Fall 2019

- This semester the Combinatorics Seminar meets on
**Thursdays**in**Snow 306**from**1-2pm**. - Please contact Jeremy Martin or Federico Castillo if you are interested in speaking.
- Good general seminar-attending advice, especially for graduate students: The "Three Things" Exercise for getting things out of talks by Ravi Vakil
- A good general resource (which we may use this semester):
*A Combinatorial Miscellany*by Anders Björner and Richard Stanley

**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