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Publications/Preprints:

Current mathematical focus:
My research interests lie in the applications of Nonstandard Analysis in general.

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      ♦  Here is a report on an unpublished project on a nonstandard analytic proof of Szemerédi's theorem in the spirit of Szemerédi's original proof.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.

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