With Vijay Higgins, Bryan Silva, Amy Somers, David Stephens, and Parth Wokhlu. In preparation.
Given a surface ∑, one defines the Kauffman Bracket "Skein" Algebra (KBSA) Sq as formal q-linear combinations of link diagrams modulo the Kauffman Bracket Skein relations. Multiplication is given by stacking diagrams. Bonahon and Wong (2012) showed that there is a miraculous "threading" homomorphism from St to Sq for specific values of t and q. Similar to the KBSA, one can also define a PGL2-web skein algebra, and it is known that there sometimes exists an injective homomorphism from the PGL2-web skein algebra to the KBSA, e.g. in the case of the three-punctured sphere. In those cases, the threading homomorphism in KBSA induces a homomorphism in the PGL2-web skein algebra. The threading operation is fairly simple to describe in the KBSA, and the goal of the project was to find a similarly simple description of the induced threading homomorphism in the PGL2-web skein algebra. To this end, we computed the threading of the algebra generators of the PGL2-web skein algebra on the three-punctured sphere. We also consider the problem of threading (a, b, c)-webs, which are a basis of the PGL2-web skein algebra on the three-punctured sphere, and present some computational tools and conjectures.
With Cullen Kurzenknabe. Submitted.
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, we proved that this phenomenon continues to hold on n-dimensional Hartogs Triangles.
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.
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.
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.
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.
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)
This historical paper talks about the controversy between Science and Religion during the post- Renaissance era. (PDF)
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)
This historical paper talks about Jules Verne, and his contributions to modern society.
Qualified for the Texas History Day state-level competition. (PDF)