-split special orthogonal groups.


     Abstract: One of the central aspects of the theory of automorphic representations is Langlands' conjecture on automorphic L-functions attached to these representations. In this talk, we will explain some of the ideas of the proofs of this conjecture via integral representations of L-functions. 


     Abstract: Typical representations appear in the Bushnell-Kutzko theory of types for studying smooth representations of p-adic reductive groups. In this talk, we will present an overview of typical representations associated with level-zero Bernstein blocks of split classical groups. 


     Abstract: The ‘Local Langlands Correspondence’ is characterized by the behaviour of certain invariants, namely, the ‘local L-functions’ and the ‘conductors of pairs’. Let F be a non-archimedean local field. In this talk, we present a systematic study of the ‘degree’ and ‘conductor’ of the L-functions for various functorial lifts associated to supercuspidal representations of the group GL(n,F). 


     Abstract: ‘Local Langlands Correspondence’ relates the representation theory of Galois groups of local fields and the representation theory of general linear groups over local fields. In this talk, we give a brief description of the ’Local Langlands correspondence’ and some of the ’invariants’ associated with this correspondence. 


     Abstract: Artin representation is one of the central objects in the study of complex representations of the Galois group of local fields. The aim of this talk is to give an introduction of Artin representation of local Galois groups using the notion of Artin cnductor of Galois representations. 


Abstract: Let F be a non-archimedean local field. An irreducible supercuspidal represenatation \pi of $GL_n(F)$ is said to be epipelagic if its Swan conductor equals 1. In this series of lectures we discuss the construction of "explicit" Langlands parameter for epipelagic representations. These lectures will be in continuation with the earlier lectures by Shiv Prakash Patel.



     Abstract: The explicit conductor formula of Bushnell, Henniart and Kutzko[BHK98] computes the conductor of a pair of supercuspidal representations of general linear groups over a non-archimedean local field. In this talk, we describe their conductor formula and also present a slightly different approach to one part of the conductor formula.