Invited Speakers

Abstract: We begin with an account of Ramanujan’s life. Histories of Ramanujan’s (earlier) notebooks and lost notebook will be given. I will then describe my involvement with Ramanujan’s notebooks and lost notebook during the past 47 years. The lecture will conclude with some examples taken from Ramanujan’s notebooks and his lost notebook. Most of the lecture will be accessible to a general audience.

Abstract: Littlewood-Richardson (LR) coefficients are arguably the most celebrated numbers in all of algebraic combinatorics. They are, by definition, the multiplicative structure constants of the ring of symmetric polynomials (in a fixed number n of variables) with respect to the basis of Schur polynomials (which are parametrized by partitions with at most n parts). They have representation theoretic (and geometric) significance: they encode how the tensor product of two irreducible polynomial representations of the general linear group GL(n) decomposes as a direct sum of irreducible representations. Considering a certain natural class of submodules of the tensor product, called the Kostant-Kumar modules, we are led to the notion of refined LR coefficients. We prove a saturation theorem for the refined LR coefficients generalising the well known result, conjectured by Klaychko and proved by Knutson-Tao, for the usual LR coefficients. This talk is based on joint work with Mrigendra Singh Kushwaha and Sankaran Viswanath.

Notwithstanding the technical terms in the above paragraph, an attempt will be made to make the talk accessible to a general audience.

Abstract: In this talk, we investigate an important family of multiplicative partition functions. For these partition functions, we treat their arithmetic properties. Among others, for these functions we give

• Explicit formula and algorithm for computation;

• Average and error term;

• Upper and lower bounds;

• Estimate of Champions numbers and large values.

The main tool is the study of their associated zeta generating functions.

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Abstract: In 2004, Ashok Agarwal initiated the study of enumeration of mock theta functions using colored partitions, mock theta functions are the last invention of Ramanujan. Later, Andrews discovery of Lost Notebook triggered an extensive research on mock theta functions. Using different combinatorial tools and analogues work to Agarwal we found various combinatorial interpretations for mock theta functions. We explore mock theta functions as q−series and examine them as generating function of restricted partition functions. In addition to this, we found some basic congruences associated with mock theta functions.

Abstract: In 1916, Riesz proved that the Riemann hypothesis is equivalent to the bound associated with the Mobius function. Around the same time, Hardy and Littlewood gave another equivalent criteria for the Riemann hypothesis while correcting an identity of Ramanujan. In this talk, we shall discuss a one-variable generalization of the identity of Hardy and Littlewood and as an application, we provide Riesz-type criteria for the Riemann hypothesis. In particular, we obtain the bound given by Riesz as well as the bound of Hardy and Littlewood. This is joint work with Archit Agarwal and Meghali Garg.

Abstract: The fourth-order reaction-diffusion equations take an inevitable space in the study of evolution equations. It describes a great number of physical phenomena like thin-film theory, lubrication theory, phase transition, and many other fields. This talk addresses the blow-up and existence of solutions of one of such fourth-order parabolic problems with nonlocal sources and the Neumann boundary condition.

Abstract: Issai Schur's famous 1926 partition theorem states that the number of partitions of $n$ into distinct parts congruent to $\pm 1$ modulo $3$ is the same as the number of partitions of $n$ such that every two consecutive parts have a difference at least $3$ and that no two consecutive multiples of $3$ occur as parts. In this talk, we consider some variants of Schur's theorem, especially their Andrews--Gordon type generating functions, from the perspective of span one linked partition ideals introduced by George Andrews. Our investigation has interesting connections with basic hypergeometric series, $q$-difference equations, computer algebra, and so on.