I worked under Dr. Hassan Safouhi to develop algorithms to efficiently compute tail probabilites using the $G_n^{(1)}$ transformation and the Slevinsky - Safouhi Formula⁶. The new algorithm reduces the complexity from the previous algorithm from $\mathcal{O}(n^4)$ to $\mathcal{O}(n)$.

Paper in progress.

During the Winter Semester of my third year, I did my undergraduate thesis under Prof. Bin Han. The title of my thesis was Linear Independence and Stability of Functions. I started off by studying preliminary material like Distribution Theory and Functional Analysis by going through Loukas Grafakos¹ Classical Fourier Analysis. Then, we proved some results which included Linear independence of compactly supported distributions and relating the Kernel of the space of semi-discrete convolutions of a distribution to the Fourier transform of the distribution.

My thesis writing has been enlightening, providing me with valuable knowledge and skills. Despite the significant time commitment, I efficiently developed my paper writing skills through guidance and feedback from my supervisor.

We also had to do a write-up on a famous classical result in our field so I chose to do it on the Poisson Summation Formula. This improved my ability to summarize complex concepts and communicate them effectively to others.

For $v=\left\{v_k\right\}_{k \in \mathbb{Z}^d} \in \ell\left(\mathbb{Z}^d\right), \; f \in \mathscr{E}^{\prime}\left(\mathbb{R}^d\right)$, we say $\{f(\cdot-k)\}_{k \in \mathbb{Z}^d}$ is linearly independent if $\sum_{k \in \mathbb{Z}^d} v_k f(\cdot-$ $k)=0 \Longrightarrow v_k=0 \; \forall k \; \in \mathbb{Z}^d$

The full thesis, including the poster, presentation and the clasical result can be accessed here [Thesis], [Poster], [Presentation], [Write-up].

Haven taken the Honors Ordinary Differential Equations (ODEs) course in the Winter '22 semester, taught by Prof. Mohammad Ali Niksirat, we were given the opportunity to work on a final project. I decided to work on proving the existence of Periodic Solutions for Second - Order Non - Autonomous Differential Equations.

$\ddot{x} =f\left(t,\; x(t),\; \dot{x} (t)\right)$ with periodic boundary conditions; $x(0)=x(\lambda),$ $\dot{x}(0)=\dot{x}(\lambda)$ where $f$ is an arbitrary function, and $\lambda \in \mathbb{R}$

Following this semester, I decided to work under Prof. Niksirat on the same project over the summer. We approached it using Degree Theory. I went through the first two chapters of Louis Nirenberg’s⁴ Topics in Non-Linear Functional Analysis. I reproved some of the theorems in the text for example the Brouwer's Fixed Point Theorem, which were the building blocks leading up to the main theorem which involved proving the existence of the solution.

The report, poster and proposal can be accessed here [Report], [Poster], [Proposal].

During the summer of 2020, after high school graduation, I discovered Professor Petra Bonfert-Taylor's course on Introduction to Complex Analysis at Wesleyan University on Coursera. This sparked my interest in Complex Analysis. In the following summer, I reached out to Prof. Manish Patnaik regarding research in the field of Analytic Number Theory, which involved studying the Moments of Quadratic Dirichlet L - Functions.

I went through the first five chapters of Michael Rosen’s¹ Number Theory in Function Fields to get a decent understanding and the suitable background required to undertake the project. The penultimate proof involved in proving the functional equation for the Dirichlet L-series was so arduous that the idea of congregating around the problem took up most of my time. As this was my first research experience, it helped me in becoming more mathematically mature and also gave me a good insight into how research is usually done.

The report and proposal can be accessed here [Report], [Proposal].