# David Urbanik

Graduate student interested in Math, CS, and Physics.

dburbani at uwaterloo dot ca

## Auxiliary Work

### Explicit Formulas for Abelian Varieties

Abelian varieties are fascinating objects with many extraordinary properties. They are described (in principle) by explicit polynomial equations and have a group law described (in principle) by algebraic maps. However, they are usually studied using abstract methods, and so in many cases of interest nobody can give any explicit formulas for either the varieties themselves or the maps!

Since these objects have found many useful applications in cryptography and computer science, I've been working in the direction of giving explicit formulas for them and their associated morphisms. My latest work gives explicit rational functions for the group law on hyperelliptic Jacobians of any positive genus g. Previously, this was only known for g = 1 and g = 2. See here for the paper, and here for source code which implements these equations over an arbitrary field.

### (Extra) Isogeny-based Cryptography

Here is a collection of various non-published work related to isogeny-based cryptography. This does not include my published work in this area.

### Other

This is a list of articles I have written on various miscellaneous topics, usually with the aim of presenting an original perspective on existing work. Given in reverse chronological order.

#### Abstract and Explicit Constructions of Jacobian Varieties

Masters Thesis. Contains the results of my paper on the group law for hyperelliptic Jacobians, as well as an introduction to the mathematics necessary to describe the abstract construction of the Jacobian in modern scheme-theoretic language.

#### A Brief Introduction to Schemes and Sheaves

Written to provide intuition, background, and motivation that I feel is lacking in most introductions. Best viewed as a supplement to a more detailed resource.

#### Reductions Between Families of Polynomials in Theory and in Practice

Written as an undergraduate CS thesis. Covers some results of Valiant, and some inconsequential computational work of my own.

#### A Friendly Introduction to Supersingular Isogeny Diffie-Hellman

For readers with a mathematical background of at least a first course in group theory.

#### On Notation in Multivariable Calculus

A detailed study of the function f(x,y)=x2 + y2 .

#### Quantum Physics and the Representation Theory of SU(2)

Written as a project for a class on representation theory.