DOCTORAL DEFENSE

Speaker:  Benjamin Reese

Title:  Geometry of pipe dream complexes

Masters Exam

Speaker:  Chloe Napier

Title:  TBA

Abstract:

Speaker:  William Dugan, U Mass Amherst

Title:        Faces of generalized Pitman-Stanley polytopes

Abstract:

The Pitman-Stanley polytope is a polytope whose integer
lattice points biject onto the set of plane partitions of a certain
shape with entries in {0 ,1}. In their original paper, Pitman and
Stanley further suggest a generalization of their construction depending
on $m \in {\mathbb N}$ whose integer lattice points biject onto the set
of plane partitions of the same shape having entries in  $\{ 0 , 1, ...
, m \}$. In this talk, we give further details of this
generalized Pitman-Stanley polytope, $PS_n^m(\vec{a})$,
demonstrating that it can be realized as the flow polytope of a certain
graph. We then use the theory of flow polytopes to describe the faces of
these polytopes and produce a recurrence for their f-vectors.

William Dugan is a student of Alejandro Morales who is funding this visit.

Masters Exam

Speaker:  Chloé Napier, University of Kentucky

Title:     New Interpretations of the Two Higher Stasheff-Tamari Orders

Abstract:

In 1996, Edelman and Reiner defined the two higher Stasheff-Tamari orders on triangulations of cyclic polytopes and conjectured that they are equal. In 2021, Nicholas Williams defined new combinatorial interpretations of these two orders to make the definitions more similar. He builds upon the work by Oppermann and Thomas in the even dimensional case of giving an algebraic analog to these orders using higher Auslander-Reiten Theory. He then gives a completely new result for the odd dimensional case. In this talk, we will discuss the combinatorial interpretations of the even dimensional case and motivate the odd dimensional case and algebraic analog by example. 

See schedule

https://www.ms.uky.edu/~readdy/KOI/

See schedule https://www.ms.uky.edu/~readdy/KOI/

KOI Combinatorics Lectures

Speaker:  Richard Ehrenborg, University of Kentucky

Title:         Sharing pizza in n dimensions

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 A₁ⁿ 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.

https://www.ms.uky.edu/~readdy/KOI/

KOI Combinatorics Lectures

Speaker:  Eric Katz, Ohio State University

Title:         Models of Matroids

Abstract:

Matroids are known for having many equivalent cryptomorphic definitions, each offering a different perspective. A similar phenomenon holds in the algebraic geometric approach to them. We will discuss different ways of viewing matroids, motivated by group actions, intersection theory, and K-theory, each of which has been generalized from the realizable case to a purely combinatorial approach.

https://www.ms.uky.edu/~readdy/KOI/

Masters Exam

Speaker:  Ford McElroy, University of Kentucky

Title:         The Eulerian Transformation and Real-Rootedness

Abstract:

Many combinatorial polynomials are known to be real-rooted. Many others are conjectured to be real-rooted. The Eulerian Transformation is a map from A:R[t] --> R[t] generated by A(t^n)= A_n(t), the nth Eulerian polynomial. Brenti (1989) conjectured that the Eulerian Transformation preserves real-rootedness. In the 2022 paper The Eulerian Transformation by Brändén and Jochemko, they disprove Brenti's conjecture and make one of their own. In the talk, we will look at

 (i) polynomial properties related to real-rootedness, 

(ii) Brändén and Jochemko's counterexample to Brenti's conjecture

(iii) evidence Brändén and Jochemko provide to support their conjecture.

KOI Combinatorics Lectures

Speaker:  Lei Xue, University of Michigan

Title:  A proof of Grünbaum's Lower Bound Conjecture for polytopes, lattices, and strongly regular pseudomanifolds

Abstract:

In 1967, Grünbaum conjectured that any d-dimensional polytope with d+s ≤ 2d vertices has at least φ_k(d+s, d) = {d+1 choose k+1} + {d choose k+1} - {d+1-k \choose k+1} \] k-faces. In the talk, we will prove this conjecture and discuss equality cases. We will then extend our results to lattices with diamond property (the inequality part) and to strongly regular normal pseudomanifolds (the equality part). We will also talk about recent results on d-dimensional polytopes with 2d+1 or 2d+2 vertices.

https://www.ms.uky.edu/~readdy/KOI/