**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 . 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 , the topological field is totally disconnected and not locally compact. To rectify these issues, one replaces (or, ) with the Berkovich projective line, which is compact, Hausdorff, path connected space and in which 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 Mathematics

**Abstract :**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 , they count set partitions with specified block sizes. At , they count standard tableaux of specified shape. At 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 -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.

**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.

**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 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.

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

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

for every unitary on .

Abstract : The -series associated to a classical modular form (of weight and level ) denoted by

where denotes the Fourier coefficient of (in its -series expansion around ) is an analytic function on the right half plane and can be analytically continued to the whole . The non-trivial zeros of this function lie inside the critical strip . The analogue (GRH) of the Riemann Hypothesis in this context states that they all lie on the critical line itself.

The following region

where is an absolute constant, is currently known to be a zero-free region for . 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: 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 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.