Talks Given:
June 2024: Inter IISER-NISER Mathematics Meet (IINMM) 2024, IISER Thirubhananthapuram, India. Dates: June 26-28, 2024.
Title: Swan conductor and slope of Galois representations.
Abstract: The ‘Swan conductor’ of a smooth, complex, semisimple local Galois representation is an additive invariant of the representation under Local Langlands Correspondence (LLC). Another related but more intrinsic invariant is ‘slope’ of a Galois representation. In this talk, we will present the behaviour of Swan conductors under various functorial lifts. We will also describe the slope of adjoint Galois representation. This is a work in progress with Dr. Arindam Jana.
June 2024: International Conference on Lie Algebra and Number Theory, NIT Calicut, India. Dates: June 10-14, 2024.
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.
May 2024: Representation Theory and Related Geometry, University of Georgia, Athens, GA, Dates: May 27-31, 2024.
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.
May 2024: 36th Automorphic Forms Workshop, Oklahoma State University, USA, Dates: May 20-24, 2024.
Title: The doubling construction for integral representations of L-functions.
Abstract: In this talk. we will present an overview of the doubling construction of integral representation of L-functions. An explicit construction will be given for quasi-split special orthogonal groups.
Short course- Introduction to Topology, Ramakrishna Mission Vivekananda Centenary College Rahara, India, Dates: February 06-10, 2024.
On the bounds of Swan conductor of functorial lifts, IIT Guwahati, Date: December 24, 2023.
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.
Ramanujan tau function and modern mathematics, IIT Roorkee, India, Date: July 22, 2023.
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.
Short course- Introduction to modular forms, SRM University, Andhra Pradesh, India, Date: June 26 - July 15, 2023.
Short course- Calculus of several variables, Ramakrishna Mission Vivekananda Centenary College Rahara, India, Date: March 066-10, 2023.
Symmetry in Mathematics, IIIT Vadodara, India, Date: December 23, 2021.
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.
Depth zero representations and their Hecke algebras, Seminar series on Representation Theory (online), Date: November 13, 2021.
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.
Contragredient representations of p-adic groups, Inter IISER-NISER Mathematics Meet (IINMM) 2021, Date: July 13, 2021, IISER Tirupati, India.
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.
Duality theorem for p-adic general spin groups, Algebra and Number Theory Seminar, Yale University, USA. Date: April 27, 2021.
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.
Unification in Mathematics, Pi Day and International Mathematics Day, CAIAS Bangalore, India. Date: March 13, 2021.
Integral representations of L-functions, IISER-NISER Mathematics Webmeet 2020, IISER Mohali, India. Date: July 13, 2020.
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.
Contragredient representations of p-adic groups, Mathematics seminar at IIT Delhi, India, Date: November 14, 2019.
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.
The doubling construction for integral representations of L-functions, Representation Theory seminar at Bar-Ilan University, Israel, Date: April 02, 2019.
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.
Integral representations of L-functions, Representation Theory Seminar at Bar-Ilan University, Israel, Date: March 26, 2019.
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.
Typical representations of p-adic reductive groups, Seminar at Bar-Ilan University, Israel, Date: November 15, 2017.
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.
Degree and Conductor of Certain Local L-functions, Colloquium at TIFR Mumbai, September 08, 2016.
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).
On Some Invariants of Local Langlands Correspondence, Colloquium at ISI Kolkata, Date: December 28, 2015.
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 to this correspondence.
Artin Representation and Artin Conductor, Students' seminar at TIFR Mumbai, Date: September 29, 2015.
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.
Conductors of Functorial Lifts, Mumbai-Pune Number Theory, IIT Bombay, Mumbai.
On a Conductor Formula of Bushnell, Henniart & Kutzko, 28th Journées Arithmé- tiques JA2013, St-Martin d’Hères, Grenoble.
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.