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This page, which is intended to summarize my past, current, and future research projects, is under construction. Please check out the section <Articles> to see my existing completed projects.
Until I get a chance to write more concretely in this space, the titles and abstracts for two of my recent talks (from early June, 2023) at my undergraduate alma mater (ISI, Bangalore) could be a decent short description of a good portion of my mathematical work.
Talk 1: Nonstandard mathematical thinking
Abstract: When Leibniz worked on developing Calculus in the 1600s, he used the concept of infinitesimals which mathematicians were not able to devise firm foundations for until the 1960s, which is when Abraham Robinson found a way, using the now-available tools from mathematical logic, to rigorously work with infinitesimals. By then however, we had already abandoned the infinitesimal approach in our mathematics education and research, except for using it as a heuristic, since the epsilon-delta approach to analysis had already been successful in providing a rigorous foundation to analysis. The strength of nonstandard analysis lies in its ability to rigorize various kinds of otherwise soft heuristic arguments. Remarkably, the same ideas in nonstandard analysis that rigorize the use of infinitesimals in classical analysis can also be used in many non-analysis subjects, as illustrated by the eventual applications of the theory in diverse areas (probability theory, mathematical economics, combinatorial number theory, mathematical physics, to name a few).
This talk will introduce the audience to the nonstandard way of thinking about standard/classical mathematical objects. The first object we will think about is the number line, whose non-standard interpretation allows us to "see" infinitesimals and understand calculus in that framework. Eventually, we will encounter other objects, including (as time permits) metric spaces, topological spaces, Hilbert spaces, probability spaces, etc. No prior knowledge of mathematical logic will be assumed --- we will explain the ideas in an intuitive and philosophical manner that avoids some technical details without compromising on understanding.
This talk will serve as a prerequisite for a follow-up talk in the Probability Seminar that will focus on some recent work of the speaker that used nonstandard analysis to understand and generalize some foundational results in probability theory.
Talk 2: Some recent applications of nonstandard analysis in Probability Theory
Abstract: This talk will describe how nonstandard analytic thinking is useful for the probability theorist. In particular, we will illustrate this by presenting the intuition behind some recent work of the speaker in the following two directions:
(i) A classical result of Maxwell says that the projection (say, onto the first k coordinates, where k is fixed,) of a point sampled uniformly from a centered sphere in high dimension behaves asymptotically like the standard k-dimensional Gaussian vector, if the sphere's radius is scaled appropriately with respect to its dimension. A recent work of Sengupta generalized this result to the situation where we sample points not from the full high-dimensional sphere, but from its "sections" obtained by intersecting the spheres with certain affine planes (thus the sections look like circles on the sphere). We will explain how to think about this generalization nonstandardly and how it leads naturally to a further generalization.
(ii) A classical result of de Finetti says that any infinite sequence of exchangeable coin tosses is, in a certain sense, a mixture of independent and identically distributed coin tosses. Here, we are calling a collection of random variables exchangeable if their finite-dimensional distributions are invariant under permutations. Since it was formulated, de Finetti's theorem has been recognized as an important result for the foundations of Bayesian statistics. It was eventually generalized to exchangeable sequences of random variables taking values in more general state spaces, most notably by Hewitt and Savage whose generalization showed that the state space can be any Polish space. We will present a nonstandard way to think about the simplest situation (the coin toss) and show how that idea essentially works in showing that de Finetti's theorem holds true for all Hausdorff state spaces as long as the underlying distribution of the random variables is Radon. This shows that the topological requirements on the state space in the literature so far were somewhat artificial, as all probability measures on Polish spaces are automatically Radon.
Here are slides for a talk on April 18, 2023 at UPenn Logic Seminar. (This talk mainly summarized Ely's 2010 mathematics education/philosophy of mathematics paper "On nonstandard student conceptions about infinitesimals")