Fall 2020

- This semester the Combinatorics Seminar meets on
**Fridays, 3-4pm Central Time (UTC -5 through October; UTC -6 in November).** - All meetings will take place on Zoom: meeting #963 7381 0272, passcode 970019
- Please contact Jeremy Martin if you are interested in speaking or attending.
- Good general seminar-attending advice, especially for graduate students: The "Three Things" Exercise for getting things out of talks by Ravi Vakil
- Also consider attending the Graduate Online Combinatorics Colloquium.

**Friday 8/26**

Organizational meeting

**Friday 9/4**

No seminar

**Friday 9/11**

No seminar (please attend the AMS meeting!)

**Friday 9/18**

Marge Bayer

*Lattice Polytopes from Schur Polynomials*
(Recording)

__Abstract:__
Given a polynomial (in any number of variables) we define a
polytope, called the Newton polytope, which is the convex hull of the
exponent vectors occurring (with nonzero coefficient) in the polynomial.
Information about Newton polytopes can shed light on families of
polynomials. In this talk I report on a study of some combinatorial
properties of the Newton polytopes of Schur polynomials, which enumerate
certain tableaux.
This work was the result of a GRWC 2019 project, and is joint with
Bennet Goeckner, Su Ji Hong, Tyrrell McAllister, McCabe Olsen, Casey Pinckney, Julianne Vega and Martha Yip.

**Friday 9/25**

Mark Denker

*Hopf Monoids: An Overview*
(Slides |
Recording)

**Friday 10/2**

Jeremy Martin

*Hopf Monoids: An Overview (II)*
(Slides |
Recording)

**Friday 10/9**

Jeremy Martin

*Hopf Monoids: An Overview (III)*
(Slides |
Recording)

**Friday 10/16**

Kevin Marshall

*A Hopf Monoid on Set Families*
(Slides |
Recording)

**Friday 10/23**

Jacob White (U. Texas, Rio Grande Valley)

*Combinatorial Hopf monoids and flag f-vectors*
(Recording -- second half only, sorry!)

__Abstract:__ A combinatorial Hopf monoid in species provides an
algebraic framework for understanding many polynomial and quasisymmetric
function invariants in combinatorics. In this talk, we will discuss the
problem of determining when the quasisymmetric functions associated to a
combinatorial Hopf monoid are related to the flag \(f\)-vector of a family
of relative simplicial complexes. We also discuss inequalities we obtain
for the quasisymmetric functions in this situation, and describe some
new examples of quasisymmetric functions, and combinatorial Hopf
monoids. If there is time, we will also discuss \(F\)-positivity.

**Friday 10/30**

Galen Dorpalen-Barry (U. Minnesota)

*Cones of Hyperplane Arrangements through the Varchenko-Gel'fand Ring*
(Slides |
Recording)

__Abstract:__
The coefficients of the characteristic polynomial of an arrangement in a
real vector space have many interpretations. An interesting one is
provided by the Varchenko-Gel'fand ring, which is the ring of functions
from the chambers of the arrangement to the integers with pointwise
multiplication. Varchenko and Gel'fand gave a simple presentation for
this ring, along with a filtration whose associated graded ring has its
Hilbert function given by the coefficients of the characteristic
polynomial. We generalize these results to cones defined by
intersections of halfspaces of some of the hyperplanes. Time permitting,
we will discuss Varchenko-Gel'fand analogues of some well-known results
in the Orlik-Solomon algebra regarding Koszulity and supersolvable
arrangements.

**Friday 11/6**

Jose Bastidas (Cornell)

*An interesting quotient of the Hopf monoid of generalized permutahedra*
(Recording)

__Abstract:__ We consider the Hopf monoid \(\Pi\) of generalized permutahedra modulo certain valuation relations.
Following McMullen's construction of the Polytope Algebra, we endow the resulting spaces \(\Pi[I]\) with the structure of
finite-dimensional graded algebras. The interaction between the algebra and the Hopf monoid structure leads to an
interesting question about certain "double-eigenspaces", whose dimensions turn out to count permutations with a
given number of cycles and excedances. Time permitting, we will discuss a similar result for the family of type B
generalized permutahedra.

**Friday 11/13**

Jonathan Montaño (New Mexico State University)

*Mixed volumes = Mixed multiplicities*
(Recording)

__Abstract:__ We show that the mixed volumes of arbitrary convex
bodies are equal to mixed multiplicities of graded families of monomial
ideals. This is joint work with Yairon Cid-Ruiz.
(Recording)

**Friday 12/4**

Yannic Vargas (Venezuelan Scientific Research Center)

*Hadamard product of free monoids and universal Hopf monoid*
(Recording)

__Abstract:__ The Hadamard product \(*\) is a basic operation on species which mirrors the familiar Hadamard
product of power series. Aguiar and Mahajan introduced a new operation on species \(\cdot\), based on set
compositions, which intertwines with the Hadamard product via the free monoid functor. In particular, this
provides an explicit basis for the Hadamard product of two free monoids in terms of bases of the factors. Using
the product \(\cdot\), we define the notion of \(\star\)-character of a Hopf monoid. In this work we show that the
category of such elements has a terminal object, which maps via the Fock functor to a Hopf algebra based on
permutations, related to the notion of permutation patterns.

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

Last updated Mon 12/7/20