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 research scholars, postdoctoral fellows, and advanced graduate students.

We plan to make it a biannual event. 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 first edition features three lecture series:

• Quantum doubly stochastic maps and their asymptotics by Prof. B V Rajaram Bhat

• Geometry of Coxeter groups by Prof. Radhika Gupta

  Coxeter groups are an abstraction of reflection groups, i.e. subgroups of isometries of Euclidean space that are generated by finitely many reflections. Coxeter completely classified all finite Coxeter groups in 1935. These groups find applications in many diverse areas of mathematics. In this mini-course, we will study Coxeter groups (with a focus on infinite ones) from the perspective of geometric group theory. Specifically, we will build the lectures towards showing that every Coxeter group acts properly and cocompactly on a non-positively curved simply connected space (also called a CAT(0) space). We will then use this action to derive some group theoretic properties of Coxeter groups. The lectures will roughly be divided as:

  • Introduction to geometric group theory and Coxeter groups
  • Introduction to CAT(0) geometry
  • Every Coxeter group is CAT(0)
  • Applications

• Matrix positivity and its preservers, with connections to (1) symmetric functions and majorization, and (2) distance geometry by Prof. Apoorva Khare

  • Lecture 1: Matrix positivity preservers in all dimensions: Schoenberg and Rudin
  • Lecture 2: Matrix positivity preservers in fixed dimension: maximizing Schur polynomial ratios
  • Lecture 3: The history of majorization inequalities: from Newton and Muirhead to Jack and Macdonald
  • Lecture 4: Distance geometry - What is common to GPS triangulation, Euclidean metric embeddings, and Heron's formula in n dimensions?

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