Some title

Some Person

USomething

Wednesday

Sept. 3, 2014

8:00-8:50am

Vincent Hall 313

Some Abstract. $e^x$

TBA

No seminar- 1st week of class

Friday

Sept. 7, 2018

3:35-4:25pm

Vincent Hall 570

No seminar- 1st week of class

Friday

Sept. 7, 2018

3:35-4:25pm

Vincent Hall 570

Ehrhart polynomial of a polytope plus dilating zonotope

Sam Hopkins

UMN

Friday

Sept. 14, 2018

3:35-4:25pm

Vincent Hall 570

In earlier joint work with Pavel Galashin, Thomas McConville, and Alex Postnikov, we introduced certain directed graphs, depending on a deformation parameter $k$, whose vertex sets are the weight lattice of a root system. We showed that there is a natural way to label the connected components of these directed graphs across values of $k$ so that the number of points in each connected component is a polynomial in $k$, which we termed an "Ehrhart-like polynomial" in analogy with the Ehrhart polynomials of lattice polytopes. We conjectured that these Ehrhart-like polynomials have nonnegative integer coefficients. This conjecture prompts us to study the polynomial which counts the number of lattice points in the Minkowski sum of a permutohedron and a dilating regular permutohedron, or more generally in the Minkowski sum of a polytope and a dilating zonotope. We extend Stanley's well-known formula for the Ehrhart polynomial of a zonotope to give a formula for this polynomial. And we then use this formula, together with a subtle integrality property of slices of permutohedra, to give a positive, combinatorial formula for the Ehrhart-like polynomials mentioned above. This is joint work with Alex Postnikov.

Science Fiction 2018

Adriano Garsia

UC San Diego

Friday

Sept. 21, 2018

3:35-4:25pm

Vincent Hall 570

In a paper with François Bergeron titled *Science Fiction and Macdonald Polynomials* we put together a package of "heuristics" that give a precise boolean splitting of the Garsia-Haiman module $M_\gamma$ for $\gamma \vdash n+1$ as an $S_n$ module. The most remarkable conjectures in that paper are still open to this date. They give a simple explicit formula for the Frobenius characteristic of $M_{\alpha_1} \cap M_{\alpha_2} \cap \cdots \cap M_{\alpha_k}$, when the partitions $\alpha_1,\alpha_2,\ldots,\alpha_k$ are immediate predecessors of $\gamma$. In this talk we review these conjectures and present recent progress on this remarkably fascinating subject. This is joint work with Guoce Xin. Slides from this talk are available online here.

Pattern inventory polynomials for consonances and dissonances

Octavio Agustín-Aquino

Universidad Tecnológica de la Mixteca

Friday

Sept. 28, 2018

3:35-4:25pm

Vincent Hall 570

Dennis White obtained formulas from the Pólya-Redfield theory for counting patterns with a certain group of automorphisms. We apply this to the problem of counting the so-called strong dichotomies, which are self-complementary rigid patterns of equitempered scales with an even number of tones. Strong dichotomies are models for the selection of consonances and dissonances in counterpoint, and are of capital importance for Guerino Mazzola's counterpoint theory. Furthermore, pattern inventory polynomials for strong dichotomies exhibit, in certain cases, a kind of cyclic sieving phenomenon, and we conjecture that there is a nice regularity for the cases when this holds true. This work was partially supported by a grant from the Niels Hendrik Abel Board.

The Rogers-Ramanujan identities- a historical view

Dennis Stanton

UMN

Friday

Oct. 5, 2018

3:35-4:25pm

Vincent Hall 570

I will survey proofs of these identities, starting with Rogers in 1894. Included are excursions into q-series, partitions, combinatorics, Lie algebras, symmetric functions, statistical mechanics, probability and finite fields. I shall conclude with recent refinements in joint work with O'Hara.

