You can find some of my formal mathematical writing here. I may write blog posts to expand on them at some point, but if you are interested in any of these topics - jump right in!

Modular Curves as Moduli Spaces

Modular curves are quotients of the upper half space by a subgroup of the full modular group SL2(Z). These objects are compact Riemann surfaces, which immediately endows them with topological, differential, and algebraic structure. Relating modular curves to a problem means we can bring a breadth of mathematics long with it. For number theorists, modular curves are useful for (at least) two problems: studying modular forms and studying torsion of elliptic curves. The focus of this paper is the latter.

Hilbert Class Field & Applications

The object of class field theory is to show how the abelian extensions of an algebraic number field K can be determined by elements drawn from a knowledge of K itself; or, if one prefers to present things in dialectical terms, how a field contains within itself the elements of its own transcending.

-Chevalley (translated)

The Levi-Civita Connection

The aim of this project is to investigate and understand the Levi-Civita connection. We motivate this exploration in two ways. The first is by viewing the Levi-Civita as a necessary component for the generalization of curvature to manifolds. The other will focus on a simpler geometric concept - the straight line. We will demonstrate two different methods of generalizing the straight line to manifolds; it will turn out that the Levi-Civita connection is the natural way to reconcile these two points of view. Finally, we will end by stating the Fundamental Theorem of Riemannian Geometry, the theorem guaranteeing the existence and uniqueness of the Levi-Civita connection.

GR-NTRU: UNDERSTANDING THE SECURITY OF LATTICE-BASED CRYPTOSYSTEMS THROUGH GROUP RINGS

Originally conceived in 1996 by authors Hoffstein, Pipher, and Silverstein, the N th-degree Truncated Ring Unit (NTRU) cryptosystem rivals common cryptosystems such as RSA in terms of speed and security. In pursuit of a deeper understanding of NTRU, we explore a generalization of the cryptosystem using group rings, known as GR-NTRU. This perspective allows for the formula- tion of a new kind of attack on NTRU-like cryptosystems. In particular, via representation theory, one can decompose a group ring into smaller matrix rings. This decomposition can greatly impact the computational complexity of lattice-based attacks on NTRU-like cryptosystems. We present a summary of how this attack affects GR-NTRU for certain classes of groups, and we end with a detailed example for the group S3.