Some title

Some Person

USomething

Wednesday

Sept. 3, 2014

8:00-8:50am

Vincent Hall 313

Some Abstract. $e^x$

Coxeter transformation on cominuscule posets

Emine Yıldırım

UQAM

Friday

Sept. 15, 2017

3:35-4:25pm

Vincent Hall 570

Let $P$ be a cominuscule poset which can be thought of as a parabolic analogue of the poset of positive roots of a finite root system. Let $J(P)$ be the poset of order ideals of $P$. In this talk, we will investigate the periodicity of the Coxeter transformation on the poset $J(P)$, and show that the Coxeter transformation has finite order for two of the three infinite families of cominuscule posets, and the exceptional cases. Our motivation comes from a conjecture by Chapoton which states that the Coxeter transformation has finite order on the poset $J(R)$ when $R$ is the poset of positive roots of a finite root system. Our solution is formulated in representation theory of finite dimensional algebras, and we will further discuss the results within the same context.

Random Flag Complexes and Asymptotic Syzygies

Jay Yang

UWisconsin

Friday

Sept. 22, 2017

Vincent Hall 016

We use the probabilistic method to construct examples of conjectured phenomenon about asymptotic syzygies. In particular, we use the Stanley-Reisner ideals of random flag complexes to construct new examples of Ein and Lazarsfeld's nonvanishing for asymptotic syzygies and of Ein, Erman, and Lazarsfeld's conjectural on the asymptotic normal distribution of Betti numbers.

Cyclic symmetry in the Grassmannian

Steven Karp

Michigan

Friday

Sept. 29, 2017

3:35-4:25pm

Vincent Hall 570

The Grassmannian $Gr(k,n)$ is the space of $k$-planes in $C^n$. Its totally nonnegative part is the subset where all Plücker coordinates are real and nonnegative. There is a natural action of the cyclic group of order $n$ on $Gr(k,n)$ which preserves its totally nonnegative part. This 'cyclic symmetry' is prominent in the combinatorics of the totally nonnegative Grassmannian. I will discuss some surprising properties of the fixed points of the cyclic action. I will also present joint work with Pavel Galashin and Thomas Lam, which uses the cyclic action to show that the totally nonnegative part of $Gr(k,n)$ is homeomorphic to a ball.

Recent results in enumeration

Dennis Stanton

UMN

Friday

Oct. 6, 2017

3:35-4:25pm

Vincent Hall 570

I will survey some of my recent joint results in enumeration, including (1) integer partitions (2) enumeration over finite fields (3) orthogonal polynomials (4) posets. I will indicate open directions for each of these areas. No proofs will be given. This is joint work with Fulman, Guralnick, Ismail, Kim, Lewis, O’Hara, Rains, and Reiner.

Representation stability: A case study

Cihan Bahran

UMN

Friday

Oct. 13, 2017

3:35-4:25pm

Vincent Hall 570

The (ordered) configuration space of the complex plane has ties into various areas of mathematics. Church-Farb showed that, as the number of points in the configuration increases, the corresponding action of the symmetric group in cohomology "stabilizes" in a certain way. I will explain this phenomenon in the case of $H^1$ for the complex plane, and then talk about generalizations to other manifolds (Church) and some recent developments in the stable range.

