Commutative Algebra seminar

Speaker: Aditya Dwivedi (IIT Bombay)

Host: Tony Puthenpurakal

Title: Briançon-Skoda theorem in positive characteristics.

Time, day and date: 4:00:00 PM, Tuesday, April 15

Venue: Ramanujan Hall

Abstract: Briançon-Skoda theorem was initially proved for regular local rings by Lipman and Sathaye in 1981. Lipman and Teissier further generalized it to refer to arbitrary rings that have pseudo-rational singularities. In positive characteristics, the notion of ring having pseudo-rational singularities translates to the ring being F-rational, which is a property about tight closure. We will both state and prove the positive characteristic version of the Briançon-Skoda theorem in this talk. The result is due to Aberbach and Huneke.

Number Theory Seminar
Speaker: Prof. Atul Dixit (IIT Gandhinagar)
Host: Krishnan Sivasubramanian
Title: Mordell-Tornheim zeta functions and functional equations of Herglotz-Zagier type functions
Time, day and date: 2:00:00 PM, Thursday, April 17
Venue: Ramanujan Hall
Abstract: In this talk, we will present our recent results on a generalization of the Mordell-Tornheim zeta function, in particular, the two- and three-term functional equations that it satisfies. This function is intimately connected with a new extension of the Herglotz-Zagier function F(x). The function F(x) is instrumental in Zagier's version of the Kronecker limit formula for real quadratic fields. Radchenko and Zagier recently studied arithmetic properties of F(x), in particular, their special values and functional equations coming from Hecke operators. One of our results on this extension not only gives the well-known two-term functional equation of F(x) as a special case but also those of Ishibashi functions, which were sought after for over twenty years. A grand generalization of an integral considered by Herglotz as well as its companion due to Muzzaffar and Williams, which involves generalized Fekete polynomials and character polylogarithms, is obtained. By deriving a functional equation for this generalization, we are able to get doubly infinite families of functional equations whose two special cases were recently obtained by Choie and Kumar. This is joint work with Sumukha Sathyanarayana and N. Guru Sharan.

There is a PDE seminar talk by Anamika Purohit from IIT Gandhinagar.
Please find the details below.

Venue: Ramanujan Hall, 17th April, 3:00 pm.

Title: An inverse problem for a time-dependent convection-diffusion equation

Abstract:  In this talk, we study partial and local data inverse problems
for the time-dependent convection-diffusion equation in a bounded domain.
For the partial data problem, we show that the time-dependent convection
and the density terms can be uniquely recovered up to the natural gauge
from the knowledge of the Dirichlet to Neumann map measured on a small
open subset of the boundary.
In the local data inverse problem, where a part of the boundary is treated
to be inaccessible, upon assuming the inaccessible part is flat, we seek
the unique determination of the time-dependent convection and the density
terms from the knowledge of the boundary data measured outside the
inaccessible part. In the process, we show a natural gauge in the
perturbations, proving that this is the only obstruction in the uniqueness