Next Seminar

Title: Introduction to Hilbert modular forms and its determination by square-free Fourier coefficients.
Speaker: Rishabh Agnihotri
Date and Time: September 10, 2021 at 03:30PM
Venue: Seminar Hall, Kerala School of Mathematics
Abstract: We introduce two notions of Hilbert modular forms namely classical and adelic. After that we see the relation between them. We also talk about the determination of adelic hilbert modular forms. More concretely we discuss the following result.

Let $\mathbf{f}$ be as above with $C_{\mathbf{f}}(\mathfrak{m})$ denote its Fourier coefficients. Then there exists a square-free ideal $\mathfrak{m}$ with $N(\mathfrak{m})\ll k_0^{3n+\epsilon}N(\mathfrak{n})^{\frac{6n^2+1}{2}+\epsilon}$ such that $C_{\mathbf{f}}(\mathfrak{m})\neq 0$. The implied constant depends only on $\epsilon, F$.

The ‘KSoM Seminar Series’ is a fortnightly seminar series usually held on Fridays at 03:30PM.
These lectures are usually research seminars, wherein the speakers will lecture on their recent research work.

Past seminars:

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: 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: Arakelov Geometry of Modular Curves $X_0(p^2)$.
Speaker: Chitrabhanu Chaudhuri, NISER Bhubaneshwar
Date and Time: March 12, 2021 at 03:30PM
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: 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.