Theoretical Statistics and Mathematics Unit

Seminar and Colloquium of the week


Title : The oldest problem
Speaker : Chandan Singh Dalawat (Harish Chandra Research Institute)
Time : January 24, 2023 (Tuesday), 03:30 PM
Venue : Auditorium
Abstract : Which rational numbers are expressible as the area of a right-angled triangle all whose sides have rational lengths ? This elementary problem was posed more than a thousand years ago but remains unsolved to this day, although a lot of progress has been made in the last few decades. We will give an accessible introduction to this problem (called the congruent number problem in the literature) and present some old and new results towards its elusive solution.


Title : The Dimer Model in 3 dimensions
Speaker : Nishant Chandgotia (TIFR-CAM)
Time : January 30, 2023 (Monday), 02:00 PM
Venue : Auditorium
Abstract : The dimer model, also referred to as domino tilings or perfect matching, are tilings of the Z^d lattice by boxes exactly one of whose sides has length 2 and the rest have length 1. This is a very well-studied statistical physics model in two dimensions with many tools like height functions and Kasteleyn determinant representation coming to its aid. The higher dimensional picture is a little daunting because most of these tools are limited to two dimensions. In this talk I will describe what techniques can be extended to higher dimensions and give a brief account of a large deviations principle for dimer tilings in three dimensions that we prove analogous to the results by Cohn, Kenyon and Propp (2000). This will be a slide talk.


Title : The stable algebraic measure tree diffusion
Speaker : Roman Gambelin (INRIA-France)
Time : January 30, 2023 (Monday), 03:15 PM
Venue : Auditorium
Abstract : : In this talk, we introduce a sequence of Markov chains on cladograms with a fixed number of leaves and study its limit as the number of leaves tends to infinity. We will see that, when placed in a suitable space of trees, the chains converge to a diffusion which is symmetric for the law of a stable tree. This is an extension of a problem posed by D. Aldous in 2000 and relies on the theory of algebraic measure trees developed by A. Winter and W. Löhr in 2018 to solve this particular problem. During the talk, we will extend this theory and bridge the gap with another notion called "mass-structural equivalence" of weighted real trees which was developed concurrently by N. Forman for the same purpose.


Title :Cones in Banach spaces and Positive Operators
Speaker : TES Raghavan (University of Illinois - Chicago)
Time : January 31, 2023 (Tuesday), 03:15 PM
Venue : Auditorium
Abstract : The motivation for studying cones in real Banach spaces stems from the proof of the Perron-Frobenius theorem for non-negative irreducible matrices. The positive orthant in R^n is a convex cone with interior, and non-negative irreducible matrices for suitable powers map any non-null vector in the orthant to an interior point of the orthant. We can as well consider any convex cone or a linear semi-group in real Banach spaces and study the spectral properties of linear operators that leave various cones invariant. Yet another motivation for studying such positive operators comes from the minimax theorem of von Neumann and the general minimax theorem of Ky Fan. In studying war duels, one is lead to certain integral equations with positive kernel and the unique optimal density strategy turns out to be the eigenfunction for the spectral radius. The talk will summarize the salient theorems of Krein and Rutman, Bonsall, and the connections to game theory.

[Upcoming Seminar] [Past Seminar] [Stat Math Unit] [Indian Statistical Institute]