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Next Seminar

Title: Non-archimedean dynamics in dimension one
Speaker: Niladri Patra
Affiliation: Tata Institute of Fundamental Research, Mumbai
Date and Time: December 09, 2022 at 03:30 PM
Venue: Seminar Hall
Abstract: In the beginning of the twentieth century, complex analysis gave rise to complex dynamics, which is the study of self-iterations of rational maps defined over complex numbers. Often the study of complex dynamical objects boil down to questions that are of arithmetic nature. This generally motivates the study of arithmetic dynamics. Arithmetic dynamics is roughly divided into two parts, dynamics over global fields and dynamics over local fields. Dynamics over local fields, which is also called non-archimedean dynamics or p-adic dynamics in the literature, has another motivation coming from p-adic analysis. The motivation is to build a theory parallel to complex dynamics with field of complex numbers replaced with \mathbb{C}_{p}. The theory that follows is called classical p-adic dynamics. One finds certain anomalies between the theories of complex dynamics and classical p-adic dynamics. These anomalies mostly arise from the fact that unlike \mathbb{C}, the topological field \mathbb{C}_{p} is totally disconnected and not locally compact. To rectify these issues, one replaces \mathbb{C}_{p} (or, \mathbf{P}^{1}(\mathbb{C}_{p})) with the Berkovich projective line, which is compact, Hausdorff, path connected space and in which \mathbf{P}^{1}(\mathbb{C}_{p}) embeds. In this talk, we will introduce discrete, complex, classical p-adic dynamics, dynamics on the Berkovich projective line and mention some of the parallels between complex dynamics and dynamics on the Berkovich projective line.

 

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 : Some geometric structures on principal Lie 2-group bundles over Lie groupoids.
Speaker: Praphulla Koushik
Affiliation: IISER Pune
Date and Time: June 24, 2022 at 03:30 PM
Venue: Seminar Hall, Kerala School of MathematicsAbstract : In this talk we introduce the notion of principal Lie 2-group bundle over a Lie groupoid; as a generalisation of the notion of classical principal (Lie group) bundle over a smooth manifold. Motivated by the idea of Atiyah sequence for G-bundles over manifolds, we introduce the notion of Atiyah sequence for Lie 2-group bundle over Lie groupoids. We then see some notion of (strict) connection and semi-strict connections on such principal bundles. This is a joint work with Saikat Chatterjee and Adittya Chaudhuri

Title: Set partitions, tableaux, and subspace profiles under regular split semisimple matrices
Speaker: Amritanshu Prasad
Affiliation: The Institute of Mathematical Sciences
Date and Time: December 10, 2021 at 03:30 PM
Venue: Seminar Hall, Kerala School of Mathematics
Abstract: We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. At 1, they count set partitions with specified block sizes. At 0, they count standard tableaux of specified shape. At -1 they count standard shifted tableaux of a specified shape. These polynomials are generated by a new statistic on set partitions (called the interlacing number) as well as a polynomial statistic on standard tableaux. They allow us to express q-Stirling numbers of the second kind as sums over standard tableaux and as sums over set partitions. In a special case, these polynomials coincide with those defined by Touchard in his study of crossings of chord diagrams.
This lecture was based on joint work with Samrith Ram: https://arxiv.org/abs/2112.00479.
Title:Lie algebras associated to closed curves on surfaces. 
Speaker: Arpan Kabiraj
Affiliation: IIT Palakkad
Date and Time: November 26, 2021 at 03:30PM
Venue: Seminar Hall, Kerala School of Mathematics

Abstract:We will discuss various Lie algebras associated to closed curves on orientable surfaces (possibly with boundary and punctures) introduced by Goldman and Wolpert in the 80’s. If time permits, we will discuss a relation between these Lie algebras and skein algebras of three-manifolds.

Title: Introduction to Hilbert modular forms and its determination by square-free Fourier coefficients.
Speaker: Rishabh Agnihotri
Affiliation: HRI, Prayagraj (Allahabad)
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.
Title:  Class group of real cyclotomic fields.
Speaker: Mohit Mishra
Affiliation: HRI, Prayagraj (Allahabad)
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.
Speaker:Krishnendu Gongopadhyay
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.