Here you can find papers, preprents, and notes. All may contain errors; please email me at skyler@skylermarks.com with any thoughts you may have.

Publications

Preprints

Notes

Skyler Marks, Elliptic Curve Cryptography, Notes, April 19, 2025.

The digital world is kept secure by cryptography. The idea behind most modern cryptographic systems is that if I have a public number (some sort of seed) and a secret number (usually called a private key) I can perform some operation with them to generate a third number (which usually called a public key). The public key is easy to generate by combining a seed and a private key, but hard to generate any other way; this allows us to verify the authenticity of a private key very easily. One such operation involves finding collinear points on an elliptic curve, giving rise to elliptic curve cryptography. This talk introduced elliptic curves over finite fields, explained how we can use their geometry to define an appropriate operation, and touched on why this operation is appropriate for cryptography. Slides for the talk can be found here.

Skyler Marks, An Invitation to Algebraic Geometry, Notes, April 19, 2025.

In high school algebra 1 and 2, we study the theory of single variable polynomial equations. In linear algebra, we study systems of linear equations, or polynomial equations with no exponents greater than 1. Algebraic Geometry combines these disciplines to study polynomial equations (in particular, their solutions) in many variables. This theory is useful as it is specific enough that we can compute with it, yet general enough that it applies to many problems we care about. The promised invitation will be extended by way of plane conics and cubics. After some definitions and preliminaries, we will review a family of classical results regarding plane conics (quadratic polynomials in two variables.

We will begin by classifying conics into families who’s members are “alike”. We will then leverage this classification to study the intersections of two conics. Our conclusion to this first act will be a detailed discussion of the number of conics passing through n points in the plane.

A follow up talk may consist of an attempt to generalize this theory to the case of plane cubics; polynomials of degree three in two variables. Through this endeavor we will introduce results from classical algebraic geometry; these may (if time permits) include Bézout’s theorem, intersection theory, and blowups; of course, for these developments, we must introduce projective geometry and some more advanced language. Finally, our third act will discuss in vague terms the position of stacks and moduli spaces in the picture, together with the categorical and algebraic foundation they necessitate. As such, these notes are incomplete.

Skyler Marks, The Kodaira Embedding Theorem, Notes, January 8, 2025.

Chow’s Theorem and GAGA are renowned results demonstrating the algebraic nature of projective manifolds and, more broadly, projective analytic varieties. However, determining if a particular manifold is projective is not, generally, a simple task. The Kodaira Embedding Theorem provides an intrinsic characterization of projective varieties in terms of line bundles; in particular, it states that a manifold is projective if and only if it admits a positive line bundle. We prove only the ‘if’ implication in this paper, giving a sufficient condition for a manifold bundle to be embedded in projective space. Along the way, we prove several other interesting results. Of particular note is the Kodaira-Nakano Vanishing Theorem, a crucial tool for eliminating higher cohomology of complex manifolds, as well as Lemmas 6.2 and 6.1, which provide important relationships between divisors, line bundles, and blowups. Although this treatment is relatively self-contained, we omit a rigorous development of Hodge theory, some basic complex analysis results, and some theorems regarding Cech cohomology (including Leray’s Theorem).

Skyler Marks, A Categorical Development of Right Derived Functors, Notes, May 12, 2024.

Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm of pure abstract category theory in their development of the field, leveraging the Freyd-Mitchell embedding theorem or similar results, or otherwise using set-theoretic language to augment a general categorical discussion. This paper seeks to demonstrate that - while it is not necessary for most mathematicians’ purposes - a development of homological concepts can be contrived from purely categorical notions. We begin by outlining the categories we will work within, namely Abelian categories (building off additive categories). We continue to develop cohomology groups of sequences, eventually culminating in a development of right derived functors. This paper is designed to be a minimalist construction, supplying no examples or motivation beyond what is necessary to develop the ideas presented.