Session Information (Location: Bruin Reception Room)

Topics in Contemporary Discrete Mathematics

Session Organizer: Jesús De Loera (University of California, Davis)
Session Speakers: Anastasia Chavez (University of California, Davis), Diego Cifuentes (Max-Planck Institute for Mathematics in the Sciences), Laura Escobar (University of Illinois at Urbana-Champaign), Jose Israel Rodriguez (University of Chicago), and Pablo Soberon (Northeastern University).

LOCATION: Bruin Reception Room

2:00 - 2:25 Laura Escobar (University of Illinois at Urbana-Champaign)
2:30 - 2:55 Jose Israel Rodriguez (University of Chicago)
3:00 - 3:30 Break
3:30 - 3:55 Diego Cifuentes (Max-Planck Institute for the Mathematics in the Sciences)
4:00 - 4:25 Anastasia Chavez (University of California, Davis)
4:30 - 4:55 Pablo Soberon (Northeastern University)

Anastasia Chavez (University of California, Davis)
Title: Dyck Paths and Positroids from Unit Interval Orders
Abstract: It is well known that the number of non-isomorphic unit interval orders on $[n]$ equals the $n$-th Catalan number. Using work of Skandera and Reed and work of Postnikov, we show that each unit interval order on $[n]$ naturally induces a rank $n$ positroid on $[2n]$. We call the positroids produced in this fashion \emph{unit interval positroids}. We characterize the unit interval positroids by describing their associated decorated permutations, showing that each one must be a $2n$-cycle encoding a Dyck path of length $2n$. We also give a combinatorial description of the $f$-vectors of unit interval orders. This is joint work with Felix Gotti.

Diego Cifuentes (Max-Planck Institute for Mathematics in the Sciences)
Title: Graphical structure in polynomial systems: Chordal networks
Abstract: The sparsity structure of a system of polynomial equations or an optimization problem can be naturally described by a graph summarizing the interactions among the decision variables. It is natural to wonder whether the structure of this graph might help in computational algebraic geometry tasks (e.g., in solving the system). In particular, the notion of chordality and treewidth play a pivotal role in related areas such as numerical linear algebra, database theory, constraint satisfaction, and graphical models. Our main contribution is the introduction of a new representation of structured polynomial systems: “chordal networks”. Chordal networks provide a computationally convenient decomposition of the system into simpler (triangular) polynomial sets, while maintaining its underlying graphical structure. We illustrate through examples from different application domains that algorithms based on chordal networks can significantly outperform existing techniques.

Laura Escobar Vega (University of Illinois at Urbana-Champaign)
Title: Brick varieties and polytopes
Abstract: The n-dimensional associahedron is a simple polytope with vertices corresponding to the triangulations of a convex (n+3)-gon. Pilaud and Santos defined the brick polytopes and used them to construct the associahedron. I will define the brick variety, show how these polytopes are constructed from this variety, and give consequences of the construction.

Jose Israel Rodriguez (University of Chicago)
Title: Factoring graphs, matrices, and polynomials as tensor products
Abstract: The tensor or Kronecker product of two matrices is well-known. The tensor product of two graphs is one whose adjacency matrix is given by the tensor product of the adjacency matrices of the respective graphs. The tensor product of two (univariate) polynomials is one whose companion matrix is given by the tensor product of the companion matrices of the respective polynomials. These tensor products are category-theoretic products in the respective categories. We discuss how graphs, matrices, and polynomials may be factored into irreducible factors with respect to these tensor products. We use the Newton--Girard formulas and homotopy continuation to find the decomposition. This is joint work with Lek-Heng Lim.

Pablo Soberon (Northeastern University)
Title: A gem in discrete geometry: Tverberg's theorem
Abstract: Tverberg's theorem is a central result of combinatorial geometry. It has far-reaching applications and variations, and has been a source of interesting open problems for over fifty years. During this talk I will describe some of my favorite variations, the different ideas involved in their proofs and how they have shaped my mathematical journey.

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