About me

I am a postdoc in the Non-linear algebra group at MPI-MIS Leipzig working with Bernd Sturmfels. Before that I was a Fuqua Research Assistant professor at the University of Miami and a Semester Postdoctoral Fellow at ICERM during the Topology in Motion program. I got my PhD at the University of Washington advised by Isabella Novik.
 

In August 2020 I will join the mathematics department at PUC-Chile as an Assistant Professor. 

I grew up in Colombia and studied mathematics at Universidad de los Andes. There I wrote an undergraduate thesis under the supervision of Federico Ardila

 Here is a link to my CV

 

 

Research and activities

My main research interests are in algebraic, geometric and topological combinatorics. Most of my work has been focussed in understanding three questions: 

  1.  What is the relationship between the combinatorial structure of a polytope and the geometry its embeddings?

  2. Why are linear orders ubiquitous in matroid theory? 

  3. Matroids enjoy several convenient properties to study them with tools from topological combinatorics. However, the toolkit requires that we study more complexes than just matroids. What are the combinatorial properties of these complexes? 


I am part of the organizing committee of the Cibercoloquio Latinoamericano de Matemáticas. A weekly online meeting where some distinguished mathematicians who speak spanish present some aspects of the work to advanced undergraduates or early career graduate students. 


Check out the following cool conferences. (DUE TO THE COVID SANITARY CRISIS BOTH EVENTS HAVE BEEN POSTPONED TO 2021)

  • Encuentro Colombiano de Combinatoria (ECCO 2020), Bogota, Colombia, June 2020. 

  • Chow Lectures. Leipzig, May 2020. 



 

Publications and preprints

 

Joint work with Alex Heaton.

Preprint.

In this paper we address the issue that different shelling orders of a simplicial complex can lead to substantially different combinatorial decompositions of the face poset.  We establish a connection between the geometry of matroid polytopes  and the above problem problem for independence complexes of matroids. In particular, we interpret some classical results of Björner geometrically and show that internal activity can be studied and deformed by using our polyhedral methods. We connect our new ideas with two old conjectures and present a series of questions that aim at making progress toward those two conjectures. Finally we introduce a SAGE package develop to experiment with a large new class of shelling orders and their associated 'activities'. 

Joint work with Federico Castillo.

 

Submitted.

The goal of the paper is to study simplicial complexes with a fixed homotopy type (i.e a wedge of a fixed number of equidimensional spheres). The main result is that there are only finitely such complexes in various classes of simplicial complexes associated to  matroids (i.e independence complex, order complex of the lattice of flats, broken circuit complexes). Using these results we outline new upper and lower bound programs for matroid complexes based on a novel set of parameters.

To appear in Journal of Combinatorial Theory Series A

We address the similarities between shifted simplicial complexes and matroid independence complexes by extending the class of ordered matroid to certain relaxed setting in which the total order of the ground-set plays an important role. The resulting classes explain many of similarities and provide a flexible combinatorial setting to study matroid invariants recursively. Notably, the Tutte polynomial extends naturally to this setting.

Joint with Karim Adiprasito and Eran Nevo.

Geometric and Functional Analysis, 26(2016), no. 2, 359-378.

We resolve a conjecture of Kalai relating approximation theory of convex bodies by simplicial polytopes to the face numbers and primitive Betti numbers of these polytopes and their toric varieties. The proof uses higher notions of chordality.  We also give asymptotically tight lower bounds on the g-numbers of the approximating polytopes in terms of their Hausdorff distance from the convex body whenever the boundary is a C^2 embedding of the sphere.
 

Joint work with: Karim Adiprasito  and Eran Nevo.

Proceedings of the American Mathematical Society, 144(2016), no. 8, 3317-3329.

We generalize the fundamental graph-theoretic notion of chordality for higher dimensional simplicial complexes by putting it into a proper context within homology theory. Then we extend some of the classical results of graph chordality to this generality, including the fundamental relation to the Leray property and chordality theorems of Dirac.

Joint work with Federico Ardila and Federico Castillo.

Electronic Journal of Combinatorics, 23 (2016), #P3.8.

We study the external activity complex of an ordered matroid as defined by Ardila and Boocher. We show that the complex is shellable and notice that a rich family of shelling orders is governed by the linear extensions of the Int/Ext poset defined by Las Vergnas for any ordered matroid.

Joint work with Steven Klee

Advances in Applied Mathematics, 67 (2015), 1-19.

We provide an extension of a conjecture of Stanley (1977) that predicts that the h-vector of a matroid is a pure O-sequence. Our extension is combinatorial and reduces to finitely many cases in fixed rank. For matroids of rank at most four we provide algorithm that proves our conjecture by constructing an explicit multicomplex. 

Relevant code for the computer assisted part of the proof can be found here.

Projects

Here is a list of projects that are in preparation and should become papers soon.

Hopf monoids of ordered simplicial complexes. 

Joint with Federico Castillo and Jeremy Martin.

