Registration: There is no registration fee for the conference, and all are invited to attend the talks. To register to attend the conference in person, click here. To register to attend the conference virtually over Zoom, click here.
Arrival-Departure Dates: The expected arrival date is Sunday, May 15. The conference begins Monday morning. Most attendees will depart either Friday, May 20 or Saturday, May 21. The conference will end at 5:30pm on Friday.
Hotels: Hotels near campus include The Graduate Hotel, Days Inn, and Hilton Garden Inn. Another possibility is Airbnb. The above link for the Graduate Hotel includes the OPAC-specific days and discounted rate. For the Days Inn, participants can call (612) 623-9303 at any time and ask to make a reservation under "OPAC Workshop" for a discounted rate.
Parking: For those who plan to drive to the conference, paid public parking is available at facilities and meters throughout the campus. For more information about parking, see this link and also this map. Note that the conference will take place in Smith Hall, on the East Bank part of campus.
Funding: There was limited funding available to support participants attending the conference. The deadline to apply for funding was January 15, 2022 and has passed.
Talk Schedule(all times in Central Time):
Monday, May 16
Tuesday, May 17
Wednesday, May 18
Thursday, May 19
Friday, May 20
P. Di Francesco
Location of Talks: All talks will take place in Smith Hall 100. Smith Hall is directly across the quad from Vincent Hall, where the math department is located.
Streaming and Recording of Talks: The talks are being streamed online. Click here and register to obtain the Zoom link. The talks are also being recorded. Click here to watch the videos on YouTube.
Talk Titles(click + for the abstracts; click titles for slides):
A quintessential problem that arises in algebraic combinatorics is to prove that a given polynomial expands nonnegatively into a certain basis. A classic example of this is the Littlewood--Richardson Rule for expanding a Schur polynomial into the Schur basis. The coefficients in this case give the multiplicity of the irreducible decomposition of tensor products in the general linear group, describe the decomposition of restrictions of modules in the symmetric group, and enumerate points in a suitable triple intersection of Grassmannian Schubert varieties. Many important problems of this nature persist today, such as expanding the Kronecker product of Schur functions into the Schur basis (corresponding to giving the irreducible decomposition of tensor products in the symmetric group) and expanding the product of Schubert polynomials into the Schubert basis (corresponding to enumerating points in intersections of arbitrary Schubert varieties). Motivated by these open problems, I’ll review powerful combinatorial structures, such as crystal graphs and dual equivalence graphs, that might be useful in approaching these positivity problems, and point to new directions that could lead to new solutions to these long standing positivity problems.
Rogers-Ramanujan identities have many connections and one of them is representation theory. Thanks to this connection, it is expected that $A_n$-Rogers Ramanujan identities exist. The case $n=1$ corresponds to the Andrews-Gordon and Andrews-Bressoud identities. Foda and Welsh had the brilliant idea to give another proof of these identities using cylindric partitions with two columns. It is conjectured that cylindric partitions with $n+1$ columns are the good combinatorial objects to attack the $A_n$ problem. In this talk I will explain recent progress in the case $n=2$ and will state conjectures for the general case.
We consider the two-dimensional 20-Vertex "ice" model on the triangular lattice with special "domain wall" type boundary conditions on various domains. Like the square lattice (6-Vertex) version, this model has a fascinating combinatorial content. We review the known exact results and conjectures relating its configurations to domino tilings of certain plane domains.
Quiver mutations are certain transformations of finite oriented graphs (quivers). They provide the underlying combinatorial structure for cluster algebras of skew-symmetric type. I will survey various open problems concerning quiver mutations, from a purely combinatorial perspective.
Positroid varieties are remarkable subsets of the Grassmannian introduced by Knutson--Lam--Speyer building on the work of Postnikov in total positivity. I will survey various open problems arising from the interplay between positroid varieties and cluster algebras, knot theory, statistical mechanics, and Catalan combinatorics.
Khovanov and Rozansky defined in 2005 a triply graded link homology theory which generalizes HOMFLY-PT polynomial. In this talk, I will outline some known results and structures in Khovanov-Rozansky homology, describe its connection to q,t-Catalan combinatorics and present several geometric models for some classes of links.
J. Huh: Open problems on Lorentzian polynomials
Conjecturally, skew-Schur polynomials, Schur P polynomials, Schubert polynomials, homogeneous components of Grothendieck polynomials, key polynomials, and degree polynomials of Bernstein-Gelfand-Gelfand all become Lorentzian after normalizations. I will present these and some other open problems on Lorentzian polynomials. Joint work with Petter Brändén, and with Jacob Matherne, Karola Mészáros, and Avery St. Dizier.