Basis shape loci and the positive Grassmannian

Cameron Marcott

University of Waterloo

Friday

Oct. 12, 2018

3:35-4:25pm

Vincent Hall 570

We study the set of $k$-dimensional planes in $\mathbb{R}^n$ admitting a basis of vectors with prescribed supports. We describe conditions on the prescribed support shape for when this set of planes has the expected dimension in the Grassmannian, and for when this set of planes intersects the positive Grassmannian in its full dimension.

Optimal switching sequence for switched linear systems

Qie He

UMN

Friday

Oct. 19, 2018

3:35-4:25pm

Vincent Hall 570

We study the following optimization problem over a dynamical system that consists of several linear subsystems: Given a finite set of n-by-n real matrices and an n-dimensional real vector, find a sequence of K matrices, each chosen from the given set of matrices, to maximize a convex function over the product of the K matrices and the given vector. This simple problem has many applications in operations research and control, yet a moderate-sized instance is challenging to solve to optimality for state-of-the-art optimization software. We propose a simple exact algorithm for this problem. The efficiency of our algorithm depends heavily on whether the given set of matrices has the oligo-vertex property, a concept we introduce for a finite set of matrices. The oligo-vertex property captures how the numbers of extreme points for a sequence of K convex polytopes related to the given matrices grow with respect to K. We derive several sufficient conditions for a set of matrices to have this property, and pose several open questions related to this property.

High acyclicity of p-subgroup complexes for the symmetric groups

Cihan Bahran

UMN

Friday

Oct. 26, 2018

3:35-4:25pm

Vincent Hall 570

For any finite group $G$ and a fixed prime $p$, the topology of the poset of nontrivial elementary abelian p-subgroups contains significant information about the representations of $G$ in characteristic $p$. For the symmetric group, this poset and its homology remain rather mysterious. A reasonable conjecture is that as we consider larger symmetric groups, the associated posets should get topologically more connected. I will rephrase this conjecture as a representation stability phenomenon ala Church-Ellenberg-Farb, and provide evidence for it by exhibiting high acyclicity of certain subposets.

Enumerating factorizations in $\mathrm{GL}_n \mathbb{F}_q$

Graham Gordon

University of Washington

Friday

Nov. 2, 2018

3:35-4:25pm

Vincent Hall 570

Recently, many people, including Huang, Lewis, Morales, Reiner, and Stanton, have enumerated certain factorizations in $\mathrm{GL}_n \mathbb{F}_q$. Much of their results are essentially $q$-analogues of factorization enumerations coming from $\mathfrak{S}_n$ in the $q \to 1$ sense. I think this work constitutes a few significant puzzle pieces in an enormous, really difficult puzzle. I will talk about another piece (or fraction of a piece) of this puzzle that I can contribute, which is an enumeration of factorizations of the identity into a product of regular-elliptic elements. Ideally, this is $q$-analogous to a result of Stanley from the 1981, which enumerates factorizations of the identity into a product of $n$-cycles. Time permitting, I will talk about my approach and/or the geometry that corresponds to this work.

Cyclic Sieving, Necklaces, and Bracelets

Eric Stucky

UMN

Friday

Nov. 16, 2018

3:35-4:25pm

Vincent Hall 570

One of the many interpretations of Catalan numbers is that $C_n$ counts necklaces with $n$ white beads and $n+1$ black beads. Motivated by this, we generalize the Catalan numbers and their $q$-analogues to many other necklaces. In this talk, we will discuss two surprising properties of these generalizations. First, they exhibit a $q=-1$ phenomenon with respect to necklace reflection, which can be extended to a cyclic sieving phenomenon for more exotic symmetries. Second, they conjecturally satisfy a "parity-unimodality" property, which in classical cases is a combinatorial shadow of a certain $\mathfrak{sl}_2$ representation.

No seminar- Thanksgiving Break

Friday

Nov. 23, 2018

3:35-4:25pm

Vincent Hall 570

Grassmann Pentagram Maps and Non-commutative Integrable Systems

Nicholas Ovenhouse

Michigan State University

Friday

Nov. 30, 2018

3:35-4:25pm

Vincent Hall 570

The pentagram map is a discrete dynamical system on the space of plane polygons. Gekhtman, Shapiro, Tabachnikov, and Vainshtein used the combinatorics and Poisson geometry associated to certain networks/quivers on surfaces to prove that this map is integrable. Recently, Mari-Beffa and Felipe introduced a version of the pentagram map on Grassmannians, and found a Lax representation. I will generalize the techniques of Gekhtman et. al. to the Grassmann case, and show this gives a "formal" integrable system in a non-commutative algebra.

