Kerala School of Mathematics conducts regular research seminars wherein research groups and individual researchers shall talk about the current developments in their topic of research.

Title:  Class group of real cyclotomic fields.
Speaker: Mohit Mishra
Date and Time: July 21, 2021 at 03:30PM
Venue: Seminar Hall, Kerala School of Mathematics
Abstract: For every finite extension of rational numbers, there is a group associated to it called the “Class Group”. Class group is a very mysterious object and there is no (infinite) family known with a prescribed class group. In 1979, G. Cornell proved that every finite abelian group $G$ can be realized as a subgroup of a class group of infinitely many cyclotomic (totally imaginary) fields. In this talk, we will prove the analogue of this result for real cyclotomic fields. This is a joint work with L.C. Washington and R. Schoof.

Title:  Obstruction Theory in Algebra and Topology.
Speaker : Bibekananda Mishra
Date and time: 10.00 am to 10.45 am, 17 th June 2021.
Online platform: Zoom
Abstract: Let P be a projective module of finite rank on a ring A. What is the precise obstruction for P to have a free component (i.e. P \cong Q \oplus A)? This question, very much of algebraic flavour, is intricately related to the question of whether the vector bundles over smooth manifolds have non-vanishing sections on them.  We will see in this talk certain invariants, called Nori homotopy groups, both in the algebraic as well as topological context, which gives us an effective description of the obstructions involved.

Title:   A degeneration of the Compactified Jacobian of irreducible nodal curves.
Speaker:  Subham Sarkar
Date and time:      12.00 pm to 12.45 pm, 17 th June 2021
Online platform: Zoom
Abstract:  For each k ≥ 1, We construct an algebraic degeneration of the compactified Jacobian of a nodal curve X_k with k-nodes, over a suitable dense subset of the k-fold product of the normalisation X_0 of X_K.
A special fiber is isomorphic to the Jacobian of X_0 and k-fold product of a rational nodal curve.
We prove that the total space is a quasi-projective variety with k-th product of normal crossing singularity.
As an application we computed the mised Hodge number of the cohomogy of compactified Jacobian.

Title: Shifted convolution sums and sign changes of Fourier coefficients of certain automorphic  forms.
Speaker: Lalit Vaishya
Date and time: 3.00 pm to 3.45 pm, 17 th June 2021
Online platform: Zoom

Abstract: Briefly, we present some of our work which deal with some problems in the theory of automorphic forms. In the first part, we discuss some problems on shifted convolution sums associated to Hecke Maass cusp forms (non holomorphic cusp forms), holomorphic Hecke eigen (cusp) forms and obtain the estimates. In the second part, we prove a quantitative result about sign changes of Fourier coefficients of Hecke eigenform supported at positive integers represented by a primitive integral positive binary quadratic from of negative discriminant having class number 1. We also study the average behavior of Fourier coefficients of Hecke eigenforms supported at positive integers represented by a primitive integral positive definite binary quadratic form of negative discriminant having class number 1. As a consequence, we prove that there are infinitely many sign change of sequence of Fourier coefficients supported at positive integers represented by these binary quadratic forms.

Title: On the Topology of Complex Projective Varieties
Speaker: NimaRose Manjila
Affiliation: IISER Pune
Date and Time: April 09, 2021 at 02:00PM
Venue: Seminar Hall, Kerala School of Mathematics

Abstract: We use Morse Theory and Lefschetz Pencil to find the Topology of Complex Projective Curves and generalise this idea to prove Lefschetz Theorem. Other results include a proof of Poincare Duality and Riemann Hurwitz theorem for Ramified Covers of Curves.

Title: Real Unipotent Elements in Classical Lie Groups.
Affiliation: IISER Mohali
Date and Time: April 08, 2021 at 03:30PM
Venue: Seminar Hall, Kerala School of Mathematics.
Abstract: Real elements are those elements in a group which are conjugate to their own inverses. Real elements appear naturally at different branches of mathematics. These elements are also known as `reversible’ elements in the literature. These elements are closely related to the so-called strongly real elements in a group which are products of two involutions. After giving a brief exposition on real elements in groups, I shall discuss classification of real unipotent elements in classical Lie groups which is part of a joint work with Chandan Maity.

Title: A generalized modified Bessel function and explicit transformations of certain Lambert series
Speaker:Rahul kumar
Date and Time: March 26, 2021 at 04:00PM
Venue: Seminar Hall, Kerala School of Mathematics.

