**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:**Prime numbers

**Speaker:**Kalyan Chakraborty

**Affiliation:**Harish-Chandra Research Institute, Prayagraj

**Venue:**Seminar Hall, KSoM

**Date and Time:**September 01, 2023 at 3:30 PM

**Abstract:**Several historical questions regarding prime numbers are still unsolved. These include Goldbach’s conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes that differ by two. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors.

**Title:**p-numerical semigroups and their fundamental properties

**Speaker:**Takao Komatsu

**Affiliation:**Zhejiang Sci-Tech University

**Venue:**Seminar Hall, KSoM

**Date and Time:**August 05, 2023 at 3:30 PM

**Abstract:**Consider the linear Diophantine equation , where are positive integers and is a non-negative integer. When we want to know the non-negative solutions, the number of solutions is finite if and only if . For a given non-negative integer , there exists the largest integer such that the number of solutions is at most . We call this largest integer the -Frobenius number. When , the -Frobenius number is the classical Frobenius number, which is the central role of the famous linear Diophantine problem of Frobenius. In this talk, we introduce the basic concept of the -numerical semigroup and mention the fundamental properties. We describe how the -Frobenius number and related values are obtained.

**Title:**Group actions in non-commutative geometry.

**Speaker:**Safdar Quddus

**Affiliation:**Indian Institute of Science, Bangalore

**Venue:**Seminar Hall, KSoM

**Date and Time:**July 31, 2023 at 12:00 PM

**Abstract:**Historically, it is well known that group action changes the homological, differential, and topological properties of manifolds. The aim of this talk is to discuss the action of finite groups on some non-commutative differential and algebraic spaces. We shall first discuss the (co)homological properties of non-commutative and quantum torus then observe how the quotient spaces resulting from the actions of discrete subgroups of behave, and then shall discuss the flip actions on the non-commutative sphere. We shall see how the tools are developed in non-commutative geometry to understand and tackle the group action in general non-commutative spaces.

**Title:**Rational points on hyperelliptic curves.

**Speaker:**Om Prakash

**Affiliation:**Kerala School of Mathematics

**Venue:**Seminar Hall, KSoM

**Date and Time:**July 28, 2023 at 2:30 PM

**Abstract:**Hyperelliptic curves which can be viewed as generalisation of elliptic curves are a special class of algebraic curves. The theory of elliptic curves is sort of well understood and has a vast literature. However, the theory of hyperelliptic curves has not received as much attention by the research community; and most results concerning hyperelliptic curves which appear in the literature on algebraic geometry are couched in very general terms. In this talk we give an overview of results concerning rational points on hyperelliptic curves and present our recent joint work with Kalyan Chakraborty.

**Title:**Hyperbolicity of the Curve Graph

**Speaker:**George Shaji

**Affiliation:**Department of Mathematics, The University of Utah

**Venue:**Seminar Hall, KSoM

**Date and Time:**July 26, 2023 at 11:00 AM

**Abstract:**We will briefly explore the “Guessing Geodesics Lemma”, and see its immense usefulness by applying it to the curve graph of a surface. Specifically, we shall show that the Curve Graph of a finite type surface (of sufficient complexity) is hyperbolic using “Unicorn Paths” or “Bicorn Paths”. This method provides an extremely beautiful, diversely applicable and significantly simpler alternative to the Masur-Minsky machinery used to prove the same fact much earlier.

**Title:**M-ideals in the Algebra of bounded analytic functions

**Speaker:** Sreejith Siju

**Affiliation:**Kerala School of Mathematics

**Venue:**Seminar Hall, KSoM

**Date and Time:**July 25, 2023 at 3:00 PM

**Abstract:**The distinguished class of subspaces of a Banach space known as M -ideals is a significant tool in the study of the geometry of Banach spaces and approximation theory. The M-ideals in the continuous functions space and the disk algebra are fairly understood. In this talk we will discuss some basic facts regarding M-ideals and some recent results on M-ideals in the algebra of bounded analytic functions. This talk is based on a joint work with Deepak and Jaydeb.

**Title:**On a class of Lebesgue-Ramanujan-Nagell equations

**Speaker:**Azizul Hoque

**Affiliation:**Department of Mathematics, Rangapara College

**Venue:**Seminar Hall, KSoM

**Date and Time:**July 21, 2023 at 11:00 AM

**Abstract:**Finding the solutions of an exponential Diophantine equation is a classical problem and even now it is a very active area of research. One of the most important equations of this type is so-called Lebesgue-Ramanujan-Nagell equation, where is a fixed positive integer. In this talk, we will discuss some interesting results concerning the integer solutions of certain generalizations of the above equation. This talk is based on some joint works with K. Chakraborty and K. Srinivas.