Catalan Functions and k-Schur functions

Anna Pun

Drexel University

Friday

Dec. 7, 2018

3:35-4:25pm

Vincent Hall 570

Li-Chung Chen and Mark Haiman studied a family of symmetric functions called Catalan (symmetric) functions which are indexed by pairs consisting of a partition contained in the staircase (n-1, ..., 1,0) (of which there are Catalan many) and a composition weight of length n. They include the Schur functions, the Hall-Littlewood polynomials and their parabolic generalizations. They can be defined by a Demazure-operator formula, and are equal to GL-equivariant Euler characteristics of vector bundles on the flag variety by the Borel-Weil-Bott theorem. We have discovered various properties of Catalan functions, providing a new insight on the existing theorems and conjectures inspired by Macdonald positivity conjecture.
A key discovery in our work is an elegant set of ideals of roots that the associated Catalan functions are k-Schur functions and proved that graded k-Schur functions are G-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We exposed a new shift invariance property of the graded k-Schur functions and resolved the Schur positivity and k-branching conjectures by providing direct combinatorial formulas using strong marked tableaux. We conjectured that Catalan functions with a partition weight are k-Schur positive which strengthens the Schur positivity of Catalan function conjecture by Chen-Haiman and resolved the conjecture with positive combinatorial formulas in cases which capture and refine a variety of problems.
This is joint work with Jonah Blasiak, Jennifer Morse and Daniel Summers.

No seminar- final exams

Friday

Dec. 14, 2018

3:35-4:25pm

Vincent Hall 570

TBA

No seminar- 1st week of class

Friday

Jan. 25, 2019

3:35-4:25pm

Vincent Hall 570

TBA

Enumerating Linear Systems on Graphs

David Perkinson

Reed College

Friday

Feb. 1, 2019

3:35-4:25pm

Vincent Hall 570

To play the dollar game on a graph, start by assigning to each vertex a number of dollars of either wealth or debt. From this initial state, called a "divisor", the vertices lend and borrow with their neighbors according to chip-firing rules in an attempt to reach a debt-free state. The set of all possible winning states is the "complete linear system" of the divisor. We are interested in determining its cardinality. In the figure below, each vertex represents one of the 201 winning positions resulting from giving ten dollars to one vertex of the cycle graph on five vertices. This is joint work with Sarah Brauner and Forrest Glebe.

Whitney Numbers for Cones

Galen Dorpalen-Barry

UMN

Friday

Feb. 8, 2019

3:35-4:25pm

Vincent Hall 570

An arrangement of hyperplanes dissects space into connected components called chambers. A nonempty intersection of halfspaces from the arrangement will be called a cone. The number of chambers of the arrangement lying within the cone is counted by a theorem of Zaslavsky, as a sum of certain nonnegative integers that we will call the cone's "Whitney numbers of the 1st kind". For cones inside the reflection arrangement of type A (the braid arrangement), cones correspond to posets, chambers in the cone correspond to linear extensions of the poset, and these Whitney numbers refine the number of linear extensions. We present some basic facts about these Whitney numbers, and interpret them for two families of posets.

Plabic R-Matrices

Sunita Chepuri

UMN

Friday

Feb. 15, 2019

3:35-4:25pm

Vincent Hall 570

Postnikov's plabic graphs in a disk are used to parametrize totally positive Grassmannians. One of the key features of this theory is that if a plabic graph is reduced, the face weights can be uniquely recovered from boundary measurements. On surfaces more complicated than a disk this property is lost. In this talk, we investigate a certain semi-local transformation of weights for plabic networks on a cylinder that preserves boundary measurements. We call this a plabic R-matrix. We explore the properties of the plabic R-matrix, including the symmetric group action it induces on plabic networks and its underlying cluster algebra structure.

