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Federico Ardila-Mantilla

Publications and source records attributed to Federico Ardila-Mantilla.

9 recordsLinked to original sources
arXiv2026

Combinatorics of the Cosmohedron

The cosmohedron was recently proposed as a polytope underlying the cosmological wavefunction for $\text{Tr}(Φ^3)$ theory. Its faces were conjectured to be in bijection with Matryoshkas, which are obtained from a subdivision of a polygon by sequentially wrapping groups of polygons into larger polygons. In this paper we prove the correctness of this construction, and elucidate its combinatorial structure. Cosmohedra generalize to a wider class of $\mathcal{X}$ in $Y$ polytopes, where we chisel a polytope from the family $\mathcal{X}$ at each vertex of a polytope $Y$. We sketch a new application of these chiseled polytopes to the physics of ultraviolet divergences in loop-integrated Feynman amplitudes.

math.CO↗
arXiv2026

Tree metrics and log-concavity for matroids

We show that a set function $ν$ satisfies the gross substitutes property if and only if its homogeneous generating polynomial $Z_{q,ν}$ is a Lorentzian polynomial for all positive $q \le 1$, answering a question of Eur-Huh. We achieve this by giving a rank 1 upper bound for the distance matrix of an ultrametric tree, refining a classical result of Graham-Pollak. This characterization enables us to resolve two open problems that strengthen Mason's log-concavity conjectures for the number of independent sets of a matroid: one posed by Giansiracusa-Rincón-Schleis-Ulirsch for valuated matroids, and two posed by Dowling in 1980 and Zhao in 1985 for ordinary matroids.

math.CO↗
arXiv2024

Markov chains, CAT(0) cube complexes, and enumeration: monotone paths in a strip mix slowly

We prove that two natural Markov chains on the set of monotone paths in a strip mix slowly. To do so, we make novel use of the theory of non-positively curved (CAT(0)) cubical complexes to detect small bottlenecks in many graphs of combinatorial interest. Along the way, we give a formula for the number c_m(n) of monotone paths of length n in a strip of height m. In particular we compute the exponential growth constant of c_m(n) for arbitrary m, generalizing results of Williams for m=2, 3.

math.CO↗
arXiv2024

The tropical critical points of an affine matroid

We prove that the number of tropical critical points of an affine matroid (M,e) is equal to the beta invariant of M. Motivated by the computation of maximum likelihood degrees, this number is defined to be the degree of the intersection of the Bergman fan of (M,e) and the inverted Bergman fan of N=(M/e)*, where e is an element of M that is neither a loop nor a coloop. Equivalently, for a generic weight vector w on E-e, this is the number of ways to find weights (0,x) on M and y on N with x+y=w such that on each circuit of M (resp. N), the minimum x-weight (resp. y-weight) occurs at least twice. This answers a question of Sturmfels.

math.CO↗
arXiv2024

Intersection theory of matroids: variations on a theme

Chow rings of toric varieties, which originate in intersection theory, feature a rich combinatorial structure of independent interest. We survey four different ways of computing in these rings, due to Billera, Brion, Fulton--Sturmfels, and Allermann--Rau. We illustrate the beauty and power of these methods by giving four proofs of Huh and Huh--Katz's formula $μ^k(M) = deg_M(α^{r-k} β^k)$ for the coefficients of the reduced characteristic polynomial of a matroid $M$ as the mixed intersection numbers of the hyperplane and reciprocal hyperplane classes $α$ and $β$ in the Chow ring of $M$. Each of these proofs sheds light on a different aspect of matroid combinatorics, and provides a framework for further developments in the intersection theory of matroids. Our presentation is combinatorial, and does not assume previous knowledge of toric varieties, Chow rings, or intersection theory.

math.CO↗
arXiv2020

Todxs cuentan: cultivating diversity in combinatorics

This article describes the efforts of the SFSU-Colombia Combinatorics Initiative to build a research and learning community between California and Colombia. It seeks to broaden and deepen representation in mathematics, based on four underlying principles: 1. Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries. 2. Everyone can have joyful, meaningful, and empowering mathematical experiences. 3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs. 4. Every student deserves to be treated with dignity and respect.

math.HO↗
arXiv2020

Todxs cuentan: building community and welcoming humanity from the first day of class

Everyone can have joyful, meaningful, and empowering academic experiences; but no single academic experience is joyful, meaningful, and empowering to everyone. How do we build academic spaces where every participant can thrive? Audre Lorde advises us to use our differences to our advantage. bell hooks highlights the key role of building community while addressing power dynamics. Rochelle Gutiérrez emphasizes the importance of welcoming students' full humanity. This note discusses some efforts to implement these ideas in a university classroom, focusing on the first day of class.

math.HO↗
arXiv2020

Todxs cuentan in ECCO: community and belonging in mathematics

The Encuentro Colombiano de Combinatoria (ECCO) is an international summer school that welcomes students and researchers with a wide variety of mathematical and personal experiences. ECCO has taught us a lot about what it might mean to truly find community and belonging in a mathematical space. The goal of this article is to share a few of the lessons that we have learned from helping to build it.

math.HO↗