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.

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

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/

DOUBLE HEADER, PART II, 1:45 - 2:30 pm

Speaker:  Yannic Vargas, TU Graz

Title:  Hopf algebras, species and free probability

Abstract:

Free probability theory, introduced by Voiculescu, is a non-commutative probability theory where the classical notion of independence is replaced by a non-commutative analogue ("freeness"). Originally introduced in an operator-algebraic context to solve problems related to von Neumann algebras, several aspects of free probability are combinatorial in nature. For instance, it has been shown by Speicher that the relations between moments and cumulants related to non-commutative independences involve the study of non-crossing partitions. More recently, the work of Ebrahimi-Fard and Patras has provided a way to use the group of characters on a Hopf algebra of "words on words", and its corresponding Lie algebra of infinitesimal characters, to study cumulants corresponding to different types of independences (free, boolean and monotone). In this talk we will give a survey of this last construction, and present an alternative description using the notion of series of a species.

Yannic Vargas is visiting Martha Yip.

Tea in POT 745 at 3:14 pm = π time

Speaker:  Mihai Ciucu, Indiana University

Title:  Cruciform regions and a conjecture of Di Francesco

Abstract:

The problem of finding formulas for the number of tilings of lattice regions goes back to the early 1900's, when MacMahon proved (in an equivalent form) that the number of lozenge tilings of a hexagon is given by an elegant product formula. In 1992, Elkies, Kuperberg, Larsen and Propp proved that the Aztec diamond (a certain natural region on the square lattice) of order n has 2^n(n+1)/2 domino tilings. A large related body of work developed motivated by a multitude of factors, including symmetries, refinements and connections with other combinatorial objects and statistical physics. It was a model from statistical physics that motivated the conjecture which inspired the regions we will discuss in this talk.

In recent work on the twenty vertex model, Di Francesco was led to a conjecture which states that the number of domino tilings of a certain family of regions on the square lattice is given by a product formula reminiscent of the one giving the number of alternating sign matrices. These regions, denoted T_n, are obtained by starting with a square of side-length 2n, cutting it in two along a diagonal by a zigzag path with step length two, and gluing to one of the resulting regions half of an Aztec diamond of order n-1. Inspired by the regions T_n, we construct a family of cruciform regions C_m,n^a,b,c,d generalizing the Aztec diamonds and we prove that their number of domino tilings is given by a simple product formula. Since (as it follows from our results) the number of domino tilings of the region T_n is a divisor of the number of tilings of the cruciform region C_2n-1,2n-1^n-1,n,n,n-2, the special case of our formula corresponding to the latter can be viewed as partial progress towards proving Di Francesco's conjecture.

Speaker:  Steven Karp, Notre Dame

Title:  q-Whittaker functions, finite fields, and Jordan forms

Abstract:

The q-Whittaker symmetric function associated to an integer partition is a q-analogue of the Schur symmetric function. Its coefficients in the monomial basis enumerate partial flags compatible with a nilpotent endomorphism over the finite field of size 1/q. We show that considering pairs of partial flags and taking Jordan forms leads to a probabilistic bijection between nonnegative-integer matrices and pairs of semistandard tableaux of the same shape, which we call the q-Burge correspondence. In the q -> 0 limit, we recover a known description of the classical Burge correspondence (also called column RSK). We use the q-Burge correspondence to prove enumerative formulas for certain modules over the preprojective algebra of a path quiver.

This is joint work with Hugh Thomas.