Math Research

Painting by Sebastián Iriarte

The foundation of my professional background lies in Pure Mathematics, with a focus on Algebraic Geometry, Discrete Geometry, and Combinatorics. These fields interact into what is known as Tropical Geometry, a modern area in mathematics that has produced many advancements in both pure and applied mathematics.

My research centers on the combinatorial structures underlying algebro-geometric objects, including Okounkov bodies, toroidal embeddings, and linear series. These areas present unique combinatorial challenges that are explored through the lens of valuations. Valuations are mathematical tools that measure size along a stratified scale, and facilitate the transformation from algebraic to tropical geometry.

While mathematical valuations differ from economic valuations, both concepts play a role in constructing tropical objects. For example, in microeconomics, consider a market with indivisible goods: the utility function that dictates an agent's behavior can be represented as a tropical polynomial, where the coefficients are determined by the agent's valuations. You can find more detailed discussions on this topic in the applied section of this website.

Work

• Geometry of higher rank valuations and variations of Newton-Okounkov bodies (joint with Omid Amini): With the aim of studying variation of Okounkov bodies, we study higher rank valuations in function fields of algebraic varieties and describe the underlying geometry behind this set of valuations. See also these slides of a talk on the topic I gave at the Joint Mathematics Meeting 2024 in San Francisco.
• Higher rank polyhedral geometry I: Here I present an extension of the usual theory of polyhedral geometry suitable for describing the objects in higher rank tropical geometry.

Theses

• PhD Thesis - Polyhedral, Tropical and Analytic Geometry of Higher Rank: A general set of results extending some frameworks in algebraic combinatorics to higher dimensional inputs that are lexicographically ordered. For an overview you can follow these slides.
• Master Thesis - Toric varieties and Tropical geometry: An exposition on some results on toric varieties and tropical geometry, including an overview of the proof of the semistable reduction theory in characteristic 0.
• Master Research Report on the Weil conjectures (Portuguese): A thorough exposition of the conjectures and their underlying historical context, together with a proof of them in the case of elliptic curves.

Applied Research

Painting by Sebastián Iriarte

Since earning my Bachelor's degree in Applied Mathematics, I have been frequently exposed to and interested in various forms of mathematical applications. Here, I provide a summary of some recent projects I have been involved in.

Economics (2023)

During my postdoctoral research in mathematics, I directed an engineering thesis at the intersection of economics and tropical geometry, drawing upon the work of economist Paul Klemperer in auction theory. His approach seeks to allocate multiple goods among various agents in a single auction, where agents submit bids based on their valuations of the goods. The function that maximizes their utility corresponds to a polynomial within the sum-max algebra framework.

By employing tools from tropical geometry, we interpret fundamental economic issues, such as the existence of a Walrasian competitive equilibrium, as geometric questions regarding utility functions. This initial framework was expanded to examine how shifts in valuation influence auction outcomes, utilizing concepts from higher-rank tropical geometry.

• Original Thesis of My Student: The complete engineering thesis exploring the integration of tropical geometry with auction theory. In colaboration with Universidad de Santiago de Chile.
• Slides from 2023 SIAM Conference Talk: Presentation materials from my talk at the Society for Industrial and Applied Mathematics Conference in Lafayette, LA.

Artificial Intelligence(2024)

I recently became involved in a project focused on the interpretability of machine learning models as part of the Supervised Program for AI Alignment Research (SPAR). This research estimates the local learning coefficient (LLC), within the framework of singular learning theory, for various AI architectures by analyzing the eigenvalues of the Hessian matrix. By applying this method to several toy models, including a Modular Addition Transformer and different neural networks like the MNIST MLP and Matrix Factorization MLP, we quantified an effective estimation of their model complexity. Our findings demonstrated that Hessian eigenvalue analysis effectively bounds the LLC, providing valuable insights into the flexibility and robustness of these models.

• Final AI Project Report (PDF): Comprehensive report detailing the methodology, experiments, and findings of the AI interpretability project.

Resume

Painting by Francisca Nuñez

For a detailed resume in PDF, click here.

Background Overview

I began my academic journey in the Engineering Department at Universidad de Chile, where I spent the first two years (2013-2014). Afterward, I transitioned to Brazil, completing a Bachelor's in Applied Mathematics with a specialization in Scientific Computing at Universidade Federal de Rio de Janeiro (2014-2017). Simultaneously, I pursued a Master’s in Pure Mathematics at IMPA.

Following this, I spent five years studying in France. During this time, I earned a Master's in Mathematics at Sorbonne Université (2017-2018), followed by a PhD in Mathematics (2018-2022). My doctoral research was conducted under the supervision of Omid Amini (École Polytechnique) and Marco Maculan (Sorbonne Université).

After completing my PhD, I joined the University of Texas at Austin as a Postdoctoral Researcher (2022-2024). In this role, I conducted research in mathematics; taught mathematics at the School of Natural Sciences; spent an academic semester at Stanford University and; actively participated in national and international conferences.

I have completed 41 courses in STEM, 26 of which were at the graduate level. A full list of the courses I've taken and taught in my career is here.

Teaching

• Lecturer - M408D, Integral and Multivariable Calculus years 2022/2023 and 2023/2024
  The University of Texas at Austin, College of Natural Science
• Teacher Assistant - Analysis LU2MA260, year 2021/2022
  Sorbonne Universite, licence de mathématiques
• Teacher Assistant - Numerical Analysis LU3MA232, year 2021/2022
  Sorbonne Universite, licence de mathématiques
• Teacher Assistant - Topology and Differential Geometry MAA302 year 2021/2022
  École Polytechnique, Bachelor of Science.
• Teacher Assistant - Linear Algebra MAA101 years 2018/19, 2019/2020, 2020/2021
  École Polytechnique, Bachelor of Science.

Courses

2018

• Elliptic Curves
• Model Theory

2017

• Algebraic Number Theory
• Commutative Algebra
• Advanced Algebraic Geometry
• Geometry of Algebraic Varieties
• Homological Algebra
• Scheme Theory
• Modular Forms

2016

• Algebraic Function Fields
• Elliptic Curves Seminar
• Probability Theory
• Classical Algebraic Geometry
• Research class - Project on the Weil conjectures
• Introduction to the Langlands Program
• Modular Forms

2015

• Abstract Algebra
• Complex Analysis
• Multivariable Real Analysis
• Analysis on Manifolds
• Algebra II
• Algebraic Curves

2014

• Ordinary Differential Equations
• Multivariable Calculus
• Classical Mechanics
• Discrete Mathematics for Computing
• General Topology
• Real Analysis
• Combinatorics and Graph Theory
• Measure Theory
• Functional Analysis

2013

• Algebra
• Precalculus
• Newtonian Physics
• Introduction to Engineering I
• Python Programming
• Linear Algebra
• Differential and Integral Calculus
• Newtonian Systems
• Introduction to Engineering II
• Chemistry

Elements

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