Seminar
Speaker: Dr Shubham Jaiswal (IISER Pune)
Host: Shripad Garge
Title: Inverse Galois problem and root clusters
Time, day and date: 11:00:00 AM, Tuesday, April 1
Venue: Online (https://meet.google.com/nce-cbfi-apc)
Abstract: This talk will discuss the two topics mentioned in the title. In the first part we discuss our work on the inverse Galois problem. In the second part we will introduce the notion of root clusters and discuss our contribution to the topic. This is a joint work with Prof. Chandrasheel Bhagwat.

Analysis Seminar.

Date, time and Venue: April 2, 2025, 3.30 pm, Ramanujan Hall of
Mathematics department

Speaker: Mr. Nitin Tomar

Title: Operators associated with various domains in $\mathbb{C}^n$

Abstract: This talk explores operator theory on several important domains
in $\mathbb{C}^n$, focusing on dilation, decomposition and model theory.
We examine the interplay between operator theory and complex geometry,
particularly through spectral sets and distinguished varieties. The
domains of our interest include the polyannulus $\mathbb{A}_r^n$, bidisc
$\mathbb{D}^2$, biball $\mathbb{B}_2$, symmetrized bidisc $\mathbb{G}_2$
and pentablock $\mathbb{P}$. The first part examines operator theory on
the polyannulus $\A_r^n$, where we explore $ \mathbb{A}_r^n
$-contractions, $ \mathbb{A}_r^n $-unitaries, and $ \mathbb{A}_r^n
$-isometries, providing characterizations and decomposition results. We
also study rational dilation for certain classes of $ \mathbb{A}_r^n
$-contractions and identify minimal spectral sets for various classes of
operators related to annulus. In particular, we characterize the class of
$ \mathbb{A}_r $-contractions using a variety in the biball. The second
part focuses on the pentablock, where we define $ \mathbb{P}
$-contractions, $ \mathbb{P} $-unitaries, and $ \mathbb{P} $-isometries,
and develop canonical decomposition results for $\Pe$-contractions. We
establish conditions for dilating $ \mathbb{P} $-contractions to $
\mathbb{P} $-isometries and explore related operators in the biball and
symmetrized bidisc, identifying relations between these domains at
operator theoretic level. In the final part, we investigate distinguished
varieties in the bidisc and symmetrized bidisc, toral polynomials,
$\Gamma$-distinguished polynomials and operator pairs annihilated by such
polynomials. We provide necessary and sufficient conditions for dilating
toral contractions to toral unitaries and characterize $ \Gamma
$-distinguished $\Gamma$- contractions admitting a dilation to
$\Gamma$-distinguished $\Gamma$-unitaries.

Commutative Algebra seminar
Speaker: Samarendra Sahoo (IIT Bombay)
Host: Tony Puthenpurakal
Title: The Auslander-Reiten Conjecture
Time, day and date: 4:00:00 PM, Tuesday, April 08
Venue: Ramanujan Hall
Abstract: The Auslander-Reiten conjecture, proposed in 1975, states that if Exti(M,M)=Exti(M,R)=0 for all i≥1, then a finitely generated module M over a Noetherian commutative ring R must be projective. In the last two lectures, we discussed how the conjecture holds if the injective dimension of Hom(M,M) or Hom(M,R) is finite. In the next lecture, we will examine a result by T. Araya, which states that for Gorenstein rings, it suffices to verify the conjecture for rings of dimension at most one. If time permits, we will also discuss recent work by D. Ghosh and M. Samanta on the finite complete intersection dimension of M and Hom(M,R).

Mathematics Colloquium
Speaker: Apoorva Khare (Indian Institute of Science)
Host: Dipendra Prasad
Title: Determinants with any smooth function reveal all Schur polynomials
Time, day and date: 4:00:00 PM, Wednesday, April 09
Venue: Ramanujan Hall
Abstract: Cauchy's identity (1840s) expands the determinant of the matrix $f[{\bf u}{\bf v}^T]$, where $f(t) = 1/(1-t)$ is applied entrywise to the $n \times n$ rank-one matrix $(u_i v_j)$. This was generalized by Frobenius (1880s). In a different century and context, Loewner (1960s) showed the vanishing of the initial Taylor coefficients of $\det f[t \cdot {\bf u}{\bf u}^T]$, where $f$ is a smooth function. This theme also appears recently in the 2010s in matrix analysis, for $f$ a polynomial.

This talk aims to bring this algebra and analysis together, by expanding $\det f[t \cdot {\bf u}{\bf v}^T]$ for all power series $f$. Time permitting, we will go from determinants to immanants for any character of the symmetric group, for bosonic/fermionic variables $u_i$ and $v_j$. (Partly based on joint works with Alexander Belton, Dominique Guillot and Mihai Putinar; with Siddhartha Sahi; and with Terence Tao.)

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

Speaker: Subhajit Roy, IIT Madras

Time: Friday, 15 April 2025, 11:30 AM

Title: On Fractional Orlicz-Hardy Inequalities

Abstract: In this talk, we establish fractional Orlicz-Hardy inequalities
for arbitrary Young functions that satisfy the doubling condition.
Furthermore, we identify the critical cases for each Young function and
prove fractional Orlicz-Hardy inequalities with logarithmic corrections.
Finally, we investigate geometric variants of these inequalities in
various domains, including bounded Lipschitz domains, domains above the
graph of a Lipschitz function, and the complements of bounded Lipschitz
domains.

Joining link: (Zoom Meeting)

https://us06web.zoom.us/j/84821721651?pwd=qcmEWktb3HTk4mB3pvEWzCsR4Y6Caz.1
Meeting ID: 848 2172 1651
Passcode: 795798