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.

Galen Dorpalen-Barry

UMN

Friday

Feb. 8, 2019

3:35-4:25pm

Vincent Hall 570

TBA

Sunita Chepuri

UMN

Friday

Feb. 15, 2019

3:35-4:25pm

Vincent Hall 570

TBA

Joel Lewis

George Washington University

Friday

Feb. 22, 2019

3:35-4:25pm

Vincent Hall 570

TBA

TBA

Friday

Mar. 1, 2019

3:35-4:25pm

Vincent Hall 570

TBA

Pavlo Pylyavskyy

UMN

Friday

Mar. 8, 2019

3:35-4:25pm

Vincent Hall 570

TBA

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

Christian Gaetz

MIT

Friday

Mar. 29, 2019

3:35-4:25pm

Vincent Hall 570

TBA

Molly Lynch

North Carolina State University

Friday

Apr. 5, 2019

3:35-4:25pm

Vincent Hall 570

TBA

TBA

Friday

Apr. 12, 2019

3:35-4:25pm

Vincent Hall 570

TBA

Andrew Berget

Western Washington University

Friday

Apr. 19, 2019

3:35-4:25pm

Vincent Hall 570

TBA

TBA

Eric Sommers

UMass Amherst

Friday

Apr. 26, 2019

3:35-4:25pm

Vincent Hall 570

TBA

Friday

May 3, 2019

3:35-4:25pm

Vincent Hall 570

TBA

No seminar- final exams

Friday

May 10, 2019

3:35-4:25pm

Vincent Hall 570

TBA

- 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.
- Student Combinatorics and Algebra Seminar meets on Thursdays.