Counting factorizations in complex reflection groups

Joel Lewis

George Washington University

Friday

Feb. 22, 2019

3:35-4:25pm

Vincent Hall 570

In this talk, I'll discuss ongoing work with Alejandro Morales generalizing a 30-year old result of Jackson on permutation enumeration: we consider the enumeration of arbitrary factorizations of a Coxeter element in a well generated finite complex reflection group, keeping track of the fixed space dimension of the factors. As in the case of the symmetric group, the factorization counts are ugly, so the goal is to choose a basis for the generating function in which the answer is nice. In the case of the infinite families of monomial matrices, we accomplish this via combinatorial arguments; a notion of transitivity of a factorization appears for the "type D" group G(m, m, n). I'll also describe some puzzling partial results in the exceptional cases.

Quotients of symmetric polynomial rings deforming the cohomology of the Grassmannian

Darij Grinberg

UMN

Friday

Mar. 1, 2019

3:35-4:25pm

Vincent Hall 570

One of the many connections between Grassmannians and combinatorics is cohomological: The cohomology ring of a Grassmannian $Gr(k, n)$ is a quotient of the ring $S$ of symmetric polynomials in $k$ variables. More precisely, it is the quotient of $S$ by the ideal generated by the $k$ consecutive complete homogeneous symmetric polynomials $h_{n-k+1}, h_{n-k+2}, ..., h_n$. We propose and begin to study a deformation of this quotient, in which the ideal is instead generated by $h_{n-k+1} - a_1, h_{n-k+2} - a_2, ..., h_n - a_k$ for some $k$ fixed elements $a_1, a_2, ..., a_k$ of the base ring. This generalizes both the classical and the quantum cohomology rings of $Gr(k, n)$. We find two bases for the new quotient, as well as an $S_3$-symmetry of its structure constants, a "rim hook rule" for straightening arbitrary Schur polynomials, and a fairly complicated Pieri rule. We conjecture that the structure constants are nonnegative in an appropriate sense (treating the $a_i$ as signed indeterminate), which suggests a geometric or combinatorial meaning for the quotient. There are multiple open questions and opportunities for further research.

Affine matrix-ball construction and its relation to representation theory

Dongkwan Kim

UMN

Friday

Mar. 8, 2019

3:35-4:25pm

Vincent Hall 570

In 1985, Shi found a generalization of the Robinson-Schensted algorithm to (extended) affine symmetric groups and described their Kazhdan-Lusztig cells in terms of combinatorics. Recently, Chmutov, Lewis, Pylyavskyy, and Yudovina developed its generalization, called the affine matrix-ball construction (abbreviated AMBC). It provides a bijection from an (extended) affine symmetric group to the set of triples $(P, Q, \rho)$ where $P$ and $Q$ are row-standard Young tableaux of the same shape and $\rho$ is an integer vector satisfying certain inequalities. In this talk, I will briefly explain this algorithm, and discuss how this is related to representation of (extended) affine symmetric groups, especially the asymptotic Hecke algebras introduced by Lusztig. This work is joint with Pavlo Pylyavskyy.

Cell Decompositions for Rank Two Quiver Grassmannians

Dylan Rupel

Michigan State University

Friday

Mar. 15, 2019

3:35-4:25pm

Vincent Hall 570

A quiver Grassmannian is a variety parametrizing subrepresentations of a given quiver representation. Reineke has shown that all projective varieties can be realized as quiver Grassmannians. In this talk, I will study a class of smooth projective varieties arising as quiver Grassmannians for (truncated) preprojective representations of an n-Kronecker quiver, i.e. a quiver with two vertices and n parallel arrows between them. The main result I will present is a recursive construction of cell decompositions for these quiver Grassmannians motivated by the theory of rank two cluster algebras. If there is time I will discuss a combinatorial labeling of the cells by which their dimensions may conjecturally be directly computed. This is a report on joint work with Thorsten Weist.