**Title:**Selmer group associated to the group of degree zero cycles on abelian varieties

**Speaker:**Kalyan Banerjee

**Affiliation:**Vellore Institute of Technology, Chennai

**Venue:**Seminar Hall, KSoM

**Date and Time:**July 11, 2023 at 3:00 PM

**Abstract:**In this talk we will look into a construction of the Selmer group of degree zero cycles on abelian varieties and will prove that it is finite.

**Title:** Genuinely ramified maps and the Galois group of generic projection

**Speaker:** Parameswaran A J

**Affiliation:** TIFR, Mumbai

**Venue:** Seminar Hall, KSoM

**Date and Time:** April 13, 2023 at 12:00 PM

**Abstract:**We show that the Galois closure of a degree d genuinely ramified covering of smooth projective irreducible curves over an algebraically closed field is the symmetric group if is a Morse map.

**Title:** Jordan derivations and -power property in rings

**Speaker:** Shakir Ali

**Affiliation:** Aligarh Muslim University

**Venue:** Seminar Hall, KSoM

**Date and Time: **February 08, 2023 at 12:00 PM

**Abstract:**Let be an associative ring(algebra) with center . Every associative ring can be turned into a Lie ring(algebra) by introducing a new product , known as the Lie product. So we may regard simultaneously as an associative ring(algebra) and as a Lie ring(algebra). An additive mapping is said to be a derivation on if holds for all . An additive mapping is said to be a Jordan derivation if holds for all . A function is called a centralizing on if holds for all . In the special case where for all , is said to be commuting on . The study of such mappings was initiated by E.C. Posner [ Proc. Amer. Math. Soc. 8(1957), 1093-1100]. In 1957, he proved that if a prime ring has a nonzero commuting derivation on , then is commutative. An analogous result for centralizing automorphisms on prime rings was obtained by J.H. Mayne [Canad. J. Math. 19 (1976), 113-115].

In this talk, we will discuss the recent progress made on the topic and related areas. Moreover, some examples and counterexamples will be discussed for questions raised naturally.

**Title:** Inversion formulas for the -function around elliptic points.

**Speaker:** Badri Vishal Pandey

**Affiliation:** Universität zu Köln, Germany

**Venue:** Seminar Hall, KSoM

**Date and Time:** February 02, 2023 at 12:00 PM

**Abstract:**Recently, Hong, Mertens, Ono, and Zhang proved a conjecture of Căldăraru, He, and Huang that expresses the Taylor series of the modular -function around the elliptic points and as rational functions arising from the signature and cases of Ramanujan’s theory of elliptic functions to alternative bases. We extend these results and give inversion formulas for the -function around and arising from Gauss’ hypergeometric functions and Ramanujan’s theory in signatures and .

**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.

**Tittle :**RESULTS AND MODELS. AN ILLUSTRATION THROUGH SUMS OF CUBES

**Speaker :**Jean-Marc Deshouillers

**Affiliation :**University of Bordeaux, France.

**Date and Time :**24-09-2022 at 03;30 PM

**Venue :**Seminar hall, Kerala School of Mathematics

**Abstract :**

RESULTS AND MODELS. AN ILLUSTRATION THROUGH SUMS OF CUBESNumber theory has the knack for phrasing easily understandable statements which are hard to prove.An archetype is Goldbach’s problem (1742), which is still unsolved :every even integer larger than 4 is a sum of two primes.The root of the talk is Waring’s problem (1770), which states – for cubes – thatevery integer is a sum of at most 9 cubes.It has been proved in 1909-1912, but we expect much more, namelyevery sufficiently large integer is a sum of 4 cubes.In the frame of this understandable question, we shall illustrate how mathematiciansproveweaker statements, leadcomputation,buildmodelsto comfort their belief in statements they cannot prove.

