Research Talks
10/2021, Goethe University Frankfurt and Bochum Joint Seminar - Polyhedral Geometry of Pivot Rules (slides)
- We introduce two new polytopal constructions for studying pivot rules for the simplex method that we call the pivot rule polytope and neighbotope. These extend the ideas of monotone path polytopes to a much broader class of pivot rules called NW-Pivot Rules. In fact, we argue that NW-pivot rules are sufficient for understanding the run-time of the simplex method in general. We describe the combinatorics of these constructions and relate them to two similar constructions: the monotone path polytope and neighbotope. Furthermore, we exhibit the permutahedron, associahedron, and Pitman-Stanley polytope as examples of these constructions. Based on joint work with Jesús De Loera, Niklas Lütjeharms, and Raman Sanyal.
09/2021, Hausdorff Institute of Mathematics Workshop on Tropical Geometry and the Geometry of Linear Programming Invited Talk - Modifications of the Shadow Vertex Pivot Rule (slides, video)
- I introduced two new pivot rules for solving linear programs on 0/1 polytopes that we call the Slim Shadow vertex pivot rule and the Ordered Shadows Vertex pivot rule. The simplex method with the slim shadow vertex pivot rule takes at most n non-degenerate steps to solve a LP, where n is the number of variables. Furthermore, it performs optimally on 0/1 polytopes with a fixed number k of nonzero coordinates and achieves a bound of at most k. The simplex method with the Ordered Shadows Vertex pivot rule takes at most dimension many non-degenerate steps to solve a 0/1 LP, which is best possible. This talk was based on joint work with Jesús De Loera, Sean Kafer, and Laura Sanità.
04/2021, (Polytop)ics: Recent advances on polytopes - Monotone Paths on Cross-polytopes (slides)
- This talk is an exposition on my paper of the same name submitted jointly with my advisor Jesús De Loera. The main results of the paper are a complete combinatorial characterization of the monotone path polytopes of cross-polytopes for a generic orientation.
02/2021, UC Davis Student-Run Research Seminar - Monotone Paths on Polyhedral Unit Balls (slides)
- This talk was an exposition of the monotone paths on cross-polytopes paper targeted at an audience of graduate students who may not know any polytope theory.
10/2019, UC Davis Student-Run Research Seminar - The Square Peg Problem for Two Curves
08/2019, MAA Mathfest 2019 - The Square Peg Problem for Two Curves (slides)
- These two talks both covered the main results of my senior thesis. Namely, there's a famous open question in topology that asks whether every simple closed curve inscribes a square. Namely, do there exist 4 points on any simple closed curve such that their convex hull is a square. This is a surprisingly difficult problem that people have attacked for more than a century. I instead stated and proved an analogous question for pairs of simple closed curves.
Expository Talks
03/2022, UC Davis Math 246 Final Project - Weight Orders of Groebner Bases and State Polytopes (slides)
- We describe a construction due to Mora and Robbiano of a polytope from any ideal of a variety called the state polytope. The vertices of this polytope correspond exactly to reduced Groebner bases of the associated ideal, and the normal fan can be constructed using weight orders. We also discuss universality of weight orders and some remarkable results for state polytopes of linear and toric ideals including a connection to the secondary polytope. This talk is based on Chapters 1 and 2 of Groebner Bases and Convex Polytopes by Sturmfels and the paper Groebner Bases of Toric Varieties also by Sturmfels.
05/2021, Bowdoin College Undergraduate Topology Course Invited Talk - Topology and Game Theory (slides)
- I provided a brief introduction to game theory and played a few games. Then I covered the necessary topology to provide Nash's proof of the existence of mixed Nash equilibria using Brouwer's fixed point theorem. The last few slides then introduce the Borsuk-Ulam Theorem and provide applications including a proof of the ham sandwich theorem and necklace splitting.