Publications/Preprints:

- "Limiting Probability Measures" (
*submitted*) | preprint arXiv:1901.10507 - "Gaussian Radon Transform as an integral over an infinite sphere" (
*in preparation*)

Current mathematical focus:

My research interests lie in the applications of**Nonstandard Analysis**
in general.

My research interests lie in the applications of

Currently, I have been focusing on applications to probability theory.
More specifically, in a recent work, I used "integration on
hyperfinite-dimensional spheres" to model the distribution of
finite-dimensional projections of points on certain spheres as the
dimension increases to infinity. If the radii of these spheres are scaled
properly, then the limiting distribution is Gaussian. This is a classical
result with its origins in the works of Boltzmann and Maxwell on the
kinetic theory of gases, and later revisited by Poincaré in another
context.

In my work, I used nonstandard methods to construct a natural
asymptotic limiting probability space for certain sequences of probability
spaces and obtained some general results regarding limits of integrals in
terms of integrals on the limiting space. These results allow us to prove
a generalization of the aforementioned classical result of Maxwell,
Boltzmann, and Poincaré with the advantage of a different perspective
provided by nonstandard methods. An **arXiv preprint** of this work
can be found here.

Recently, Ambar Sengupta (and even more recently, Peterson-Sengupta) have used standard analysis to
study asymptotic behavior of integrals over spheres sliced by certain
affine subspaces, which is a generalization of the above situation. The
nonstandard machinery developed in the above work is naturally amenable to
this generalization as well. This is the topic of one of my ongoing
projects where I am working on getting stronger asymptotic results in this
setting. *A paper on this topic (which will also include some
discussion on the physical interpretations) is in preparation. *

Other interests that I am NOT actively working on right now:

Aside from the above, I enjoy exploring (and plan to work on some of
these in future projects) the following areas:

--- Combinatorial Number Theory
(more specifically, different measures of density of subsets of natural
numbers, and structural properties such as existence of
arithmetic/geometric progressions of prescribed lengths, etc., of those
subsets).

♦ Here is a report on an unpublished project on a
nonstandard analytic proof of Szemerédi's theorem in the spirit of
Szemerédi's original proof. This was worked on during a week-long workshop
on Nonstandard
Methods in Combinatorial Number Theory at the American Institute of Mathematics.

--- Model Theory (nonstandard
analysis can actually be described as applied model theory in some
senses).

--- Ergodic Theory (I am
mainly interested in its number theoretic applications, though I am also
interested in the "nonstandard formulations" of many purely ergodic
theoretic concepts).

--- Ramsey Theory (in
particular, ultrafilter methods in Ramsey Theory).

--- other topics in standard Probability
Theory, as well as the
probabilistic method applied to combinatorics.

Philosophy:

Other than being generally philosophically minded (and a sceptic of
sorts), my interest in philosophy grew out of working in mathematical
logic (model theory) and being curious to understand how philosophers
study/view the same topics in logic. Being a dual mathematics-philosophy
student, it is natural for me to care about the history and philosophy of
mathematics as well. Here
is a research article on the history and epistemology of infinitesimals
(and infinites) that I wrote for a Philosophy course named "Scientific
Knowledge" in 2016. I hope to polish on this and work more along this line
in future.