**About the speaker**

**Zoom Meet link:**

https://us06web.zoom.us/j/84175273748?pwd=T2dsNCtaa0MxRHBvZ2hEVGdVVHVSZz09

Meeting ID: 841 7527 3748

Passcode: 545014

**Title:**Zero free regions of spectral averages -functions

**Speaker:**Sandeep E. M.

**Affiliation:**Indian Statistical Institute, North East center.

**Date and Time:**September 12, 2022 at 03:30 PM

**Venue:**Seminar Hall, Kerala School of Mathematics

**Abstract:**In this talk, I would describe a recent result on the zero-free region of a weighted average of -functions of integral weight Hecke eigenforms and if time permits, some remarks on the same for those corresponding to weight zero Hecke-Maass cusp forms (level ). The first is a joint work with Satadal Ganguly.

**Title:**Families of quadratic fields with -rank at least .

**Speaker:**Azizul Hoque

**Affiliation:**Rangapara College

**Date and Time:**August 2, 2022 at 03:30 PM

**Venue:**Seminar Hall, Kerala School of Mathematics

**Abstract:**Constructing number fields with large -rank has proved to be a challenging practical problem, due in part to the fact that we believe such examples to be very rare. There is a conjecture (folklore) that the -rank of is unbounded when runs through the quadratic fields. It was Quer who constructed imaginary quadratic fields with -rank equal to , and this result still stands as the current record. We will discuss two methods for constructing quadratic fields with large -rank. We will show that for every large positive real number , there exists a sufficiently large positive constant such that the number of quadratic fields with -rank at least and absolute discriminant is . If time permits, we will construct a parametric family of real (resp. imaginary) quadratic fields with -rank at least .

**Title:**On a Conjecture of Iizuka

**Speaker:**Azizul Hoque

**Affiliation:**Rangapara College

**Date and Time:**August 1, 2022 at 03:30 PM

**Venue:**Seminar Hall, Kerala School of Mathematics

**Abstract:**For any odd prime number , we will construct an infinite family of quadruples of imaginary quadratic fields of the form , , and whose class numbers are all divisible by a given odd integer . This provides a proof of a particular case of Iizuka’s conjecture (in fact a generalization of it).

**Title:**Snippets from the history of Science in Ancient India.

**Speaker:**T. R. Govindarajan

**Affiliation:**The Institute of Mathematical Sciences

**Date and Time:**July 14, 2022 at 03:30 PM

**Venue:**Seminar Hall, Kerala School of Mathematics

**Abstract:** There are several achievements and there are failures too. Will sketch a few of these in Astronomy, Mathematics and Architecture. Will present some unsolved questions in our understanding too. Will end up with some discussions on what we gave the outside world and what we learnt.

**Title :**Physics and Maths of Braids and Knots:

**Speaker:**T. R. Govindarajan

**Affiliation:**The Institute of Mathematical Sciences

**Date and Time:**July 13, 2022 at 03:30 PM

**Venue:**Seminar Hall, Kerala School of Mathematics

**Abstract:** Braids and Knots have fascinated both Physicists and mathematicians for more than a century. I will present several developments in both the areas as well as new biological questions and end with unanswered questions.

**Title :**-units in Recurrence Sequences

**Speaker:**Sudhansu Sekhar Rout

**Affiliation:**Institute of Mathematics & Applications, Bhubaneswar

**Date and Time:**June 30, 2022 at 03:30 PM

**Venue:**Seminar Hall, Kerala School of Mathematics

**Abstract:**In this talk, we give various finiteness results concerning terms of recurrence sequences representable as a sum of -units with a fixed number of terms. We prove that under certain (necessary) conditions, the number of indices n for which allows such a representation is finite, and can be bounded in terms of the parameters involved. In this generality, our result is ineffective, i.e. we cannot bound the size of the exceptional indices. We also give an effective result, under some stronger assumptions.

**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:**On Partial Differential Equations and Diffusion Processes.

**Speaker:**Rajeeva Karandikar

**Affiliation:**Chennai Mathematical Institute.

**Date and Time:**May 26, 2022 at 11:00 AM

**Venue:**Seminar Hall, Kerala School of Mathematics.

**Abstract:** In this talk we will describe connections between second order partial differential equations and the associated Markov processes. This connection has been an active area of research for several decades.

**Title:**Iwasawa theory and Galois groups

**Speaker:**Sujatha Ramdorai

**Affiliation:**University of British Columbia.

**Date and Time:**May 26, 2022 at 03:30 PM

**Venue:**Seminar Hall, Kerala School of Mathematics.

**Abstract:** Understanding the Galois group of the field of rational numbers and of its finite extensions is one of the central problems in Number Theory. Over the last five centuries, this has led to the development of several areas within Pure Mathematics. In this talk, we shall discuss how Iwasawa theory has contributed to studying this problem.