Abstract: An exact transformation, which we call a master identity, is obtained for the series P∞n=1 σa(n)e−ny for a ∈ C and Re(y) > 0. As corollaries when a is an odd integer, we derive the well-known transformations of the Eisenstein series on SL2 (Z), that of the Dedekind eta function as well as Ramanujan’s famous formula for ζ(2m + 1). Corresponding new transformations when a is a non-zero even integer are also obtained as special cases of the master identity. These include a novel companion to Ramanujan’s formula for ζ(2m+ 1).Although not modular, it is surprising that such explicit transformations exist. The Wigert-Bellman identity arising from the a = 0 case of the master identity is derived too. The latter identity itself is derived using Guinand’s version of the Vorono ̈ı summation formula and an integral evaluation of N. S. Koshliakov involving a generalization of the modified Bessel function Kν(z). Koshliakov’s integral evaluation is proved for the first time. It is then generalized using a well-known kernel of Watson to obtain an interesting two-variable generalization of the modified Bessel function. This generalization allows us to obtain a new transformation involving the sums-of-squares function rk(n). This is joint work with Atul Dixit and Aashita Kesarwani.

Title: Weak Mordell-Weil Theorem for Chow groups over global function fields
Speaker:Kalyan Banerjee
Date and Time: March 26, 2021 at 03:00PM
Venue: Seminar Hall, Kerala School of Mathematics.
Abstract: The classical weak Mordell-Weil theorem for an abelian variety A over a number field K says that A(K)/nA(K) is finite for any integer n bigger than 1. This has further consequence that the group A(K) of K-rational points on A is finitely generated.  In this talk we are going to consider a variety X defined over the algebraic closure of a function field of a smooth projective curve and consider the group of degree zero cycles modulo rational equivalence on this variety denoted by A_0(X). We are going    to consider the question analogous to the weak Mordell-Weil theorem for the Galois invariants of A_0(X), that is whether the group A_0(X)^G/nA_0(X)^G is finite, where G is the absolute Galois group of the function field and n is an integer bigger than 1. We are going to prove this analogue under some assumption on the variety X.

Title: Representation theory of finite groups – an Introduction – II.
Speaker: Hassain M
Affiliation: Kerala School of Mathematics
Date and Time: March 17, 2021 at 02:00PM
Venue: Seminar Hall, Kerala School of Mathematics.
Abstract: Let $G$ be a finite group. A $n$-dimensional representation of $G$ is a homomorphism from $G$ to the group $\mathrm{GL}_n(\mathbb C)$ of $n\times n$ invertible matrices over $\mathbb C.$ In this talk, I will discuss some interesting examples and basic results in representation theory of finite groups.

Title: Arakelov Geometry of Modular Curves $X_0(p^2)$.
Speaker: Chitrabhanu Chaudhuri, NISER Bhubaneshwar
Date and Time: March 12, 2021 at 03:30PM
Venue: Seminar Hall, Kerala School of Mathematics  (online lecture)
Abstract: I shall outline the construction of a semisimple and minimal regular model for $X_0(p^2)$ over an appropriate number field. This will be a regular scheme over the spectrum of the ring of integers of that number field, such that the fibres are complete curves with at worst nodal singularities and satisfying certain stability conditions. The generic fibre of the model is isomorphic to $X_0(p^2)$. The purpose of this construction is to use the theory developed by Shou-Wu Zhang, using Arakelov theory,  for proving an effective version of a conjecture by Bogomolov in this special case of modular curves $X_0(p^2)$.

Title: Representation theory of finite groups – an Introduction – I.
Speaker: Hassain M
Affiliation: Kerala School of Mathematics
Date and Time: March 10, 2021 at 02:00PM
Venue: Seminar Hall, Kerala School of Mathematics.
Abstract: Let $G$ be a finite group. A $n$-dimensional representation of $G$ is a homomorphism from $G$ to the group $\mathrm{GL}_n(\mathbb C)$ of $n\times n$ invertible matrices over $\mathbb C.$ In this talk, I will discuss some interesting examples and basic results in representation theory of finite groups.

Title: Noncommutative Korovkin Theory.
Speaker: Arunkumar C. S.
Affiliation: Kerala School of Mathematics
Date and Time: February 26, 2021 at 03:00PM
Venue: Seminar Hall, Kerala School of Mathematics

Abstract: In this talk, we will introduce the Hyper Rigidity of operator systems in $C^*$-algebras as a non commutative analogue of Korovkin sets in the space of continuous functions, $C[0,1]$.  Also, we point out one of our recent results, and a couple of open  questions along this direction.

Title: A certain kernel function for -values of half-integral weight Hecke eigenforms.
Speaker: Sreejith M. M.
Affiliation: Kerala School of Mathematics
Date and Time : February 05, 2021 at 03:00 p.m.
Venue : Seminar Hall, Kerala School of Mathematics
Abstract: In this talk we will derive a non-cusp form of weight $k+1/2$ ($k\geq2$, even) for $\Gamma_0 (4)$ in the Kohnen plus space whose Petersson scalar product with a cuspidal Hecke eigenform $f$ is equal to a constant times the $L$ value $L(f,k-1/2).$  We also prove that for such a form $f$ and the associated form $F$ under the $D^{\text{th}}$ Shimura-Kohnen lift the quantity $\frac{a_f(D)L(F,2k-1)}{\pi^{k-1}\langle f,f\rangle L(D,k)}$ is algebraic.