No seminar- Spring Break

Friday

Mar. 22, 2019

3:35-4:25pm

Vincent Hall 570

TBA

A combinatorial duality and the Sperner property for the weak order

Christian Gaetz

MIT

Friday

Mar. 29, 2019

3:35-4:25pm

Vincent Hall 570

A poset is Sperner if its largest antichain is no larger than its largest rank. In the 1980's, Stanley used the Hard-Lefschetz Theorem to prove the Sperner property for strong Bruhat orders on Weyl groups. I will describe joint work with Yibo Gao in which we prove Stanley's conjecture that the weak Bruhat order on the symmetric group is also Sperner, by exhibiting a combinatorially-defined representation of $\mathfrak{sl}_2$ respecting the structure of the weak and strong orders. I will explain how this representation gives rise to a combinatorial duality between the weak and strong Bruhat orders and leads to a strong order analogue of Macdonald's reduced word identity for Schubert polynomials.

Topological combinatorics of crystal posets

Molly Lynch

North Carolina State University

Friday

Apr. 5, 2019

3:35-4:25pm

Vincent Hall 570

Crystal bases were first introduced by Kashiwara when studying modules of quantum groups. Each crystal base has an associated directed, edge colored graph called a crystal graph. In many cases, these crystal graphs give rise to a natural partial order. In this talk, we study crystal posets associated to highest weight representations. We use lexicographic discrete Morse functions to connect the Möbius function of an interval in a crystal poset with the relations that exist among crystal operators within that interval. We will discuss some further directions for this work.

Some congruences for sums of binomial coefficients

Moa Apagodu

Virginia Commonwealth University

Friday

Apr. 12, 2019

3:35-4:25pm

Vincent Hall 570

In a recent beautiful but technical article, William Y.C. Chen, Qing-Hu Hou, and Doron Zeilberger developed an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequences, namely those (like the Catalan and Motzkin sequences) that are expressible in terms of constant terms of powers of Laurent polynomials. We first give a leisurely exposition and then extend it in two directions. The Laurent polynomials may be of several variables, and instead of single sums we have multiple sums.

K polynomials and matroids

Andrew Berget

Western Washington University

Friday

Apr. 19, 2019

3:35-4:25pm

Vincent Hall 570

This talk is my about 2009 UMn PhD thesis problem which I have finally solved, only 10 years after the fact. The set-up was to take $n$ vectors, tensor them together in $n!$ ways, and consider their span as a representation of the symmetric group $S_n$. The problem was to see if the character of this representation could be determined knowing only the matroid of the $n$ starting vectors. In this talk I present an affirmative solution to this problem, including an explicit generating function that computes the character from the matroid. The long overdue solution took a lengthy detour through representation theory, algebraic geometry and, fusing the two, equivariant K-theory. I will keep the technical aspects to a minimum, and focus on explaining how this problem in combinatorics led to a tractable geometric problem.

Generalized Bott-Samelson resolutions for Schubert varieties

TBAEric Sommers

UMass Amherst

Friday

Apr. 26, 2019

3:35-4:25pm

Vincent Hall 570

This talk focuses on generalized Bott-Samelson resolutions of Schubert varieties. These resolutions are iterated $G/P$-bundles (for different parabolic subgroups $P$ of the algebraic group $G$). Special cases have appeared in the work of Zelevinsky, Wolper, Ryan and several other authors. After introducing these resolutions and some of their properties, we discuss a possible best resolution for a given Schubert variety $X(w)$, based on a combinatorial condition on the inversion set of the Weyl group element $w$. If time permits, we will discuss a computer program that calculates local intersection cohomology (i.e., Kazhdan-Lusztig polynomials) from these resolutions. This is joint work with Jennifer Koonz.