**Title:**Black holes through different windows

**Speaker:**Ajith Parameswaran

**Affiliation:**International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore

**Date and Time:**May 25, 2022 at 03:30 PM

**Venue:**Seminar Hall, Kerala School of Mathematics.

**Abstract:** Made entirely of curved spacetime, black holes are among the most enigmatic objects in the Universe. Although early theoretical ideas on objects like black holes go back to the eighteenth century, the first rigorous mathematical formulation of a black hole was made by Karl Schwarzschild in 1915, based on Einstein’s General Theory of Relativity. A variety of astronomical observations made in the last century confirmed that black holes are not just theoretical constructs — the Universe is littered with them. Recently, a variety of astronomical observations have started to probe the detailed nature of black holes. This talk will provide a summary of this exciting journey.

**Title:**Factorization of Integers using Rational Sieve

**Speaker:**Manika Gupta

**Affiliation:**Kerala School of Mathematics

**Date and Time:**April 20, 2022, 11.00 AM

**Venue:**Seminar Hall, Kerala School of Mathematics.

**Abstract:**We present a discussion on different algorithms for the factorization of integers, focussing on the rational sieve. The rational sieve uses sieve theoretic methods to find the prime factors of a number. It is a special case of the General Number Field Sieve (GNFS) which is the most efficient classical algorithm for factoring ‘large’ integers. We also present a brief overview of the quadratic sieve and GNFS.

**Title:**A lower bound on number of representation of even numbers as sum of an odd prime and product of at most two primes

**Speaker:**Om Prakash

**Affiliation:**Kerala School of Mathematics

**Date and Time:**April 19, 2022, 03.00 PM

**Venue:**Seminar Hall, Kerala School of Mathematics.

**Abstract:**This talk is about proving a lower bound on the number of representation of large enough even numbers as sum of an odd prime and a product of at most two primes.

**Title:**TBA

**Speaker:**Pole Vennela

**Affiliation:**Kerala School of Mathematics

**Date and Time:**April 19, 2022, 11.00 AM

**Venue:**Seminar Hall, Kerala School of Mathematics.

**Abstract:**TBA

**Title:**Erdös Distance Problem

**Speaker:**Viswanathan S

**Affiliation:**Kerala School of Mathematics

**Date and Time:**April 18, 2022, 04.00 PM

**Venue:**Seminar Hall, Kerala School of Mathematics.

**Abstract:**In this talk we shall introduce Erdös Distance Conjecture. This is now a glowing hot topic in mathematics. Starting with the basic sketch of the problem we shall explore certain partial bounds for the conjecture culminating in Moser’s construction. This problem has now invited tools from several fields of mathematics, some include – Fourier analysis, graph theory, and even information theory

**Title:**Bounded Gaps between Primes

**Speaker :**Viswanathan S

**Affiliation:**Kerala School of Mathematics

**Date and Time:**April 18, 2022, 11.00 AM

**Venue:**Seminar Hall, Kerala School of Mathematics.

**Abstract:** In this talk we shall sketch the idea of Goldston, Pintz, and Yildirim that paved a way to attack the twin prime conjecture by establishing a connecting between equidistribution of primes in certain congruence classes and the gaps between primes (via sieve methods). Their idea later inspired Yitang Zhang and a chain of other mathematicians including Terrence Tao and James Maynard to reduce the infinitely occurring gap between successive primes which led to major breakthroughs in that field.

**Title:**Integers free of large prime factors

**Speaker:**Pole Vennela

**Affiliation:**Kerala School of Mathematics

**Date and Time:**April 17, 2022, 04.15 PM

**Venue:**Seminar Hall, Kerala School of Mathematics.

**Abstract: ** In this talk we are going to discuss about the number \psi(x,y) of integers < x and free of prime factors > y has been given satisfactory estimates in the regions y < (log X)3/4-6

**Title:**The Paley-Weiner Theorem

**Speaker:**Manika Gupta

**Affiliation:**Kerala School of Mathematics

**Date and Time:**April 17, 2022, 02.30 PM

**Venue:**Seminar Hall, Kerala School of Mathematics.

**Abstract:**In this talk, a proof of the Paley-Weiner theorem is presented. The theorem relates decay properties of a function at infinity with analyticity of its Fourier transform. The theorem plays a crucial role in the proof of the main large sieve inequality. We shall also discuss the background and other applications of this result.