Chip-firing for root systems

Sam Hopkins

MIT

Friday

Oct. 20, 2017

3:35-4:25pm

Vincent Hall 570

Propp recently introduced a variant of chip-firing on the infinite path where the chips are given distinct integer labels and conjecture that this process sorts certain (but not all) initial configurations of chips. Hopkins, McConville, and Propp proved Propp's sorting conjecture. We recast this result in terms of root systems: the labeled chip-firing game can be seen as a “vector-firing” process which allows the moves $\lambda \to \lambda + \alpha$ for $\alpha \in \Phi^+$ whenever $\langle \lambda, \alpha^\vee \rangle = 0$, where $\Phi^+$ is the set of positive roots of a root system of type $A_{2n-1}$. We give conjectures about confluence for this process in the general setting of an arbitrary root system. We show that the process is always confluent from any initial point after modding out by the action of the Weyl group (an analog of unlabeled chip-firing in arbitrary type). We also show that if we instead allow firing when $\langle \lambda, \alpha^\vee \rangle \in [-k-1,k-1]$ or $[-k,k-1]$, we always get confluence from any initial point. Moreover, in these two settings, the set of weights with given stabilization has a remarkable geometric structure related to permutohedra. This geometric structure leads us to define certain “Ehrhart-like” polynomials that conjecturally have nonnegative integer coefficients.

This is joint work with Pavel Galashin, Thomas McConville, and Alex Postnikov.

Component preserving mutations: building up maximal green sequences from sub-quivers

Eric Bucher

Michigan State

Friday

Oct. 27, 2017

3:35-4:25pm

Vincent Hall 570

Quiver mutation is a operation one can define on a directed graph that has shown to model the behavior of a large variety of mathematical objects. We will discuss a bit about quiver mutation, and the proceed to exploring quivers for a special sequence of mutations called maximal green sequences. The aim of the talk is to discuss recent work that allows one to build maximal green sequences for larger quivers by looking at "component preserving" sequences on induced subquivers. These new techniques have allowed us to construct maximal green sequences for large families of quivers where their existence was previously unknown.

Robinson-Schensted-Knuth via quiver representations

Hugh Thomas

UQAM

Friday

Nov. 3, 2017

3:35-4:25pm

Vincent Hall 570

The Robinson-Schensted-Knuth correspondence is a many-faceted jewel of algebraic combinatorics. In one variation, it provides a bijection between permutations of $n$ and pairs of standard Young tableaux with the same shape, which is a partition of $n$. In another (more general) version, it provides a bijection between fillings of a partition $\lambda$ by arbitrary non-negative integers and fillings of the same shape $\lambda$ by non-negative integers which weakly increase along rows and down columns (i.e., reverse plane partitions of shape $\lambda$). I will discuss an interpretation of RSK in terms of the representation theory of type $A$ quivers (i.e., directed graphs obtained by orienting a path graph). This allows us to generalize RSK to other Dynkin types (plus a choice of minuscule weight), and is related to periodicity results for piecewise-linear toggling. I will not assume familiarity with either RSK or with quiver representations. This is joint work with Al Garver and Becky Patrias.

The Mullineux involution and wall-crossing for the rational Cherednik algebra

Galyna Dobrovolska

Columbia

Friday

Nov. 10, 2017

3:35-4:25pm

Vincent Hall 570

The Mullineux involution on $p$-regular Young diagrams corresponds to the operation of taking the tensor product with the sign representation on modules for the symmetric group in characteristic $p$. In my talk I will report on some results towards proving R. Bezrukavnikov's conjectures on wall-crossing for the rational Cherednik algebra in positive characteristic, which was linked to the Mullineux involution by I. Loseu.

Identities for symmetric skew Grothendieck polynomials

Damir Yeliussizov

UCLA

Friday

Nov. 17, 2017

3:35-4:25pm

Vincent Hall 570

Symmetric Grothendieck polynomials can be viewed as an analog of Schur polynomials for the K-theory of Grassmannians. I will present various properties and applications for dual families of skew Grothendieck polynomials, such as skew Cauchy identities, skew Pieri rules, dual filtered Young graphs, generating series identities, some probabilistic models, and enumerative results.

Friday

Nov. 24, 2017

Chromatic Graph Homology for Brace Algebras

Vladimir Baranovsky

UC Irvine

Friday

Dec. 1, 2017

3:35-4:25pm

Vincent Hall 570

The chromatic graph homology complex $C_G(A)$ of a graded commutative algebra $A$ and a graph $G$ was originally defined by Helme-Guizon and Rong. It may be viewed as a toy version (comultiplication free version) of the Khovanov homology complex of a link (in the special case when $G$ is the Tait graph of a link diagram and $A$ is the cohomology of a sphere).

