Research and Publications

Mathematical Research (2020-current)

  1. Weak-Type Regularity and the Bergman Projection, Polymath Jr. REU (2024).

    This project concerned the Bergman Projection, which is the orthogonal projection from L2, the space of square-integrable functions, to A2, the space of square-integrable and analytic functions. By definition, the Bergman Projection is a bounded linear operator on L2 space. It is often known that the Bergman projection is bounded on Lp space for values of p different from 2. For example, the range of values (pmin, pmax) on which the Bergman Projection is bounded is already known on domains known as Hartogs Triangles. This begs the question, can more be said about the mapping properties of the Bergman Projection at the endpoints of Lp boundedness? It is known that in 2-dimensional and 3-dimensional Hartogs Triangles, the Bergman Projection satisfies a weak-type estimate at the upper endpoint pmax, but not at the lower endpoint pmin. In this project, a teammate and I succeeded in proving that the Bergman Projection on n-dimensional Hartogs Triangles indeed does not satisfy a weak-type estimate at the lower endpoint pmin. We are still working towards a proof that the Bergman Projection satisfies a weak-type estimate at the upper-endpoint pmax.

  2. An Asymptotic Analysis on the Distribution of Primes via a Continuous Extension of the Factorial Function, Mathematical Reflections 1 (2024). (PDF)

    Research Question: How can principles of Calculus extend the Factorial function to the Reals, and what is the applicability of such a function to the Asymptotic analysis of the distribution of Primes?

    This was a survey paper on analytic number theory that also served as my IB Extended Essay (a 4,000-word dissertation required to earn the IB Diploma). The first part of the paper deals with an intuitive derivation of a "real-valued" factorial function (I do not believe this derivation to be new, but I independently came up with this derivation early on in my research). This derivation leads to a unit-shift of the Gamma function (which, according to Wolfram MathWorld, was actually CF Gauss' formulation). The paper then shifts focus to a rigorous derivation of Stirling's Approximation via Laplace's Method. The climax of the paper is essentially my annotation of Srinivasa Ramanujan's beautiful two-page proof of Bertrand's Postulate (that there exists a prime p in [x, 2x] for all real x > 1). The paper only assumes high-school-level knowledge of single-variable calculus, yet ultimately reveals the non-trivial relations between number theory and calculus.

  3. Solutions for the Frobenius Number of Three Variables, Mathematical Reflections 1 (2023). (PDF)

    Abstract: In this paper we investigate a general method of solving for the Frobenius number of three coprime integers, a, b, c denoted g(a, b, c), which is the largest positive integer that cannot be expressed in the form ax+by+cz for non-negative integers x, y, z. We divide the problem into three main subcases and show how to solve for the Frobenius number in those cases and give complete solutions for all cases, except for the third case where we impose an additional condition.

Historical Research (2018-2023)

During my high school years, I was an active competitor in the Historical Paper division of the National History Day (NHD) competition series. The following is a list of the titles, descriptions and rankings (if applicable) of my papers.

  1. Goa, the Portuguese Anachronism (2023).

    This historical paper concerned the Indian Annexation of Goa, the last remaining colonial presence on the Indian subcontinent. The paper discussed various international entities' reactions in response to the Annexation of Goa as well as the effect of the said incident in the context of Cold War tensions.

  2. English Diplomacy via German Relativity: The 1919 Eclipse Expedition to Show “Light-Rays, When Near the Sun, DO NOT GO STRAIGHT” (2022).

    This historical paper concerns the diplomatic significance of the 1919 Eclipse Expedition, led by Sir Arthur Stanley Eddington, to confirm Einstein’s theory of Relativity right after WWI.

    First place in the Texas History Day 2022. Qualified for the National History Day. Appeared in The Texas Historian Journal. (PDF)

  3. A Controversy Unlike any Other: A Simple Dispute of Science and Faith, or a Politically- Fueled Power Struggle? (2021)

    This historical paper talks about the controversy between Science and Religion during the post- Renaissance era. (PDF)

  4. The Unprecedented Maxim: The Presumption of Innocence and the Revolution it Sparked on Law and Society (2020)

    This historical paper talks about the beginnings of the Presumption of Innocence and its legacy in today’s society.

    Qualified for the Texas History Day state-level competition. (PDF)

  5. From Pure Imagination to Science: Jules Verne's Literary Legacy Inspiring Triumph and Foreshadowing Tragedy in Modern Society (2019)

    This historical paper talks about Jules Verne, and his contributions to modern society.

    Qualified for the Texas History Day state-level competition. (PDF)