Speaker:  Marta Pavelka, U Miami

Title:  2-LC triangulated manifolds are exponentially many

Abstract:

We introduce "t-LC triangulated manifolds" as those triangulations obtainable from a tree of d-simplices by recursively identifying two boundary (d-1)-faces whose intersection has dimension at least d - t - 1. The t-LC notion interpolates between the class of LC manifolds introduced by Durhuus-Jonsson (corresponding to the case t = 1), and the class of all manifolds (case t = d). Benedetti-Ziegler proved that there are at most 2^(N d^2) triangulated 1-LC d-manifolds with N facets. Here we show that there are at most 2^(N/2 d^3) triangulated 2-LC d-manifolds with N facets.

We also introduce "t-constructible complexes", interpolating between constructible complexes (the case t = 1) and all complexes (case t = d). We show that all t-constructible pseudomanifolds are t-LC, and that all t-constructible complexes have (homotopical) depth larger than d - t. This extends the famous result by Hochster that constructible complexes are (homotopy) Cohen-Macaulay.

This is joint work with Bruno Benedetti. Details of the proofs and more can be found in our paper of the same title.

Marta Pavelka is a student of Bruno Benedetti.
 

Speaker:  Thomas McConville, Kennesaw State University

Title:  Lattices on shuffle words

Abstract:

The shuffle lattice is a partial order on words determined by two common types of genetic mutation: insertion and deletion. Curtis Greene discovered many remarkable enumerative properties of this lattice that are inexplicably connected to Jacobi polynomials. In this talk, I will introduce an alternate poset called the bubble lattice. This poset is obtained from the shuffle lattice by including transpositions. Using the structural relationship between bubbling and shuffling, we provide insight into Greene’s enumerative results. This talk is based on joint work with Henri Mülle. 

Thomas McConville will be visiting Khrystyna Serhiyenko

Speaker:  William Gustafson, University of Kentucky

Title:  Weak minor posets

Speaker:   Derek Hanely, University of Kentucky

Title:          Ehrhart Theory of Paving and Panhandle Matroids

Abstract:

There has been a wealth of research recently on polytopes arising from matroids. One such polytope, the matroid base polytope, is obtained as the convex hull of incidence vectors corresponding to the bases of an underlying matroid. In this talk, we will discuss a generalization of Ferroni's work on the Ehrhart theory of sparse paving matroids. In particular, we will show that the base polytope $P_M$ of any paving matroid $M$ can be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial of $P_M$, starting with Katzman's formula for the Ehrhart polynomial of a hypersimplex. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of \textit{stressed-hyperplane relaxation} introduced by Ferroni, Nasr, and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. To conclude, we will present evidence that panhandle matroids are Ehrhart positive and, as an application of the main result, compute the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes. This is joint work with Jeremy Martin, Daniel McGinnis, Dane Miyata, George Nasr, Andrés Vindas Meléndez, and Mei Yin.

Speaker:  Cyrus Hettle, Georgia Tech

Title:  Mathematically Quantifying Gerrymandering in
          Georgia’s Congressional Redistricting

Abstract:

While gerrymandering has been widely suspected in Georgia for years,
it has been difficult to quantify. We generate a large ensemble of
randomly generated non-partisan maps that are sampled from a
probability distribution which respects the geographical constraints
of the redistricting process. Using a Markov chain Monte Carlo process
and techniques involving spanning trees, we can quickly generate a
robust set of plans.

Based on historical voting data, we compare the Georgia congressional
redistricting plan enacted in 2021 with the non-partisan maps. We find
that the 2021 plan will likely be highly non-responsive to changing
opinions of the electorate, unlike the plans in the ensemble. Using
additional spatial analysis, we highlight areas where the map has been
redrawn to weaken the influence of Democratic voters.

This talk is based on joint work with Swati Gupta, Gregory Herschlag,
Jonathan Mattingly, Dana Randall, and Zhanzhan Zhao.

Discrete CATS Seminar

Speaker:  Richard Ehrenborg, University of Kentucky

Title:  Sharing Pizza in n Dimensions

Monday, February 7th 2022

2 pm POT 745

Abstract:

We introduce and prove the n-dimensional Pizza Theorem. Let ℋ be a real n-dimensional hyperplane arrangement. If K is a convex set of finite volume, the pizza quantity of K is the alternating sum of the volumes of the regions obtained by intersecting K with the arrangement ℋ. We prove that if ℋ is a Coxeter arrangement different from A_1^n such that the group of isometries W generated by the reflections in the hyperplanes of ℋ contains the negative of the identity map, and if K is a translate of a convex set that is stable under W and contains the origin, then the pizza quantity of K is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of H that we call the even restricted arrangement. We get stronger results in the case of balls. We prove that the pizza quantity of a ball containing the origin vanishes for a Coxeter arrangement ℋ with |ℋ|-n an even positive integer.

This is joint work with Sophie Morel and Margaret Readdy.

Discrete CATS Seminar

Note special day and time!

Speaker:  JiYoon Jung, Marshall University

Title:  Lattice path matroid polytopes

Abstract: A lattice path matroid is a transversal matroid
corresponding to a pair of lattice paths on the plane. A matroid base
polytope is the polytope whose vertices are the incidence vectors of
the bases of the given matroid. In this talk, we study the facial
structures of matroid base polytopes corresponding to lattice path
matroids. In the case of a border strip, we show that all faces of a
lattice path matroid polytope can be described by certain subsets of
deletions, contractions, and direct sums. In particular, we express
them in terms of the lattice path obtained from the border
strip. Subsequently, the facial structures of a lattice path matroid
polytope for a general case are explained in terms of certain tilings
of skew shapes inside the given region.

Zoom meeting number: 868 0578 9053

http://www.ms.uky.edu/~jrge/Discrete_Seminar_Fall_2021/

Speaker:  Ben Reese

                 University of Kentucky

Title:  Pipe Dream Complexes and Root Polytopes

Qualifying Exam

Committee:  M Readdy (Chair), R Ehrenborg, M Yip, Derek Young