Combinatorics of cluster structures in Schubert varieties

Melissa Sherman-Bennett

UC Berkeley/Harvard

Friday

May 3, 2019

3:35-4:25pm

Vincent Hall 570

The (affine cone over the) Grassmannian is a prototypical example of a variety with "cluster structure"; that is, its coordinate ring is a cluster algebra. Scott (2006) gave a combinatorial description of this cluster algebra in terms of Postnikov's plabic graphs. It has been conjectured essentially since Scott's result that Schubert varieties also have a cluster structure with a description in terms of plabic graphs. I will discuss recent work with K. Serhiyenko and L. Williams proving this conjecture. The proof uses a result of Leclerc, who shows that many Richardson varieties in the full flag variety have cluster structure using cluster-category methods, and a construction of Karpman to build plabic graphs for each Schubert variety. Time permitting, I will also discuss our results on cluster structures on a larger class of positroid varieties, which involve the combinatorics of "generalized" plabic graphs.

Facial weak order in hyperplane arrangements

Aram Dermenjian

UQAM

Friday

May 10, 2019

3:35-4:25pm

Vincent Hall 570

We discuss the facial weak order, a poset structure that extends the poset of regions on a central hyperplane arrangement to the set of all faces of the arrangement which was first introduced on the braid arrangements by Krob, Latapy, Novelli, Phan and Schwer. We provide various characterizations of this poset including a global one, a local one, one using covectors and a geometric one using the associated zonotope. We then show that the facial weak order is in fact a lattice for simplicial hyperplane arrangements, generalizing a result by Björner, Edelman and Zieglar showing the poset of regions is a lattice for simplicial arrangements. We end by stating some properties on the facial weak order.

Counting core partitions and numerical semigroups using polytopes

Hayan Nam

UC Irvine

Tuesday

June 4, 2019

3:35-4:25pm

Vincent Hall 113

A partition is an $a$-core partition if none of its hook lengths are divisible by $a$. It is well known that the number of $a$-core partitions is infinite and the number of simultaneous $(a, b)$-core partitions is a generalized Catalan number if $a$ and $b$ are relatively prime. In the first half of the talk, we give an expression for the number of simultaneous $(a_1,a_2,\dots, a_k)$-core partitions that is equal to the number of integer points in a polytope. In the second half, we discuss objects closely related to core partitions, called numerical semigroups, which are additive monoids that have finite complements in the set of non-negative integers. For a numerical semigroup $S$, the genus of $S$ is the number of elements in $\mathbb{N} \setminus S$ and the multiplicity is the smallest nonzero element in $S$. In 2008, Bras-Amorós conjectured that the number of numerical semigroups with genus $g$ is increasing as $g$ increases. Later, Kaplan posed a conjecture that implies Bras-Amorós conjecture. In this talk, we prove Kaplan's conjecture when the multiplicity is 4 or 6 by counting the number of integer points in a polytope. Moreover, we find a formula for the number of numerical semigroups with multiplicity 4 and genus $g$.

Chicken Nuggets and Numerical Semigroups

Ayo Adeniran

Texas A&M

Thursday

June 6, 2019

3:35-4:25pm

Vincent Hall 113

A numerical semigroup is a subset of $\mathbb{N}$ that is closed under addition, contains 0 and has finite complement in $\mathbb{N}$. There are several fundamental invariants of a numerical semigroup $S$ among which are the Frobenius number and genus of S, denoted $F(S)$ and $g(S)$, respectively. The quotient of a numerical semigroup S by a positive integer d is the set $S/d=\{x| dx \in S\}$ which is also a numerical semigroup. In this talk, I will present some recent results showing the relation between the genus of $S/d$ and the genus of S. If time permits, we will also talk about identities relating the Frobenius numbers and the genus of quotients of numerical semigroups that are generated by certain types of arithmetic progressions. This is joint work with S. Butler, C. Defant, Y. Gao, P.E. Harris, C. Hettle, Q. Liang, H. Nam, and A. Volk.

- Seminar meets on Fridays 3:35-4:25 in room 570 of Vincent Hall.
- Seminar announcement list sign-up.
- Organizers: Sam Hopkins and Vic Reiner. Website maintained by Sam Hopkins.
- Past seminar archive.
- Future seminar site for academic year 2019-2020.
- Student Combinatorics and Algebra Seminar; meets on Thursdays.