**Time** Feb 28, 09:30 - 10:30

**Speaker** Prof. Michel Waldschmidt

**Abstract** An asymptotic estimate for the number of integers which are sums of two squares involves a numerical constant which is called the Landau-Ramanujan constant. A similar constant occurs when counting the number of integers which are of the form x²+xy+y². The two binary forms x²+y² and x²+xy+y² are homogeneous versions of the cyclotomic polynomials Φ₂(t)=t²+1 and Φ₃(t)=t²+t+1. In joint works with Étienne Fouvry and Claude Levesque (2018) and with Etienne Fouvry (2020, 2022), we investigate the number of integers which are represented by binary forms in some families.

**Time** Feb 28, 11:00 - 12:00

**Speaker** Prof. Eknath Ghate

**Abstract** Galois representations have become fundamental tools with which to study problems in number theory. We shall give an overview of our recent work on the the explicit shape of the reductions of two-dimensional local Galois representations.

**Time** Feb 28, 12:15 - 13:15

**Speaker** Prof. Mrinal Kumar

**Abstract** Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. A straightforward algorithm for this problem is to just iteratively evaluate the polynomial at each of the inputs. The question of obtaining faster-than-naive (and ideally, close to linear time) algorithms for this problem is a natural and basic question in computational algebra. In addition to its own inherent interest, faster algorithms for multipoint evaluation are closely related to fast algorithms for other natural algebraic questions like polynomial factorization and modular composition.

In this talk, I will briefly survey the state of art for this problem, and discuss some recent improvements and applications.

**Time** Feb 28, 15:00 - 16:00

**Speaker** Prof. Kiran S Kedlaya

**Abstract** The zeta function of an algebraic variety over a finite field is a fundamental invariant introduced by Weil and studied extensively ever since. We discuss the fundamental open problem of efficiently computing the zeta function of a variety from an explicit presentation in terms of polynomials, the role of deep theoretical tools (etale and crystalline cohomology) in some partial results, and a potential improvement in the context of quantum algorithms.

**Time** Feb 28, 16:30 - 17:30

**Speaker** Prof. C S Rajan

**Abstract** We establish an analogue of the classical Polya-Vinogradov inequality for GL(2) over the finite field of p elements. In the process, we calculate the singular Gauss sums for this group.

We apply this to count the number of two by two integral matrices with matrix entries bounded by x, whose reduction modulo p is primitive, in the sense that the eigenvalues generate the finite field with p² elements.

This is joint work with Satadal Ganguly. I will also report on work done jointly with Sameer Kulkarni, computing the singular Gauss sums for GL(n) over finite fields.

**Time** Mar 01, 09:30 - 10:30

**Speaker** Prof. V Kumar Murty

**Abstract** We describe joint work with Aaron Chow in establishing a relationship between the problems of factoring integers and
computing Fourier coefficients of holomorphic cusp forms.

**Time** Mar 01, 11:00 - 12:00

**Speaker** Prof. U K Anandavardhanan

**Abstract** Given a group G and two Gelfand subgroups H and K of G, associated to an irreducible representation π of G, there is a notion of H and K being correlated with respect to π in G. This notion was defined by Benedict Gross in 1991. We discuss this theme and give some details in some specific examples (which are part of joint works with Arindam Jana and Basudev Pattanayak).

**Time** Mar 01, 12:15 - 13:15

**Speaker** Prof. Rafael Oliveira

**Abstract** Hyperbolicity cones are convex semialgebraic sets generalizing both polyhedral and spectrahedral cones, the latter forming the basic geometric sets from linear and semidefinite programming.
Hyperbolic polynomials, which give rise to these hyperbolicity cones, have recently found applications in several areas of mathematics, statistical physics, computer science, and optimization.
The general Lax conjecture is a fundamental question in real algebraic geometry and optimization: do hyperbolicity cones form a strict generalization of spectrahedral cones?

In this talk, we will give an introduction to hyperbolic polynomials and their cones, and raise several computational questions related to these objects, which blend algebraic complexity, real algebraic geometry and continued fractions.

**Time** Mar 01, 15:00 - 16:00

**Speaker** Prof. Stephan Baier

**Abstract** The question which integers can be represented by a quadratic form (in several variables) is an old topic in number theory. There is an abundance of results, and many famous mathematicians contributed to this field. As an example, Fermat proved that every prime congruent to 1 modulo 4 can be represented as a sum of two squares. More specifically, one may ask in how many ways a given integer can be represented by a quadratic form. I will review classical results on these questions by Fermat, Gauss, Legendre, Lagrange, Jacobi and Ramanujan. Afterwards, I will report about recent research on integers represented by indefinite forms in 3 variables, where the sizes of these variables are restricted in a certain way.

**Time** Mar 01, 16:30 - 17:30

**Speaker** Prof. Saurabh Kumar Singh

**Abstract** In this talk, we shall discuss about the early days of the Circle Method and the Subconvexity Problem. We will examine how the Circle Method developed into one of the most powerful techniques, and how Subconvexity became one of the central problems in Analytic Number Theory.