When $A$ is the de Rham algebra of a compact oriented manifold $M$, our earlier work with R. Sazdanovic has related $C_G(A)$ to the homology of the "graph configuration space" obtained by taking the cartesian power of $M$ and removing a subset of diagonals encoded by $G$.

In a recent work with M. Zubkov we investigate whether the commutativity assumption on $A$ may e relaxed to homotopy commutativity. We construct $C_G(A)$ in the case when $A$ is a brace algebra and $G$ is a planar tree. Examples of such $A$ include singular cochain algebras (McClure-Smith), Hochschild cochains of associative algebras (Gerstenhaber-Voronov) and cobar constructions on Hopf algebras (Kadeishvili, Young). We hope that the last class of examples can get us closer to an explicit model for computing invariants of codimension 2 tangles, as provided by the formalism of (stratified) factorization homology.

Canonical bases for permutohedral plates

Nick Early

UMN

Friday

Dec. 8, 2017

3:35-4:25pm

Vincent Hall 570

There is a natural construction according to which the set of all faces of an arrangement of hyperplanes can be made into a vector space, by taking linear combinations of their characteristic functions. Our space is equipped with a standard basis of polyhedral cones called permutohedral cones, studied as plates by A. Ocneanu, consisting of characteristic functions which are labeled by ordered set partitions; these are in duality with faces of the arrangement of reflection hyperplanes $x_i=x_j$. Motivated in particular by the desire to justify certain shuffle-type relations coming from quantum field theory, we construct a new, canonical basis which is compatible with one or both of two quotients: modding out by characteristic functions of (1) nonpointed cones containing doubly infinite lines, and (2) cones of codimension at least 1. The important feature here is that subsets of the basis map to bases of the quotients.

Folding and dominance: relationships among mutation fans for surfaces and orbifolds

Shira Viel

NCSU

Friday

Dec. 15, 2017

3:35-4:25pm

Vincent Hall 570

The $n$-associahedron is a well-known $n$-dimensional polytope whose vertices are labeled by triangulations of an $(n+3)$-gon with edges given by diagonal flips. The $n$-cyclohedron is defined analogously using centrally-symmetric triangulations of a $(2n+2)$-gon, or, modding out by the symmetry, triangulations of an $(n+1)$-gon with one orbifold point. The polytopes can be realized in such a way that their normal fans are the ``$\mathbf{g}$-vector fans," or ``mutation fans," for certain cluster algebras. In this talk I will justify and generalize two relationships which hold between these fans: the normal fan to the $n$-cyclohedron can be obtained by intersecting the normal fan to the $(2n-1)$-associahedron with a certain subspace, and the normal fan to the $n$-cyclohedron refines the normal fan to the $n$-associahedron. I will show that these relationships are instances of ``folding" and ``dominance," respectively, and hold more generally for mutation fans for cluster algebras modeled by surfaces and orbifolds.

Affine Growth Diagrams

Tair Akhmejanov

Cornell

Friday

Jan. 19, 2018

3:35-4:25pm

Vincent Hall 570

We introduce a new type of growth diagram, arising from the geometry of the affine Grassmannian for $GL_m$. These affine growth diagrams are in bijection with the $c_{\vec\lambda}$ many components of the polygon space Poly($\vec\lambda$) for $\vec\lambda$ a sequence of minuscule weights and $c_{\vec\lambda}$ the Littlewood-Richardson coefficient. Unlike Fomin growth diagrams, they are infinite periodic on a staircase shape, and each vertex is labeled by a dominant weight of $GL_m$. Letting $m$ go to infinity, a dominant weight can be viewed as a pair of partitions, and we recover the RSK correspondence and Fomin growth diagrams within affine growth diagrams. The main combinatorial tool used in the proofs is the $n$-hive of Knutson-Tao-Woodward. The local growth rule satisfied by the diagrams previously appeared in van Leeuwen's work on Littelmann paths, so our results can be viewed as a geometric interpretation of this combinatorial rule.

