## KU Combinatorics Seminar Fall 2019

Thursday 8/29
Organizational meeting

Thursday 9/5
Federico Castillo
Type cones for cubes

Abstract: The type cone (or deformation cone) of a polytope $$P$$ is a polyhedral cone parametrizing all weakly Minkowski summands of $$P$$. There are different descriptions of this cone; in this talk we follow McMullen's point of view and apply it to case where $$P$$ is a combinatorial cube. The main result is that the type cone of any cube is simplicial. I will explain all of these terms in the talk, assuming no prior knowledge. This is joint work with J. Doolittle, B. Goeckner, Y. Li, and M. Ross from a GRWC project.

Thursday 9/12
Federico Castillo
Gale Diagrams

Abstract: This is a hands-on introduction to Gale diagrams, which are tools to visualize high dimensional polytopes. We will go over some examples in detail to get familiarized.

Thursday 9/19
Federico Castillo
Newton polytopes of multidegrees

Abstract: The multidegree of a multiprojective variety can be seen as a polynomial encoding the intersection of the given variety with products of linear spaces. For a product of only two projective spaces, the multidegrees were classified (up to a scalar) by June Huh in 2012. Describing all possible multidegrees, even up to scalar, seem an intractable problem. We will focus on a simplified problem: we explain where this multidegrees are supported.

Thursday 9/26

Thursday 10/3
Jeremy Martin
What is... a Hopf monoid?

Abstract: Very roughly speaking, Hopf monoids provide algebraic structure to the process of putting combinatorial things together and breaking them into pieces. The things might be posets, graphs, matroids, (quasi)symmetric functions, generalized permutohedra, or something else. Hopf monoids can help understand the structure of these things and the similarities between them.

Thursday 10/10
No seminar

Thursday 10/17
No seminar

Thursday 10/24
Jeremy Martin
Positivity of Elser invariants of graphs

Abstract: Veit Elser proposed a random graph model for percolation in which physical dimension appears as a parameter. Studying this model combinatorially leads naturally to the consideration of numerical graph invariants which we call Elser numbers $$\mathsf{els}_k(G)$$, where $$G$$ is a connected graph and k a nonnegative integer. Elser had proven that $$\mathsf{els}_1(G)=0$$ for all $$G$$. By interpreting the Elser numbers as Euler characteristics of appropriate simplicial complexes called nucleus complexes, we prove that for all graphs $$G$$, they are nonpositive when $$k=0$$ and nonnegative for $$k\geq2$$. The last result confirms a conjecture of Elser. Furthermore, we give necessary and sufficient conditions, in terms of the 2-connected structure of $$G$$ for the nonvanishing of the Elser numbers. This is a project from GRWC 2018, joint with Galen Dorpalen-Barry, Cyrus Hettle, David Livingston, George Nasr, Julianne Vega, and Hays Whitlatch.

Thursday 10/31
Kevin Marshall
Generalizations of Eavesdropping Games: Greedoids and Multiple Bugs

Abstract: The eavesdropping game consists of a graph and two players, Bob the Broadcaster, and Eve the Eavesdropper. Bob and Eve each play their moves simultaneously, Bob chooses a spanning tree to broadcast, and Eve places a listening device (bug) on an edge. Eve wins if she intercepts the message, otherwise Bob wins. Clemens' Plus-One Algorithm can be used to approximate a Nash equilibrium to the original eavesdropping game. We first adapt Clemens' Plus-1 Algorithm to a more general eavesdropping game in which the underlying graph is replaced with a pure simplicial complex or greedoid. We also consider a generalization where Eve can place multiple bugs; these games come in two different categories depending on whether at least one bug, or all bugs, must be incident to Bob's spanning tree.

Thursday 11/7
Hailong Dao
Convexly generated ideals and Freiman inequality

Abstract: Let $$A$$ be a set of points in affine space whose convex hull has dimension $$d$$. A fundamental result in additive combinatorics by Freiman states that the size of $$A+A$$ is at least $$d+1|A|-\binom{d+1}{2}$$. The inequality can be rephrased as an inequality about the number of generators of the square of a monomial ideals. In this talk I will present generalizations of this result using algebraic methods, which also suggest a projective version of the original inequality.

Thursday 11/14
No seminar

Thursday 11/21
Mark Denker
Generalizing the Eagon-Reiner Theorem

Abstract: In 1990, Fröberg proved that a graph $$G$$ is chordal if and only if a certain monomial ideal $$I(G^c)$$ has a linear resolution. In this talk I discuss the connections between graphs, simplicial complexes, and monomial ideals, then give an extension of Fröberg's theorem to a different kind of graph ideals.

Thursday 11/28
No seminar (Thanksgiving)

Thursday 12/5
Federico Castillo
Everything You Wanted To Know About Linear Resolutions But Were Afraid To Ask

Thursday 12/12
TBA

For seminars from previous semesters, please see the KU Combinatorics Group page.

Last updated Thu 11/21/19