When Fomin and Zelevinsky invented cluster algebras, one motivation was to understand canonical bases in representation theory. Finite-type cluster algebras come with a natural basis given by cluster monomials. For infinite-type cluster algebras, the construction of bases containing the cluster monomials has been the subject of intense work in recent years, producing three families of bases: the "generic basis", "common triangular basis", and "theta basis". All these bases are parametrized by the tropical points in the dual cluster variety, in line with the Fock-Goncharov conjecture. On the other hand, in representation theory, we also have three families of bases: the "semicanonical basis", "canonical basis", and "Mirkovic-Vilonen basis". Of course, it is very tempting to match up these families with the ones coming from cluster algebras. I will explain known results and open problems in this direction.
The dimer model studies a natural probability measure on the space of perfect matching ("dimer covers") of an edge-weighted graph; the probability of a dimer cover is proportional to the product of its edge weights. When the graph is bipartite, a useful change in viewpoint is to view the edge weights as defining a "$C^*$ local system", that is a line bundle with connection. This leads us to consider natural generalizations using higher-rank bundles, in particular $SL_n$-local systems. Several of our open problems concern the combinatorial objects enumerated by the resulting Kasteleyn matrix.
A. Knutson: Schubert calculus and quiver varieties
I'll give a rundown of the different axes of generalization of the original Schubert calculus (intersection theory on Grassmannians), the state of the art, and which open problems seem most tractable and/or most important. Many of the recent advances have come from seeing flag manifolds (or really, their cotangent bundles) as special cases of Nakajima quiver varieties, and I'll discuss the combinatorial questions suggested in this framework. Time permitting, I may also discuss a theory of "generic pipe dreams" that interpolate between ordinary and bumpless, and some questions about them.
T. Lam: Positive geometries
Positive geometries are certain semialgebraic spaces equipped with a differential form, the canonical form. Examples of positive geometries include polytopes, positive parts of toric varieties, various kinds of totally positive spaces, and conjecturally, Grassmann polytopes and amplituhedra. I will discuss various open problems concerning the combinatorics and geometry of positive geometries and their canonical forms.
In q,t-mathematics a major object of interest is the nabla operator. This operator is applied to various symmetric functions and we are interested in explicit combinatorial expansions of the results. Several such expansions have been established: the compositional shuffle conjecture evaluates nabla applied to the modified Hall-Littlewood polynomials, the Loehr-Warrington conjecture evaluates nabla applied to the Schur polynomials, and the result of application of nabla to products of q,t-modified compete homogeneous functions was established in a recent paper by Carlsson and the author. In the talk I will present an approach based on p-tableaux which shows that all these evaluations are in fact equivalent. It is still unclear however what are the representation theoretic and geometric meanings of this. Based on a joint work with Erik Carlsson.
For over a century, the Algebraic Combinatorics has enjoyed a remarkable development due in part to the fundamental combinatorial interpretations such as the Young rule and the Littlewood-Richardson rule. It is an article of faith that other major structure constants should also have combinatorial interpretations: the Kronecker, plethystic and Schubert coefficients. But really, why should we believe that in the first place? And what exactly do we mean by a combinatorial interpretation? I will present a Computational Complexity point of view on these questions and speculate about future developments. Much of the talk is based on a recent joint work with Christian Ikenmeyer.
Kostka, Littlewood-Richardson, Kronecker, plethysm coefficients are fundamental quantities in algebraic combinatorics, yet many natural questions about them stay unanswered for more than 80 years. They lack "nice formulas", a notion which can be formalized using computational complexity theory, and thus what we can hope is to understand are their asymptotic behavior in various regimes and inequalities they could satisfy. Understanding these quantities has applications beyond combinatorics: this is closely related to understanding the [limit] behavior of tiling models in statistical mechanics and, more recently, they are involved in establishing computational lower bounds and separation of complexity classes like VP vs VNP in arithmetic complexity theory. In this talk I'll outline the above applications and then discuss the state of the art and outstanding problems related to asymptotics, positivity and complexity of structure constants focusing mostly on the Kronecker coefficients of the symmetric group.
A. Postnikov: 8½ open problems in algebraic combinatorics
We will discuss several problems related to Schubert calculus, Schur positivity, total positivity, matrix completion, rays, and x-rays.