**Title:**A lower bound on number of representation of even numbers as sum of an odd prime and product of at most two primes.

**Speaker:**Om Prakash

**Affiliation:**Kerala School of Mathematics

**Date and Time:**April 16, 2022, 11.00 AM

**Venue:**Seminar Hall, Kerala School of Mathematics.

**Abstract:**This talk is about proving a lower bound on the number of representation of large enough even numbers as sum of an odd prime and a product of at most two primes.

**Title:**Orthogonality of invariant vectors

**Speaker:**U. K. Anandavardhanan

**Affiliation:**Indian Institute of Technology Bombay

**Date and Time:**March 21, 2022 at 03:30 PM

**Venue:**Seminar Hall, Kerala School of Mathematics.

**Abstract:** This talk is about finite groups and their representation theory. Given a group and two Gelfand subgroups and of , associated to an irreducible representation of , there is a notion of and being correlated with respect to in . This notion was defined by Benedict Gross in 1991. The talk will not assume much background material. Towards the end of the talk, we’ll present some recent results regarding this theme (which are joint with Arindam Jana).

**Title:**Nice ideals, their role in ideal convergence and some thoughts

**Speaker:**Pratulananda Das

**Affiliation:**Jadavpur University

**Date and Time:**March 04, 2022 at 03:30 PM

**Venue:**Seminar Hall, Kerala School of Mathematics.

**Abstract:**In this talk we will primarily talk about the so called “nice ideals” in the realm of set theory. We will in particular describe the very basic notion of ideal convergence to see how these ideals have been found to be very useful in summability theory before moving to more deeper observations

**Title:**Nice ideals, their role in ideal convergence and some thoughts

**Speaker:**Pratulananda Das

**Affiliation:**Jadavpur University

**Date and Time:**March 03, 2022 at 03:30 PM

**Venue:**Seminar Hall, Kerala School of Mathematics.

**Abstract:**In this talk we will primarily talk about the so called “nice ideals” in the realm of set theory. We will in particular describe the very basic notion of ideal convergence to see how these ideals have been found to be very useful in summability theory before moving to more deeper observations

**Title:**On a family of elliptic curves of rank at least .

**Speaker:**Richa Sharma

**Affiliation:**PDF, Kerala School of Mathematics

**Date and Time:**04 February 2022 at 3.30 PM

**Venue:**Seminar Hall, KSoM

**Abstract:**Let be a family of elliptic curves over , where is a positive integer and are distinct odd primes. We prove that the torsion subgroup of is trivial and the -rank of this family is at least , whenever with neither nor divides .

**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.

**Speaker:**Aman Singh

**Affiliation:**Kerala School of Mathematics

**Date and Time:**November 19, 2021 at 03:30 PM

**Venue:**Seminar Hall, Kerala School of Mathematics

**Title:**Dirichlet’s Prime Distribution Theorem – II

**Abstract:**We shall see a complete proof of the fact that there are infinitely many primes in any given arithmetic progression , where .

**Speaker:**Aman Singh

**Affiliation:**Kerala School of Mathematics

**Date and Time:**November 17, 2021 at 04:00 PM

**Venue:**Seminar Hall, Kerala School of Mathematics

**Title:**Dirichlet’s Prime Distribution Theorem – I

**Abstract:**We will introduce the group of characters associated with a given finite abelian group and prove some orthogonality relations. We shall then see the Dirichlet characters and some of its properties leading up to the proof of non-vanishing of for a non-principal character .

**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:**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.

**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.

**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:**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 be a finite group. A -dimensional representation of is a homomorphism from to the group of invertible matrices over In this talk, I will discuss some interesting examples and basic results in representation theory of finite groups.

**Title:**Arakelov Geometry of Modular Curves .

**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 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 . 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 .

**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 be a finite group. A -dimensional representation of is a homomorphism from to the group of invertible matrices over 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 -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.

**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 (, even) for in the Kohnen plus space whose Petersson scalar product with a cuspidal Hecke eigenform is equal to a constant times the value We also prove that for such a form and the associated form under the Shimura-Kohnen lift the quantity 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 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 .

**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 be an adelic Hilbert cusp form of weight and level over a totally real number field . In this talk, we study the sign changes in the Fourier coefficients of 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 , the weight or the level .

**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 -convexity and matricial convexity in the setting of -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 -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 :**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 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