**Next Seminar**

**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.

**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:**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.