Much as the square grid admits a natural 2-coloring, the triangular grid admits a natural 3-coloring and one can consider trimers containing one vertex of each color. Building on work of Conway, Lagarias, and Thurston, I’ll describe a two-parameter family of finite subgraphs of the triangular grid that are analogous to Aztec diamonds and that give rise to numerous conjectural exact formulas for the number of trimer covers.
A fundamental problem at the confluence of algebraic geometry, combinatorics and representation theory is to understand the structure and vanishing behavior of the cohomology of line bundles on flag varieties. Over fields of characteristic zero, this is the content of the Borel-Weil-Bott theorem and is well-understood, but in positive characteristic it remains wide open, despite important progress over the years. I will give an overview of the subject emphasizing a variety of open questions, and discuss some recent progress.
R. Schiffler: Perfect matching problems in cluster algebras and number theory
Snake graphs and band graphs are used to construct canonical bases for cluster algebras of surface type. The perfect matchings of these graphs parametrize the Laurent polynomial expansions of the basis elements. Later it was shown that there is a bijection between (unlabeled) snake graphs and continued fractions such that the number of perfect matchings of the snake graph is equal to the numerator of the continued fraction. In this talk we will explore this connection to number theory further.
The irreducible polynomial representations $\rho^\lambda$ of $GL_n$ are indexed by integer partitions $\lambda$ with at most $n$ parts. The character of $\rho^\lambda$ is the Schur polynomial $s_\lambda$. The composition of two such representations $\rho^\lambda : GL_n \to GL_m$ and $\rho^\mu : GL_m \to GL_\ell$ is also a polynomial representation of $GL_n$ and its characters is denoted $s_\lambda[s_\mu]$, the plethysm of two Schur functions. It is an open problem to find a combinatorial interpretation of the coefficients in the Schur expansion of $s_\lambda[s_\mu]$. I will discuss some ideas on how to attack this problem using the representation theory of the algebra of uniform block permutations (see joint work with Rosa Orellana, Franco Saliola, and Mike Zabrocki).
If W is a finite Coxeter group, then there is a finite hyperplane arrangment whose regions are indexed by W. These regions describe torsion classes for the corresponding preprojective algebra, and form a complete semi-distributive lattice. Once W is infinite, the elements of W only describe a hyperplane arrangement inside the Tits cone. I will describe three strategies to develop a theory to extend W to describe the whole space: (1) Reading's theory of shards (2) ideas from the representation theory of preprojective algebras (3) Dyer's conjectures about biclosed sets of roots. This talk will be friendly to people who have not seen any of these concepts before.
H. Thomas: Combinatorics of string-like amplitudes
I will discuss a system of equations associated to a surface. The variables, of which there are typically infinitely many, are associated to homotopy classes of curves. I will discuss strategies towards solving the equations, in terms of combinatorics, hyperbolic geometry, and representation theory. In the case that the surface is a disk, the corresponding system of equations (which exceptionally has only finitely many variables) was introduced and solved in 1969 by Koba and Nielsen. It plays a key rôle in their definition of the tree string amplitude. We hope to define analogous integrals corresponding to other surfaces, which would provide higher loop order corrections to the tree string amplitude. I will explain some of the challenges involved. This talk will be based on joint work with Nima Arkani-Hamed, Hadleigh Frost, Pierre-Guy Plamondon, and Giulio Salvatori.
Hessenberg varieties are a family of subvarieties of the flag variety parametrized by two objects: a linear operator and a Dyck path, or nondecreasing step function that stays above the diagonal. They combine the geometric and combinatorial richness of Schubert varieties with that of nilpotent orbits. Specific subfamilies of Hessenberg varieties — especially Springer fibers and regular semisimple Hessenberg varieties — encode important geometric representations, leading to important applications in knot theory and other fields. In this talk, we describe ten open questions about Hessenberg varieties, including questions about their geometry, connections to webs, representation theory, and others.
The asymmetric simple exclusion process (ASEP) is a model for translation in protein synthesis and traffic flow; it can be defined as a Markov chain describing particles hopping on a one-dimensional lattice. I will give an overview of some of the connections of the stationary distribution of the ASEP to combinatorics (tableaux and multiline queues) and special functions (Askey-Wilson polynomials, Macdonald polynomials, and Schubert polynomials). I will also mention some open problems and observations about positivity in Markov chains.
Blog: An Open Problems in Algebraic Combinatorics blog, to accompany the OPAC 2022 conference and continue on afterwards, is running online here. A pdf version of the blog posts is also available here.