No seminar (please attend the AMS meeting!)
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.
Hopf Monoids: An Overview (Slides | Recording)
Hopf Monoids: An Overview (II) (Slides | Recording)
Hopf Monoids: An Overview (III) (Slides | Recording)
A Hopf Monoid on Set Families (Slides | Recording)
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.
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.
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.
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)
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