Mathematics Symposium on Celebrating International Women in Mathematics Day

On Sunday, May 12, 2024, the mathematics department of Harish-Chandra Research Institute, India will host an event to celebrate International Women in Mathematics Day.

As you may know, this day honours Maryam Mirzakhani, an Iranian mathematician and Stanford University professor who was born on May 12, 1977. In 2014, Mirzakhani became the first woman to receive the Fields Medal for her work on the geometry and dynamics of Riemann surfaces. She died from breast cancer in 2017 at the age of 40. The 12th of May, her birthday, was chosen to celebrate Women in Mathematics in her memory.

To mark this occasion, we are organising a one-day online symposium featuring five talks by our guest speakers: Dr. Pratima Hebbar, Dr. Gangotryi Sorcar, Dr. Sayani Bera, Dr. Yajnaseni Dutta and Dr. Shalini Bhattacharya.

Program Schedule on May 12

Abstract: Birth-death processes are a classical model in probability to study the evolution of biological populations. Francis Galton, the half-cousin of Charles Darwin, and the infamous originator of the eugenics movement in the Victorian era, first proposed a birth-death process to investigate the extinction of aristocratic family names in England. We will introduce the Galton-Watson processes, and discuss some long-established results regarding survival vs. extinction. We then study some generalizations of this population model. The first generalization allows individuals who have differing traits. The second generalization allows individuals to undergo spatial displacement in addition to reproduction. This model is crucial in understanding how species spread across heterogeneous, fast-changing landscapes, and applies to wildlife conservation in the context of climate change, urban planning, and the study of infectious diseases.

Abstract: A simplicial complex is flag-no-square if its faces are exactly the cliques of its 1-skeleton and there are no induced 4-cycles in its 1-skeleton. A manifold is flag-no-square (fns) if it is homeomorphic to a flag-no-square simplicial complex. In dimensions 1, 2, and 3, all manifolds are fns. In dimension 4, although it is known that a low Euler characteristic is an obstruction for being fns, not many examples of fns 4-manifolds are known. In this talk, given any even number s, we will construct a fns 4-manifold M with Euler characteristic s. We will also give an idea of how to construct k! many combinatorially distinct fns triangulations on a 4-manifold whose Euler characteristic is ak+b, where a and b are constants. This is on joint work with Eran Nevo and Daniel Kalmanovich.

Abstract: In this talk, we briefly survey a few results on basins of attraction corresponding to iterated (as well as non-iterated/non-autonomous) dynamical systems of automorphisms of $\mathbb{C}^k, k \ge 2$. These basins are known to constitute pathological classes of domains in $\mathbb{C}^k, k \ge 2$, thus connecting holomorphic dynamics and function theory of several complex variables. Further, we aim to discuss an affirmative answer to a long-standing open problem (since 2000) in this context, called the Bedford's Conjecture. This is a joint work with K. Verma.

Abstract: Given an elliptic fibration of a K3 surface, one can reglue the fibres of the elliptic fibration differently to obtain a different K3 surface. These regluings are governed by a group scheme over the base of the elliptic fibration. My plan for the talk is to tell this story. Moving from curves to 3-folds, we find some very interesting group schemes related to the intermediate Jacobian of a cubic 3-fold. I will report on a joint work in progress with Mattei and Shinder where we consider the family of cubic 3-folds obtained as the hyperplane sections of a fixed smooth cubic 4-fold. The total space this time is a hyperKähler manifold. HyperKähler manifolds are nothing but higher dimensional analogues of K3 surfaces, resulting in impressive parallels with elliptic fibrations of K3 surfaces.

Abstract: We will describe the problem of mod p reduction of p-adic local Galois representations. For two dimensional crystalline representations of the local Galois group Gal(Q¯ p|Qp), the reduction can be computed using the compatibility of p-adic and mod p Local Langlands Correspondences; this method was first introduced by Christophe Breuil in 2003. We will give a sketch of the background and history of the problem and discuss few interesting patterns in the behavior of the reduction map. Most of the above results are applicable only for odd primes or sufficiently large primes. If time permits, we will look into the curious case of p = 2, which is an ongoing work with A. Venugopal. We will compare our results with some existing results for odd primes, for example with Buzzard-Gee’s results for slope 1/2 and in general with Zigzag conjecture of Ghate.

Mathematics Symposium on Celebrating International Women in Mathematics Day

Program Schedule on May 12

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