We introduce a Hopf structure for every quasi-matroidal class of ordered simplicial complexes as described here
These Hopf structures are related to the ones for matroids and the Hopf monoid of permutohedra introduced by Ardila and Aguiar. The punchline is that, at the level of polytopes in the Hopf setting,  linear orders on the groundset correspond to a local work at the level of matroid polytopes.

Dual matroid polytopes, twisted line shellings and generalized activities. 

Joint with Alex Heaton. 

We discuss some novel techinques to shell independence complexes of matroids by using polytopes. Large classes of shellings in this new family lead to restrictiction set structures akin to classical internal activity, which can be identified with the theory of line shellings of dual matroid polytopes. The new activity theories can be constructed by sweeping the matroid basis polytopes with a series of hypeplanes and is constructed one face at a time. With this we propose an algortihm that conjecturally constructs a pure multicomplex for a given matroid. This gives a new approach to a classical conjecture of Stanley. 

 

Teaching

Courses at UM

January 2017 - May 2019

All course materials are available in Blackboard. 

  • MTH 210 - Linear Algebra - Spring 2018/2019

  • MTH 161 Calculus I - Fall 2018

  • MTH 113 Finite Math - Fall 2018/2017

  • MTH 309 Discrete Mathematics - Fall 2017

  • MTH 461 Modern Algebra - Spring 2017

  • MTH 163 Calculus III - Spring 2017

Summer 2017

I was a TA in an MSRI summer school organized by Eran Nevo and Raman Sanyal. The topics discussed in Eran's course, whose problem sessions I organized were Stanley-Reisner theory, Hodge theory for simplicial polytopes and higher rigidity theory.

 

Projects

Here is a list of projects that are in preparation and should become papers soon.

Hopf monoids of ordered simplicial complexes. 

Joint with Federico Castillo and Jeremy Martin.

We introduce a Hopf structure for every quasi-matroidal class of ordered simplicial complexes as described here
These Hopf structures are related to the ones for matroids and the Hopf monoid of permutohedra introduced by Ardila and Aguiar. The punchline is that, at the level of polytopes in the Hopf setting,  linear orders on the groundset correspond to a local work at the level of matroid polytopes.

Boundaries of the  5x5 completely positive cone.

Joint with Max Pfeffer

We study the boundary of the cone of symmertric 5x5 matrices that admit a positive Cholesky factorisation as a semialgebraic set. We describe the semialgebraic components by finding simple parametrizations, This allows us to compute the degree and that design several tests to understand, in this toy example, the challenges of factoring such matrices near the boundary. 

Talks and travel

Below are some future and past events and talks.

Selected conferences

Other conferences

  • SIAM Minisymposium: Recent trends in matroid theory. 
    Bern, Switzerland, 2019. 

  • AMS Special Session in geometric and topological combinatorics.
    Baltimore, MD, 2019. 

  • SIAM Minisymposium: Symmetric simplicial complexes. 
    Atlanta, GA, 2017. 

  • AMS Session in Alebraic and Topological methods in Combinatorics. 
    Seattle, WA, 2016. 

  • AMS Session in Topological Combinatorics.
    Memphis, TN, 2015.

  • AMS Session in Enumerative and Algebraic Combinatorics.
    Chicago, IL, 2015.

  • Underrepresented Students in Topology and Algebra. USTARS 2014. Berkeley, CA, 2014.

  • Graduate Student Combinatorics Conference. University of Minnesota, MN. 2013

Research seminars

  • Non-Linear Algbera Seminar Online, NASO, April 2020.

  • Insitut Camille Jordan - Université Lyon I, Combinatorics Seminar. Lyon, France, February 2020.

  • University of Primorska, Discrete Math Seminar. 
    Koeper, Slovenia, November 2019.

  • Bern-Fribourg-Neuchatel InterCity seminar. Universität Bern, Switzerland, October 2019.

  • Universität Osnabruck - Algebra and combinatorics seminar, Osnabruck, Germany, June 25, 2019. 

  • FU Berlin - Discrete Geometry Seminar, Berlin Germany, June 20, 2019. 

  • MPI-MIS Summer Seminar, Leipzig, Germany. June 17, 2019. 

  • PUC Chile - Coloquio, Santiago, Chile, May 2019.

  • Seminario Sabanero de Combinatoria - Bogotá, Colombia. May 2019.  

  • University of Miami - Combinatorics seminar. (several talks).

  • Cornell University - Discrete Geometry and Combinatorics Seminar. 18, 15.  

  • University of Kansas - Combinatorics Seminar. 18, 14.

  • University Washington - Combinatorics seminar. Many times. 

  • University of Kentucky - Discrete CATS seminar. 15. 

  • University of British Columbia - Discrete Math Seminar. 15

  • UC Berkeley - Combinatorics Seminar. 15. 

  • UCLA - Combinatorics Seminar. 15. 

  • University of Minnesota - Combinatorics Seminar. 15. 

  • University of Michigan - Combinatorics Seminar. 15

  • UQAM - Combinatorics Seminar. 15. 

  • SUNY Binghamton - Combinatorics Seminar. 15. 

  • Aalto University - Combinatorics Seminar. 15

Minicourses

  • Matroid theory in geometry, topology and algebra.
    Mexico City, March, 2019. 

 

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