Title: Characterization of linear maps preserving unitary conjugation.
Speaker: Dr. Shankar P.
Affiliation: Indian Statistical Institute, Bangalore
Date and Time: January 22, 2021 at 03:00PM
Venue: Seminar Hall, Kerala School of Mathematics

Abstract: Let $H$ be a complex separable Hilbert space and let $B(H)$ be the algebra of all bounded linear operators on $H$. In this talk, we discuss about what are the linear maps $\alpha:B(H) \rightarrow B(H)$ which satisfy

$\alpha(UXU^*)=U\alpha(X) U^*~~\forall~~ X\in B(H),$

for every unitary $U$ on $H$.

Title: Sign Changes in restricted coefficients of Hilbert Modular forms
Speaker: Krishnarjun K
Affiliation: Harish Chandra Research Institute, Prayagraj(Allahabad)
Date and Time: January 08, 2021 at 03:00PM
Venue: Seminar Hall, Kerala School of Mathematics
Abstract: Let $\textbf{f}$ be an adelic Hilbert cusp form of weight $k$ and level $\mathfrak{n}$ over a totally real number field $F$. In this talk, we study the sign changes in the Fourier coefficients of $\textbf{f}$ when restricted to square-free integral ideals and integral ideals in an “arithmetic progression”. In both cases we obtain qualitative results and in the former case we obtain quantitative results as well. Our results are general in the sense that we do not impose any restriction of the number field $F$, the weight $k$ or the level $\mathfrak{n}$.

Title : Some notions of non-commutative convexity
Speaker: Syamkrishnan M. S.
Affiliation: Kerala School of Mathematics
Date and Time: December 04, 2020 at 03:00PM
Venue: Seminar Hall, Kerala School of Mathematics
Abstract: In this talk, we shall introduce two non-commutative versions of the classical convexity, namely the $C^*$-convexity and matricial convexity in the setting of $C^*$-algebras. We shall discuss the similarities as well as dissimilarities between the convex sets in the classical setting with the convex sets in the non commutative case. Also, we will be discussing its connections with other areas of operator algebras.

Title: On non-vanishing of modular L functions inside the critical strip
Speaker : Sandeep E. M.
Affiliation : Kerala School of Mathematics
Date and Time : November 20, 2020 at 03:00 p.m.
Venue : Seminar Hall, Kerala School of Mathematics

Abstract : The $L$-series associated to a classical modular form $f$ (of weight $k$ and level $1$) denoted by

$L(f,s) := \sum_{n\geq 1} \frac{a_f(n)}{n^s}$

where $a_f(n)$ denotes the $n^{\text{th}}$ Fourier coefficient of $f$ (in its $q$-series expansion around $q=0$) is an analytic function on the right half plane $\{\Re(s)>\frac{k+1}{2}\}$ and can be analytically continued to the whole $\mathbb{C}$. The non-trivial zeros of this function lie inside the critical strip $(k-1)/2 < \Re(s) < (k+1)/2$. The analogue (GRH) of the Riemann Hypothesis in this context states that they all lie on the critical line  $\Re(s) = k/2$ itself.
The following region

$\sigma \geq 1-\dfrac{c}{\log (k+|t|+3)}$

where $c>0$ is an absolute constant, is currently known to be a zero-free region for $L(f,s)$. Some aspects of this non-vanishing related to my work will be discussed in this talk. This is a joint work with Prof M Manickam and Prof V Kumar Murty

Title : Elliptic Curves: Introduction and An Application
Speaker: Kalyan Chakraborty
Affiliation: Kerala School of Mathematics
Date and Time: November 06, 2020 at 03:00 p.m.
Venue: Seminar Hall, Kerala School of Mathematics
Abstract: This talk will begin with an introduction to elliptic curves. We shall then progress into the BSD conjecture and finally look into the idea of the proof of Fermat’s last theorem.

Title: An invitation to the theory of L functions
Speaker : Krishnarjun K
Affiliation : Harish Chandra Research Institute, Prayagraj(Allahabad)
Date and Time : October 23, 2020 at 03:00 p.m.
Venue : Seminar Hall, Kerala School of Mathematics

Abstract : The aim of this talk is to introduce the notion of an $L$ function and to describe a few basic properties. We shall also prove two classical theorems, one of Riemann and Dirichlet and demonstrate how techniques from complex analysis can be used to prove arithmetic results. We shall briefly touch upon current research topics of interest in the subject, if time permits.

Here is a list of the previous colloquiums and seminars