Research
My publications
_, Salvetti, L., Uncountable sets and an infinite linear order game, submitted to Topology Proceedings, 2024.
Eslava, L., Hansberg, A., _, Ventura, D., New recursive constructions of amoebas and their balancing number, submitted to Discrete Applied Mathematics, 2023.
_, Feasible Edge Replacements - Class and Algorithms (Version 1.0.0) [GitHub repository].
Caro, Y., González, I., Hansberg, A., Jácome, M., _, A. Montejano, Graphs with constant balancing number, Procedia Computer Science, 2023.
_, An alternative to back-and-forth, Topología y sus Aplicaciones, 2020.
Dissertations
The Open Coloring Axiom, MSc thesis, 2021. Advisor: Dr. Roberto Pichardo Mendoza.
Aplicaciones del lema de Rasiowa-Sikorski a la teoría de órdenes parciales y a la teoría de Ramsey, BSc thesis (in Spanish), 2019. Advisor: Dr. Roberto Pichardo Mendoza.
Open problems I'm interested in (updated Jan 2024)
Set theory
This paper proves that zero-dimensional spaces are homeomorphically determined by the isomorphism class of their lattice of uniform topologies. The general case for completely regular spaces is Problem 3.18. [I'm actively working on this].
This paper defines amoeba graphs and the very first property they prove implicitly assumes they are finite. There is no theory on infinitary amoeba graphs yet, which likely involves set theory. This has also been asked by Maya Stein. [I'm actively working on this].
Is there a generalization to the characterization of Player's II winning strategy in Baker's Game? [The answer is no]. Question 21.
Descriptive combinatorics
In this paper, it is proved that if you color the complete graph on the reals in a Baire-measurable fashion where each color class is non-meager, you can find a color-consistent member of a family of graphs where each blown-up clique is a Cantor set. The Lebesgue-measure analog of this (non-meager is replaced by non-null) is unknown, and some hints are provided in the paper.
Is there a locally checkable problem that admits a Lebesgue-measurable solutions but no Baire-measurable ones? Problem 2.14. [Likely still open].
Is the (un-pointed) Cantor topological conjugacy relation Borel? How complex is it? [Solved by someone else, it is Borel, 2023+].
Ramsey theory
A Ramsey-like result for small-dimensional simplices is proved here. The general case for higher dimensions is unknown and likely uses different techniques. [Other people are also working on this].
Analog of Hindman's theorem for concatenation in trees.
The aforementioned paper on amoeba graphs has a bunch of structural characterization problems. [Problem 4 is true].
Balanced hypergraphs based on algebraic stuff. [I'll add details and references later].
Page 6 of this proves that there are 2-adequate patterns mod m and Lemma 1 of this paper proves there are n-adequate patterns mod 2. When are there n-patterns mod m?
Any finite coloring of the integers produces monochromatic sets of the form {x, y, xy, x+y}. [Proved by someone else in 2022].
Distributed algorithms
Find more fast algorithms inspired by measurable dynamics like this one.
Other research projects and students I have mentored
Jan - Aug 2024
We leveraged advancements in Graph Neural Networks to optimize a noisy communication network with machine learning to estimate its Shannon capacity. With a masters student from Brandeis. GitHub repo.
Sept 2022 - Sept 2023
As Research Advisor, guided a HS student in developing a Python library for Ramsey-theoretical functions and presenting our project at the Canadian Undergraduate Mathematics Conference, significantly contributing to her acceptance into Brown University.
With a HS student, rediscovered a theorem in graph Ramsey theory.
More Mentoring
May 2024 - presentWith an undergrad at UofT, we study graph automorphism groups.
With an undergrad at UofT, we study variants of colored amoebas and their uses.
Masters student at the University of Genoa. Software engineering.
Sept 2020 - Jan 2021
Under the supervision of Dr. Gergely Ambrus (Alfréd Rényi Institute of Mathematics), we solved a problem about transversal properties of families of convex sets. The presentation, with MIT PhD student David Darrow, won a honorary mention at the XXXVI Coloquio Víctor Neumann Lara.
