Description: Tate's thesis
Date, time, venue: 26/08, 11:30am -- 1pm, Room 215
Hosts: Keshav Aggarwal and Mallesham Kummari
Speaker: Akash Yadav, IIT Bombay
Abstract: We will discuss the end of section 1.5, and sections 1.6 and 1.7
from Goldfeld-Hundley Vol 1. Topics are p-adic and adelic Fourier
inversion.

 Partial Differential Equations Seminar:
Speaker: Ramesh Mete (IISc. Bengaluru)
Title: The J and dHYM equations, and corresponding natural flows.
Time, Day and Date: 11:30 a.m., Tuesday, August 27
Venue: Online (Zoom): https://us06web.zoom.us/j/88652942508?pwd=5qyzxKZbRXYOSTwwDDJZpjQah0qcLl.1

Abstract: 

The J-equation and deformed Hermitian-Yang-Mills (dHYM) equation are two important examples of complex Hessian equations which have received considerable attention in last two decades. The J-equation, introduced independently by S.K. Donaldson and X.X. Chen (1999) from different viewpoint, is related to the cscK problem in K\"{a}hler geometry. The dHYM equation, introduced by Leung-Yau-Zaslow (2000), has connection to mirror symmetry in string theory. It is well-known that each equation admits a (unique) smooth solution if and only if certain cone (or sub-solution) condition holds, or equivalently, if and only if a Nakai-Moishezon type criterion holds (which is the so-called "stable" situation). In this talk, we will focus on the existence and uniqueness of (singular) solutions for both equations in the unstable case on compact K\"{a}hler surfaces and higher dimensional K\"{a}hler manifolds with Calabi symmetry using some natural flows. Based on a joint work with Dr. Ved Datar (IISc Bengaluru) and Prof. Jian Song (Rutgers University).

4. Algebraic Groups Seminar:
Speaker: Dibyendu Biswas (IIT Bombay)
Title: Bruhat Decomposition
Time, Day and Date: 4:00 p.m., Tuesday, August 27
Venue: Ramanujan Hall, Department of Mathematics
Abstract:We will study Bruhat decomposition in reductive groups.

Title: Operads and Infinite loop space theory

Time, Day and Date: 11:30 a.m,  Wednesday 28th August.

Venue: Ramanujan hall

Host: Rekha Santhanam

Speaker: Sahin Mandal

Abstract: In this series of talks we will define Operads and discuss their
properties. We will then define Segal's "gamma" space to prove
Barratt-Priddy-Quillen theorem and discuss uniqueness theorem for infinite
loop spaces machines as proven by May and Thomason.
 

Description: Spectral theory of automorphic forms
Date, time, venue: 28/08, 11:30am -- 1pm, Room 215
Hosts: Keshav Aggarwal and Mallesham Kummari
Speaker: Suraj Panigrahy and Aditi Savalia, IIT Bombay
Abstract: We'll discuss sections 1.3 -- 1.4 from Iwaniec's Topics in
classical automorphic forms. Suraj will continue with his presentation and
discuss section 1.3. Aditi will discuss section 1.4 (and 1.5 if time
allows).

Commutative Algebra Seminar:
Speaker: Tony J. Puthenpurakal (IIT Bombay)
Title: K-theoretic methods in local algebra
Time, Day and Date: 12:30 p.m., Tuesday, August 27
Venue: Room 215, Department of Mathematics
Abstract: We continue over studies in k-theory of complexes

Colloquium:

Speaker: Najmuddin Fakhruddin (School of Mathematics, TIFR, Mumbai)

Title: The Hodge theory of the KZ equations and enriched representation rings

Time, Day and Date: 4:00 p.m. on Wednesday, August 28

Venue: Ramanujan Hall

Abstract: The representation ring of a simple Lie algebra over the field of complex numbers is the free abelian group on the isomorphism classes of irreducible representations with the product structure given by decomposing the tensor product of two irreducible representations as a direct sum of irreducible representations (with multiplicities). In recent joint work with Prakash Belkale and Swarnava Mukhopadhyay we have defined certain families of "enriched" representation rings: these are free modules over the integral polynomial ring in one variable on the set of irreducible representations, with a product which specializes to the usual product when the variable of the polynomial ring is set to 1. Furthermore, the enriched "multiplicities" are polynomials with non-negative coefficients. These rings arose in our work on the Hodge theory of the Knizhnik-Zamolodchikov (KZ) equations: these are certain linear partial differential equations associated to a simple Lie algebra, a finite set of irreducible representations and an auxiliary complex number kappa, and were defined in the context of conformal field theory in the 1980s, but various aspects of these equations, in particular their monodromy, have been studied by mathematicians such as Kohno, Drinfeld and many others. Our work builds on the work of Schechtman-Varchenko and Looijenga which led to the proof that these equations for kappa rational are of Gauss-Manin type, i.e., arise from the cohomology of families of algebraic varieties. In my talk I will explain the construction of the enriched representation rings and how they are relevant to computing the ranks of the Hodge filtration on KZ local systems. The general constructions will be made explicit throughout the talk in the concrete case of the Lie algebra sl_2.
 

6. Algebraic Groups Seminar:
Speaker: Dibyendu Biswas (IIT Bombay)
Title: Nilpotent conjugacy class
Time, Day and Date: 4:00 p.m., Friday, August 30
Venue: Room 215, Department of Mathematics

Abstract: We will continue the analysis of nilpotent conjugacy classes using the book by Collingwood and McGovern.