Frontiers in Mathematics is an intensive three-day workshop in Mathematics held at Kerala School of Mathematics (KSoM), in association with the National Board for Higher Mathematics (NBHM), for advanced graduate students, research scholars and postdoctoral fellows.

It is a biannual event, held each year around March and October. Each edition is built around a simple idea: devote one full day to one topic. A leading specialist takes you from the foundations all the way to the state of the art, through a series of lectures complemented by tutorial and problem-solving sessions. It is an opportunity for focused, deep learning, and collaboration.

The lectures for this edition are as follows:

• Theory of Fourier Multipliers by Prof. Parasar Mohanty

  Abstract. In this three series lectures we will be discussing the basics of Fourier multipliers on Euclidean Groups.

  Lecture 1: Introduction to Multipliers: In this lecture, we will first briefly recall the basic properties of the Fourier transform on $\mathbb{R}^n, \mathbb{T}^n$ and $\mathbb{Z}^n$. We will then introduce the multiplier spaces $M_q^p$ and characterize the spaces $M_2^2, M_1^1$ and $M_p^1$. One can show that $M_q^p = \{0\}$ if $q < p$. For $p \ge q$, we will see that the elements of $M_q^p$ correspond to convolution operators whose kernels are tempered distributions. Finally we will prove that $M_p^p \subseteq L^\infty(\mathbb{R}^n)$.

  Lecture 2: Transference Principle: In this lecture, we will discuss transference methods and explore the relationship between the multiplier spaces $\text{of }~\mathbb{R}^{n}$, $\mathbb{T}^n$, and $\mathbb{Z}^n$.

  Lecture 3: Ball Multipliers: We will introduce Besicovitch sets and discuss one of Fefferman’s fundamental results on ball multipliers.

• Combinatorial Models in Representation Theory by Prof. K. N. Raghavan

  Characters are to representations as fingerprints are to individuals. Just as a fingerprint carries only a fragment of the information about a person yet identifies them uniquely, a character determines the representation — but does not easily give it up. In practice, retrieving structural information about a representation from its character alone is difficult: knowing someone's fingerprint tells you remarkably little about who they are as a person, only that they are this person and not another. This is precisely where combinatorial models enter. They live between representations and their characters: they carry enough information that we can recover more about a representation from them than from its character alone, yet they remain compact enough to be tractable — a middle layer that is both faithful and wieldy.

  Young tableaux are the most familiar example of such a model, and they will occupy the heart of these lectures. Through one or two carefully chosen illustrations, we will see how much of the structure of a representation can be read directly from its tableaux — how faithfully these combinatorial objects shadow the algebra. Towards the end, we hope to catch a glimpse of Littelmann's path model: a far-reaching generalization of tableaux that draws on ideas from the theory of quantum groups and from the Standard Monomial Theory of Lakshmibai, Seshadri and their collaborators. The path model reveals that the combinatorics of tableaux is not an isolated miracle, but an instance of something much deeper and more general.

• Isoperimetric inequalities by Prof. Manjunath Krishnapur

  Abstract: We start with the classical isoperimetric inequality which states that among all bodies in 3-space with a given volume, the ball has the least surface area. We shall give at least one proof. Then we discuss a few generalizations, as well as related inequalities such as the Sobolev inequality. A closely related topic is that of concentration of measure, which we shall briefly touch upon. If time permits we exhibit some surprising consequences in combinatorics. Some knowledge of measures, particularly Lebesgue measure, is assumed.

We welcome applications from motivated participants across India. Local hospitality will be provided. Limited funding is available to support travel expenses.

Registration form

More details coming soon!