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.

- Office: Vincent Hall 424
- Email: torre680 [at] umn [dot] edu

# Mathematics

## Research Interests

## Recent Papers

- In collaboration with:
- Tara Gomes,
- Christopher O'Neill, and
- Aleksandra Sobieska

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.

- In collaboration with:
- Maya Banks,
- Michael K. Brown,
- Tara Gomes,
- Prashanth Sridhar, and
- Alexandre Zotine

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.

- In collaboration with:
- Tara Gomes and
- Christopher O'Neill

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

# 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)