Parametric behavior of A-hypergeometric solutions

Christine Berkesch

UMN

Friday

Jan. 26, 2018

3:35-4:25pm

Vincent Hall 570

A-hypergeometric systems are the D-module counterparts of toric ideals, and their behavior is linked closely to the combinatorics of toric varieties. I will discuss recent work that aims to explain the behavior of the solutions of these systems as their parameters vary. In particular, we stratify the parameter space so that solutions are locally analytic within each (connected component of a) stratum. This is joint work with Jens Forsgård and Laura Matusevich.

TBA

TBA

TBA

Friday

Feb. 2, 2018

3:35-4:25pm

Vincent Hall 570

TBA

The Hall Algebra at $t = 1/q$ and Torus Knots

Adriano Garsia

UCSD

Monday

Feb. 5, 2018

1:25-2:30pm

Vincent Hall 301

and place (VinH 301)

In this talk we show that the Hall Algebra operators $Q_{km, kn}$ with $k = gcd(km, kn)$ may be given for $t=1/q$ a very simple plethystic form. This discovery yields elementary and direct derivations of several identities relating these operators at $t=1/q$ and their relation to the Rational Compositional Shuffle conjecture (now a Theorem). We also show a glimpse of how this is all related to Torus knows.

Algebras of quantum monodromy data and decorated character varieties

Leonid Chekhov

Michigan State

Friday

Feb. 9, 2018

3:35-4:25pm

Vincent Hall 570

We discuss the Riemann-Hilbert correspondence for extensions of the de Rham moduli space by allowing connections with higher order poles. We show that geometrically this corresponds to interpreting higher order poles in the connection as boundary components with bordered cusps (vertices of ideal triangles in the Poincaré metric) on the Riemann surface. We thus introduce the notion of decorated character variety. This decorated character variety is the quotient of the space of representations of the fundamental groupid of arcs by a product of unipotent Borel subgroups (one per bordered cusp). We demonstrate that this representation space is endowed with a Poisson structure induced by the Fock-Rosly bracket and show that the quotient by unipotent Borel subgroups giving rise to the decorated character variety is a Poisson reduction. We deal with the Poisson bracket and its quantization simultaneously, thus providing a quantisation of the decorated character variety. In the case of dimension 2, we also endow the representation space with explicit Darboux coordinates. We conclude with a conjecture on the extended Riemann-Hilbert correspondence for higher rank algebras (joint work with M. Mazzocco and V. Rubtsov).

Equivariant quantum cohomology of the Grassmannian via rim hooks and puzzles

Kaisa Taipale

UMN

Friday

Feb. 16, 2018

3:35-4:25pm

Vincent Hall 570

We present a non-recursive, positive combinatorial formulas for expressing the equivariant quantum product in the Schubert basis of the Grassmannian. This extends work of Bertram, Ciocan-Fontanine and Fulton, who provided a way to compute quantum products of Schubert classes of the Grassmannian by applying a combinatorial rimhook rule. Combining our equivariant rule with Knutson and Tao's puzzle rule provides an effective algorithm for computing equivariant quantum Littlewood-Richardson coefficients (polynomials). This rule requires a specialization of torus weights that is tantalizingly similar to maps in affine Schubert calculus.

Biclosed sets in representation theory

Alexander Garver

UQAM

Friday

Feb. 23, 2018

3:35-4:25pm

Vincent Hall 570

The weak order on elements of a Coxeter group appears in many mathematical contexts including geometric combinatorics, generalized associahedra, and representation theory of preprojective algebras. The weak order may be equivalently described using biclosed sets. We study lattices of biclosed sets that generalize the weak order on permutations. We show that any such lattice of biclosed sets is isomorphic to subcategories of the module category of an analogue of the preprojective algebra, which we call torsion shadows. If time permits, we will present a similar description of the shard intersection order of these lattices of biclosed sets. This is joint work with Thomas McConville and Kaveh Mousavand.

