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Student Talks 2021

Table of Contents

All expository talks given by students of ISI Bangalore in the year 2021 at our Math Club are listed below.

Fall 2021

EP2102: Polygonal Number Triples #

  • Speaker: Rahil Miraj (B. Math, 2023)
  • Abstract: Through this talk, we will give a new proof of the below theorem:
    \(\textsf{Theorem.}\) The equation \(P(s, x) + P(s, y) = P(s, z)\) has infinitely many solutions in positive integers \((x, y, z)\) for any given positive integer \(s \geq 3\), where \(P(r, k)\) is the \(k-\)th \(r-\)gonal number.
    We will provide the syntax of a computer program, written in Python, which will help us to find the solutions of the above theorem according to the choice of some variables, which we will introduce in this talk. Also we will show that Ramanujan's Door Number problem, a well known problem in Number Theory, is a special case of our theorem under Triangular Numbers, i.e., when \(s=3\) and will solve it by our method.
  • Video: Available here.
  • Date and Time: Sunday, 24th October 2021, 3:00 PM - 4:45 PM
  • Venue: Online (Zoom)

EP2101: Geometry in Statistical Models #

  • Speaker: Soham Bakshi (B. Math, 2022)
  • Abstract: In this talk we shall explore the “informative geometric” structure on parametric statistical models. Historically, C.R Rao (1945) was first to introduce Riemannian metric using the Fisher information matrix. This lead to the Fisher-Rao distance between two distributions, which is basically the geodesic distance in Riemann sense. In subsequent studies, dual connections were introduced which is now the main direction of study on statistical manifolds. This geometrization of statistics gave birth to what is known now as the field of information geometry. Applications of information geometry include information theory, asymptotic theory of statistical inference and many other related branches. In this talk, we shall discuss and motivate the basic geometric setup and some foundational works of this field.
  • Suggested Background: This talk should be accessible to anyone with a background in basic probability and statistics. I will try to keep the talk more or less self-contained. However, some knowledge in differential geometry on manifolds will be beneficial.
  • Slides: Available here.
  • References:
    • C. R. Rao, Information and the accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81–89. (1945)
    • S. Amari, Barndorff-Nielsen, Kass, Lauritzen, and C. R. Rao, Differential Geometry in Statistical Inference, Lecture Notes-Monograph Series. Institute of Mathematical Statistics. (1987)
    • S. Amari, Differential geometry of curved exponential families - curvature and information loss, Ann. Stat. 10, 375–385 (1982)
    • Nikolai N. Chentsov, Statistical decision rules and optimal inference, Monographs, American Mathematical Society (1982)
  • Video: Available here.
  • Date: Sunday, 10th October 2021, 3:00 PM - 4:45 PM
  • Venue: Online (Zoom)