Eduardo Torres Dávila

Math Ph.D. Candidate & Software Engineer

Eduardo headshot

Greetings! 👋 I am currently in my fifth year pursuing a Ph.D. in mathematics at the University of Minnesota - Twin Cities. Under the guidance of Christine Berkesch, I am mainly involved in the world of commutative algebra and algebraic geometry. In particular, I am studying differential operators and their uses to describe toric geometry.

Outside of mathematics I enjoy learning about new technologies and attempt to use this knowledge for mathematical research or building websites. Otherwise, you will find me reading a book, watching formula 1, or enjoying the outdoors.

Mathematics

Research Interests

Differential Operators
Differential operators are a generalization to derivatives studied in Calculus. Specifically, a differential operator is a map from one space of functions to another satisfying the general version of the product rule, called the Leibniz rule. Rings of differential operators offer a new perspective to ring structures and varieties.
Free Resolutions
A resolution is an exact sequence of modules that is used to define invariants characterizing the structure of a specific module. A finite resolution is one where only finitely many of the objects in the sequence are non-zero. For free resolutions, the modules in the resolution must be free modules over a base ring.
Numerical Semigroups
A numerical semigroup is the set of all non-negative integers except a finite number. Along with the set, one considers the binary operation of addition to get other elements of the numerical semigroup. For example, $S = \langle 3, 4, 5 \rangle$ is the numerical semigroup with integers that can be gotten by addings multiple $3, 4,$ and $5$'s together. $3, 4,$ and $5$ are called the generators of $S$.
Toric Varieties
A toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Toric varieties lend themselves nicely to new theories since they can be classified by polyhedral fans. Indeed, one can study the combinatorics of the fan to understand properties of the associated variety.

Recent Papers

Each numerical semigroup $S$ with smallest positive element $m$ corresponds to an integer point in a polyhedral cone $C_m$, known as the Kunz cone. The faces of $C_m$ form a stratification of numerical semigroups that has been shown to respect a number of algebraic properties of $S$, including the combinatorial structure of the minimal free resolution of the defining toric ideal $I_S$. In this work, we prove that the structure of the infinite free resolution of the ground field $\Bbbk$ over the semigroup algebra $\mathbb{k}[S]$ also respects this stratification, yielding a new combinatorial approach to classifying homological properties like Golodness and rationality of the poincare series in this setting. Additionally, we give a complete classification of such resolutions in the special case $m = 4$, and demonstrate that the associated graded algebras do not generally respect the same stratification.

We give an overview of a Macaulay2 package for computing with the multigraded BGG correspondence. This software builds on the package BGG due to Abo-Decker-Eisenbud-Schreyer-Smith-Stillman, which concerns the standard graded BGG correspondence. In addition to implementing the multigraded BGG functors, this package includes an implementation of differential modules and their minimal free resolutions, and it contains a method for computing strongly linear strands of multigraded free resolutions.

This paper is the third in a series of manuscripts that examine the combinatorics of the Kunz polyhedron $P_m$, whose positive integer points are in bijection with numerical semigroups (cofinite subsemigroups of $\mathbb{Z} \geq 0$) whose smallest positive element is $m$. The faces of $P_m$ are indexed by a family of finite posets (called Kunz posets) obtained from the divisibility posets of the numerical semigroups lying on a given face. In this paper, we characterize to what extent the minimal presentation of a numerical semigroup can be recovered from its Kunz poset. In doing so, we prove that all numerical semigroups lying on the interior of a given face of $P_m$ have identical minimal presentation cardinality, and we provide a combinatorial method of obtaining the dimension of a face from its corresponding Kunz poset.

Software

Technologies

Python

SageMath

Macaulay2

JavaScript

TypeScript

Java

Frameworks/Libraries

Matplotlib

NumPy

Astro

React

TailwindCSS

Spring

Previous Work

Amazon (June 2023 - September 2023)

Software Engineer Intern

Full stack web development using a great range of technologies. Extensive exposure to AWS services like DynamoDB, S3, and Lambda to handle and process big datasets. Constant development in the Java programming language for working on the backend infrastructure of team's services. Development on frontend applications using technologies like React to create responsive and dynamic pages for customers.

Latest Project

ArXiv Game

A simple online game where one tries to find the more recent arXiv article. Try it out yourself!

Co-developed with: Aaron Li

Teaching

Current Course

  • Math 1272: Calculus II - Graduate Teaching Assistant (Fall 2024)

Previous Courses

  • Math 1151: Precalculus II - Graduate Teaching Assistant (Spring 2024)
  • Math 1151: Precalculus II - Graduate Teaching Assistant (Fall 2023)
  • Math 1271: Calculus I - Graduate Teaching Assistant (Spring 2022)

Mentoring

  • UMN Combinatorics and Algebra REU - Graduate Teaching Assistant (Summer 2024)