U N I V E R S I T Y   OF   K E N T U C K Y
D I S C R E T E   C A T S   S E M I N A R 

Feb 2

Ben Braun

University of Kentucky

Flow polytope volumes and Ehrhart theory

The first half of this talk will be a survey regarding flow polytopes and their volumes. The Ehrhart h*-polynomial is a refinement of the volume of a lattice polytope. In the second half of this talk, I will discuss recent joint work with K. Bruegge, R. Davis, and D. Hanely regarding Ehrhart h*-polynomials of a class of acyclic directed graphs that we call extensions of bipartite graphs.

Feb 16

Martha Yip

University of Kentucky

Permuflows (for the working mathematician)

Danilov, Karzanov and Koshevoy described a combinatorial method for constructing a family of regular unimodular triangulations of flow polytopes. In recent years it has been shown that the dual graph of a DKK triangulation has the structure of lattice, and many interesting families of lattices arise from these triangulations.

We introduce a family of combinatorial objects called permutation flows. Permuflows capture the combinatorics of DKK triangulations in a compact way and highlights the connections to permutations. I will explain how these objects:

1. Correspond with cliques of routes.

2. Give the face poset of a DKK triangulation of the flow polytope.

3. Simplify the computation of the lattice of a DKK triangulation.

4. Lead to a formula for the h* polynomial of the flow polytope.

This is joint work with González D'León and Hanusa.

Feb 23

Jonah Berggren

University of Kentucky

Framing triangulations and posets from nontrivial netflow vectors

Danilov, Karzanov, and Koshevoy introduced framings on a directed acyclic graph and used them to induce regular unimodular “framing triangulations” on the unit flow polytope. The dual graphs of these triangulations have been shown to have the structure of the Hasse diagram of a lattice, generalizing many classical and modern families of lattices in combinatorics. Through a recent/ongoing work of González D’León, Hanusa, and Yip, this theory has led to a deeper understanding of h*-polynomials of certain flow polytopes.

Thus far, this study of framing triangulations and framing lattices has largely been limited to unit flow polytopes — i.e., from DAGs with one source, one sink, and netflow vector (1,0,…,0,-1). I will talk about my recent efforts to generalize beyond the unit case. First, I will give a notion of framed DAGs inducing unimodular framing triangulations in the full generality of arbitrary integer flow polytopes. I will conclude by reducing to a special case of framed DAGs, containing all theories of framing triangulations existing in the literature, to which we can give a theory of framing posets generalizing framing lattices in the unit case.

Mar 2

Emily Pickard

University of Kentucky

Degeneracy loci and permutation groups

Qualifying Exam

Any permutation sigma in S_n determines n^2 rank conditions on any n by n matrix. In doing so, sigma determines a degeneracy locus for a flagged vector bundle on a variety by imposing rank conditions on each fiber. The dimension of such loci and values of the rank can be obtained from diagrams constructed based on only the permutation itself. We will begin exploring these diagrams and loci with our standard symmetric group, and then move to explore other types of permutation groups and their corresponding diagrams. This work is based primarily on the work of William Fulton and David Anderson. 

Mar 9

George Nasr

Augusta University 

IDP for 2-partition maximal symmetric polytopes

The Integer Decomposition Property (IDP) for a polytope P essentially asks if the points in any scaled version of a polytope can be written as a sum of points in P itself. Despite a seemingly trite definition, asking if a polytope has IDP is among the many popular problems in discrete mathematics that has a breadth of applications, from solving other questions in discrete mathematics like understanding Ehrhart polynomials, to understanding properties of abstractly defined algebraic structures associated to polytopes, to optimization for integer programing problems whose constraints define a polytope. We provide a framework for which one can approach showing the integer decomposition property for symmetric polytopes. We utilize this framework to prove a special case which we refer to as 2-partition maximal polytopes in the case where it lies in a hyperplane of R^3. Our method involves proving a special collection of polynomials have saturated Newton polytope.