Double jump phase transition in a random soliton cellular automaton

Hanbaek Lyu

Ohio State

Friday

Mar. 2, 2018

3:35-4:25pm

Vincent Hall 570

In this talk, we consider the soliton cellular automaton introduced by Takahashi and Satsuma in 1990 with a random initial configuration. We give multiple constructions of a Young diagram describing various statistics of the system in terms of familiar objects like birth-and-death chains and Galton-Watson forests. Using these ideas, we establish limit theorems showing that if the first $n$ boxes are occupied independently with probability $p\in(0,1)$, then the number of solitons is of order $n$ for all $p$, and the length of the longest soliton is of order $\log n$ for $p<1/2$, order $\sqrt{n}$ for $p=1/2$, and order $n$ for $p>1/2$. Additionally, we uncover a condensation phenomenon in the supercritical regime: For each fixed $j\geq 1$, the top $j$ soliton lengths have the same order as the longest for $p\leq 1/2$, whereas all but the longest have order at most $\log n$ for $p>1/2$. As an application, we obtain scaling limits for the lengths of the $k^{\text{th}}$ longest increasing and decreasing subsequences in a random stack-sortable permutation of length $n$ in terms of random walks and Brownian excursions.

This is a joint work with Lionel Levine and John Pike.

Ice and Everything Else

Benjamin Brubaker

UMN

Friday

Mar. 9, 2018

3:35-4:25pm

Vincent Hall 570

We'll discuss how solvable lattice models (including the "square ice" model of the title) sit at the nexus of so many interesting fields of mathematics, including combinatorics, mathematical physics, representation theory, and algebraic topology, to name a few. Examples will include both the Jones polynomial for distinguishing knots and Kuperberg's proof of the alternating sign matrix conjecture. Yet other examples we'll mention are joint work with Bump and Friedberg, and a more recent paper with Schultz.

Friday

Mar. 16, 2018

Sandpiles and representation theory

Vic Reiner

UMN

TBAFriday

Mar. 23, 2018

3:35-4:25pm

Vincent Hall 570

Every graph has a subtle invariant, called its sandpile group: a finite abelian group whose size is the number of spanning trees in the graph. After reviewing this, we will discuss an analogous "sandpile group" for any representation of a finite group, motivated in part by the classical McKay correspondence. (Based on joint work with Georgia Benkart, Carly Klivans, Christian Gaetz, Jia Huang, and Darij Grinberg.)

Coxeter combinatorics of involutions with applications to geometry

Zach Hamaker

Michigan

Friday

Mar. 30, 2018

3:35-4:25pm

Vincent Hall 570

The combinatorics of Coxeter groups has been a rich area of study for many years, motivated by connections to the geometry of flag varieties and representation theory. Building on work of Richardson and Springer, a similar combinatorial theory has been developed for involutions in Coxeter groups. Their original motivation comes from the geometry of spherical varieties. We highlight several aspects of this theory, including the enumeration of reduced words for involutions and a natural characterization of the Chinese monoid. We conclude with some remarks on an apparent relationship between type A spherical varieties and the geometry of the orthogonal and Lagrangian Grassmannians. This is joint work with Brendan Pawlowski and Eric Marberg.

Three Problems on expressions of elements of Coxeter groups

Jean-Philippe Labbé

Freie Univ. Berlin

Friday

Apr. 6, 2018

3:35-4:25pm

Vincent Hall 570

In 2004, Knutson and Miller asked whether subword complexes of Coxeter groups can be realized as the boundary of simplicial convex polytopes. Subword complexes incarnate several well-known structures: associahedra, finite type cluster complexes, simplices, and even-dimensional cyclic polytopes, for instance. Apart from those, only 1 non-trivial case has been found to be polytopal; this problem requires a deep understanding of reduced expressions in Coxeter groups. In this talk, I will describe three very closely related classical problems on expressions in Coxeter groups that reveal challenging aspects of this open problem.

