"Complex analysis is the good twin and real analysis the evil one: beautiful formulas and elegant theorems seem to blossom spontaneously in the \(\mathbb{C}\)-domain, while toil and pathology rule the \(\mathbb{R}\)-domain." - Charles Pugh
Remark: For creating commutative diagrams, I swear by quiver - it's an incredibly powerful tool that has revolutionized the way I work with abstract visualizations.
"The key to understanding the arithmetic of elliptic curves and their generalizations lay in the Selmer group." - Andrew Wiles
Curious about my mathematical background? You can browse through the math courses I've taken and access my notes by clicking here. Additionally, I've curated a collection of interesting random math resources, which you can find here.
On a technical note, I do all my writing in (Neo)vim (formerly VSCode) using \( \LaTeX \) (VimTeX) on Warp terminal, and preview my documents with the Zathura pdf viewer.
Below, you'll find a selection of my presentation slides and accompanying typed expository essays on various mathematical topics that fascinate me. If you happen to stumble upon any typos or errors, please don't hesitate to reach out to me via email!| Expository Papers, Presentations, and Articles | Links |
|---|---|
| CUMC 2023 at UofT: The algebraic view on the arithmetic of elliptic curves | pdf pptx |
| CUMC 2024 at UBC: An intuitive introduction to Absolute Galois groups and its Representation | pdf pptx |
| Chess Puzzle #1 (checkmate in two!) | |
| An essay on Iwasawa Theory, The Eichler-Selberg Trace Formula and Shimura's Algebraicity Theorem | |
| Applied Cryptography in Cybersecurity | |
| Patrick Massot's tutorial (The Lean 4 tutorial project) solutions | URL |
| IISc Lean (vers. 4) program solutions | URL |
| LFTCM 2024 conference (Lean 4 vers.) problem solutions | URL |
| Natural Number Game (Lean 4 vers.) solutions | URL |
| Mathematics in Lean (Lean 4 vers. textbook) solutions | URL |
| On the weakly modular form of weight \(k\) | |
| Some Sieve Theory! | |
| Algebraic Elliptic Curves and Analytic Theory of Elliptic Functions | |
| Arithmetic Theory of Elliptic Curves | |
| Pendulums and Elliptic curve over \(\mathbb{F}_p\) | |
| Some fascinating prime numbers...! | |
| Understanding Diophantine equations using the Picard group over a algebraically closed field \(k\)? | |
| Fermat vs Waring: An Introduction to Number Theory in Function Fields | |
| A new perspective on the fundamental theorem of arithmetic | |
| Proof of Burnside's Theorem | |
| Classification of Finitely Generated Abelian Groups | |
| Is \(\pi\) transcendental? | |
| Simplicity of the alternating group \(A_n\) for \(n \ge 5\) | |
| Are primes really primes...? | |
| The Infamous IMO 1988 Question 6! | |
| Proof of Fermat's Last Theorem in One page! What...? | |
| MSRI Summer School: Automorphic Forms and the Langlands Program (problems) | |
| Abstract motivation behind Group Theory! | |
| Proof by Infinite descent...! | |
| Isomorphism between finite-dimensional vector spaces | |
| Elementary Outline on the Wiles idea for Frey's curve | |
| Does the Fermat's Last Theorem hold in \(F[x]\)? | |
| Solving quadratics w/o a formula! | |
| Idea behind EC isogeny based Public-Key Cryptography | |
| The Chinese Remainder Theorem! What about it? | |
| How the proof of FLT worked! | |
| A report on Wiles' Cambridge lecture (by K. Rubin and A. Silverberg) | |
| The legendary analytic function! | |
| Result that \(n^{\text{th}}\) root of a prime \(p\), is irrational | |
| Something that fascinated me! | |
| What the hell is a module? | |
| Galois what..., representations! | |
| Euler characteristics in Iwasawa theory and their congruences | |
| An Overview of the Taylor-Wiles Method | |
| Modularity of Rigid Galois Representation | |
| Arithmetic statistics and the Iwasawa theory of elliptic curves | |
| Unveiling the Enchanting Elegance: A Dance of Ideas in the Natural Sciences | |
| Beauty and elegance of algebra | |
| Idea's behind theoretical computer science! |
| Publications | Links |
|---|---|
| ------------------------------------------------------------------- Hopefully coming soon! ------------------------------------------------------------- |
Beal's Conjecture. Let \(A, B, C, x, y, z \in \mathbb{Z}_+\) with \(A, B, C\) pairwise coprime and \(x, y, z \ge 3\). If \(A^x + B^y = C^z\), then \(xyz \le 2\) (or \(p \mid (A, B, C)\), ie., \(A, B,\) and \(C\) have a common prime factor).
Generalized Modularity Conjecture. Every elliptic curve over an arbitrary number field, \(E_{/K}\) is modular.
Curious about some useful HTML, LaTeX and vim snippets snippets? Click here!
I recently came across a fascinating paper by Edward Frenkel, a renowned Russian-American mathematician, titled "Langlands Program and Conformal Field Theory". What struck me as particularly intriguing was how the paper demonstrated that concepts from theoretical physics can be used to prove theories within the Langlands program, a connection I hadn't previously thought possible. This paper is a must-read for mathematical physicists, and those interested in learning more about the Langlands program should check out the article "Modern Mathematics and the Langlands program" by the Institute of Advanced Study at Princeton. Additionally, "Langlands program and Physics" provides further insight into the ideas presented in Frenkel's paper.