Research Background

I was trained in Computational Algebra and Inverse Galois Theory. My thesis focused on computing the loci of genus 2 curves with ((n,n))-split Jacobians, or equivalently the Hurwitz spaces of covers (\mathbb{P}^1 \to \mathbb{P}^1) with prescribed ramification, a significant computational undertaking involving Gröbner bases, invariant theory, and invariants of group actions. I computed the cases (n=3) and (n=5), and developed a general method for higher values of (n). These equations were independently verified for the first time by A. Kumar fifteen years later and remain relevant to isogeny-based cryptography and the computation of ((n,n))-isogenies.

Extending my knowledge of Hurwitz spaces, I continued working on automorphism groups of algebraic curves, fields of moduli versus fields of definition, and computational aspects of moduli spaces. Along the way, my interests expanded to algebraic geometry, arithmetic geometry, number theory, coding theory, and cryptography. This research led, among other things, to the introduction of dihedral invariants of hyperelliptic curves, weighted greatest common divisors, and weighted heights in weighted projective spaces. More recently, I have focused on the arithmetic and geometry of weighted projective varieties and their applications to coding theory and cryptography.

Over the last decade, alongside my work in arithmetic geometry, I have become increasingly involved in artificial intelligence and machine learning. This work has resulted in the development of graded neural networks, graded transformers, and quantum weighted algebraic geometry codes. My current research seeks to bridge algebra, geometry, coding theory, and AI, using algebraic structures to better understand information, computation, and learning.

Along the way, I learned some coding theory, algebraic geometry, and number theory. In the process, I discovered dihedral invariants of hyperelliptic curves, weighted gcds and weighted heights.  In the last decade, along side my research in arithmetic geometry, I have been involved in AI and Neural Networks. I invented graded neural networks, graded transformers, quantum graded weighted codes, and continue research in these areas. 

Words of Wisdom

"This poor guy is a victim of the system, one of those people who can't write down an equation even if their life depended on it."
— A famous mathematician, speaking about a "modern algebraic geometer".

"Maybe you’ll find yourself in a mediocre department where your work is intentionally undervalued, people with much lesser research records are promoted before you, and hypocrites and frauds run wild. Don’t get discouraged; don’t give up! Remember why you got into math? It wasn’t for the money, recognition, or fame—it was for that special feeling you get when you find the perfect solution or understand a beautiful argument. That hasn’t changed, son! If you still have that magical feeling, you’re doing fine."
– From Dad.

"I can't believe they destroyed a perfectly good old farm to build this damn university."
– From an old colleague.

David Hilbert’s radio address - English translation.

“There is a secret to mathematics: do what you can, not what you dreamed of doing. And try to learn from any paper that you write.”
– John Thompson, after my dissertation defense

“Certainly the best times were when I was alone with mathematics, free of ambition and pretense, and indifferent to the world.” Langlands, in Mathematicians: An Outer View of the Inner World.
-From James Milne website