Cyclic Sieving and Cluster Duality for Grassmannian

Linhui Shen

Michigan State

Thursday

Apr. 12, 2018

12:20-1:10pm

Ford Hall 150

& place (Ford 150)

The cyclic sieving phenomenon (CSP) was defined by Reiner, Stanton, and White as a generalization of J. Stembridge’s q=-1 phenomenon. In this talk, we investigate the CSP for plane partitions under a piecewise-linear toggling operation. We place our result in the context of cluster theory for Grassmannians. This is joint work with Daping Weng (Yale University).

Dual braid monoids, Koszul algebras, and clusters

Philippe Nadeau

U. Lyon 1

Friday

Apr. 20, 2018

3:35-4:25pm

Vincent Hall 570

The dual braid monoid of a finite Coxeter group $W$ is a homogeneous monoid with group of fractions the classical braid group attached to $W$. It was defined in general by David Bessis, and possesses nice algebraic and combinatorial properties. In this talk we will study the algebra of this monoid, and show that it belongs to the class of Koszul algebras. Moreover, positive elements of the cluster complex attached to $W$ naturally index a family in the "Koszul dual" of this algebra. These elements conjecturally form a basis of this dual algebra. This is joint work with Matthieu Josuat-Vergès and Jang Soo Kim.

Young tableaux, Kazhdan-Lusztig cells, and Springer fibers

Dongkwan Kim

MIT

Friday

Apr. 27, 2018

3:35-4:25pm

Vincent Hall 570

The generalized Robinson-Schensted algorithm, introduced by J.-Y. Shi, is a surjection from the affine symmetric groups to the pairs of row-standard Young tableaux of the same shape. It was used to prove the conjecture of Lusztig on the enumeration of left cells in affine symmetric groups. On the other hand, combinatorics of row-standard Young tableaux is also closely related to the affine paving of Springer fibers. In this talk, I briefly recall these concepts and how they are related, and focus on generalizations of this picture to other classical types.

Gale-Robinson quivers, representations, and combinatorial formulas

Max Glick

Ohio State

Friday

May 4, 2018

3:35-4:25pm

Vincent Hall 570

Gale-Robinson sequences were one of the first examples of the Laurent phenomenon. Moreover, the associated quivers relate to the dimer model from physics. We investigate a family of representations of these quivers that are geared towards providing concrete information about the corresponding cluster algebras. In this way, we provide a representation theoretic explanation for known combinatorial formulas for the Gale-Robinson sequence and also obtain similar formulas for several other cluster variables. This is joint work with Jerzy Weyman.

Wildly nontransitive dice

Joe Buhler

CCR La Jolla

Monday

Jun. 11, 2018

3:35-4:25pm

Vincent Hall 570

Nontransitive triples of dice have been known for (at least) 60 years. Let $A = \{2,6,7\}$, $B = \{1,5,9\}$, $C = \{3,4,8\}$ denote dice taking each of the three indicated values with probability $1/3$. As you can (and should!) check, $A$ dominates $B$ in the sense that when each is rolled the probability that $A > B$ is greater than $1/2$. Moreover, $B$ dominates $C$, and $C$ dominates $A$, so the dominance relation is non-transitive. To make matters worse, the triple $A[2]$, $B[2]$, $C[2]$ --- where $A[2]$ denotes the sum of two rolls of $A$, etc. --- is (as you have perhaps already checked) also a nontransitive triple. However, the cyclic dominance order is exactly the reverse of the earlier one. The goal of this talk is to present sets of dice with vastly more peculiar nontransitivity properties.

- Seminar meets on Fridays 3:35–4:25 in room 570 of Vincent Hall.
- Seminar announcement list sign-up.
- Organizers: Gregg Musiker (Fall), Pavlo Pylyavskyy (Spring) and Mike Chmutov. Website maintained by Mike Chmutov.
- Past seminar archive.
- Student Combinatorics Seminar.