Semyon Alesker (Tel Aviv University)

Non-commutative determinants and valuations on convex sets

In this talk we describe an application of non-commutative determinant to (convex) analysis. More precisely, we describe a construction of continuous valuations on convex sets based on the use of non-commutative (Moore) determinant of quaternionic hermitian matrices. Finitely additive functionals on convex compact subsets of a Euclidean space are called valuations. Valuations continuous with respect to the Hausdorff metric play a central role in the theory and have high geometric interest. They have been studied in convexity for a long time. New examples of such valuations represent an interest to the theory, and the above mentioned construction provides such a new example. If time permits we will also describe an octonionic version of it.

Arkady Berenstein (University of Oregon)

Noncommutative clusters

The goal of my talk (based on joint work with Vladimir Retakh) is to introduce noncommutative clusters and their mutations, which can be viewed as generalizations of both classical and quantum cluster structures.

Each noncommutative cluster X is built on a torsion-free group G and a certain collection of its automorphisms. We assign to X a noncommutative algebra A(X) related to the group algebra of G, which is an analogue of the cluster algebra, and establish a Noncommutative Laurent Phenomenon in several algebras A(X).

``Cluster groups" G for which the Noncommutative Laurent Phenomenon holds include triangular groups of marked surfaces (closely related to the fundamental groups of their ramified double covers) and principal noncommutative tori which exist for any exchange matrix B.

Alexei Borodin (MIT)

Fourier-like bases and Integrable Probability

Mattia Cafasso (University of Angers)

Calogero-Painleve systems, their isomonodromic formulation and non-commutative monodromy data

I will present our recent results on the isomonodromic formulation of the so-called Calogero-Painleve systems. In particular, for the case of the second Painleve equation, I will illustrate the relationship between our work and the non-commutative second Painleve equation introduced by Retakh and Roubtsov, and will study its isomonodromy data (joint work with Marco Bertola and Vladimir Roubtsov).

Leonid Chekhov (Michigan State & Steklov Math. Inst., Moscow)

Mirzakhani's volumes for bordered Riemann surfaces

We address a problem of constructing generating functions for Mirzakhani's volumes of moduli spaces for Riemann surfaces with bordered cusps. As a byproduct we derive Fenchel--Nielsen coordinates having canonical Poisson brackets and prove their modular invariance.

Philippe Di Francesco (UIUC & IPhT)

Macdonald Operators and Cluster Algebra

Pavel Etingof (MIT)

Cyclotomic Double affine Hecke algebras

I'll show that the partially spherical cyclotomic rational Cherednik algebra (obtained from the full rational Cherednik algebra by averaging out the cyclotomic part of the underlying reflection group) has four other descriptions: (1) as a subalgebra of the degenerate DAHA of type A given by generators; (2) as an algebra given by generators and relations; (3) as an algebra of differential-reflection operators preserving some spaces of functions; (4) as equivariant Borel-Moore homology of a certain variety. Also, I'll define a q-deformation of this algebra, called cyclotomic DAHA. Namely, I'll give a q-deformation of each of the above four descriptions of the partially spherical rational Cherednik algebra, replacing differential operators with difference operators, degenerate DAHA with DAHA, and homology with K-theory. In addition, I'll explain that spherical cyclotomic DAHA are quantizations of certain multiplicative quiver and bow varieties, which may be interpreted as K-theoretic Coulomb branches of a framed quiver gauge theory. Finally, I'll apply cyclotomic DAHA to prove new flatness results for various kinds of spaces of q-deformed quasiinvariants. This is joint work with A. Braverman and M. Finkelberg.

Vladimir Fock (Université de Strasbourg)

Clusters and singularities

In their recent work Fomin, Pylyavskyy and Shustin observed an amazing relations between clusters and planar curve singularities. Namely they constructed quivers out of morsifications of singularities. They conjectured (and proved in many cases) that the singularities are equivalent if and only if the quivers are mutationally equivalent. We suggest a direct construction of a cluster variety starting from a singularity and a map to it from the space of versal deformations.

Michael Gekhtman (University of Notre Dame)

Cluster structures on Poisson-Lie groups

We describe a construction of exotic cluster structures in GL(n) compatible with Poisson-Lie brackets covered by the Belavin-Drinfeld classification (joint work wit M. Shapiro and A. Vainshtein).

Edinah Gnang (Johns Hopkins University)

A matrix rank-nullity theorem for third order hypermatrices

Third order hypermatrices are three dimensional analog of matrices. How do they come up in pure and applied math? What is a natural hypermatrix algebra on such hypermatrices? In addition answering these questions, the talk will describe a framework for extending the matrix rank-nullity theorem to third order hypermatrices. The talk is based on joint work with Yuval Filmus.

Claire Gilson (University of Glasgow)

Quasi Pfaffians and Integrable Systems

In this talk we shall look at the Pfaffian equivalent of a quasi-determinant. These quasi-pfaffians have identities analogous to quasi-determinant identities. We shall address the question of the connection between quasi-pfaffian identities and non-commutative integrable systems.

Rinat Kedem (UIUC)

q-multiplicity formulas, quantum Q-systems and quantum affine algebras

I will give a quick overview of the role of the quantum cluster algebras associated with Q-systems in a very old problem in physics and more recent ideas in representation theory. Once q-deformed, the combinatorial (completeness) problem is recast in terms of an algebra again, giving the relation of the quantum cluster algebra or the deformed Grothendieck ring to a quotient of a quantum (nilpotent) affine algebra, or spherical Hall algebra, in the case of type A Q-systems.

Maxim Kontsevich (IHES)

Non-commutative Poisson brackets revisited

A non-commutative theory of shifted symmetric functions

Multiparameter versions of Schur functions already appeared in the work of Macdonald. In work of Okounkov-Olshanski, a ring of shifted symmetric functions is defined which is a deformation of the symmetric functions. The ring contains shifted Schur functions which are stable under extension of the list of variables.

This talk reports on work in progress (joint with Vladimir Retakh) defining a ring of non-commutative shifted symmetric functions using the quasi-determinant approach used in defining non-commutative symmetric functions by Gelfand-Krob-Lascoux-Leclerc-Retakh-Thibon.

Aaron Lauve (Loyola University Chicago)

Rational Series in the Free Group and the Word Problem in the Free Field

We characterize rational series over the free group in terms of a Fredholm operator used by Pimsner-Voiculescu and Connes. And we develop an effective algorithm for solving the word problem in the free skew field.

Marta Mazzocco (University of Birmingham)

The geometry behind the q-Askey scheme

Alexander Molev (University of Sydney)

Equivalences between Yangian presentations

It has been an open problem since the pioneering work of Drinfeld to construct an explicit isomorphism between the R-matrix presentation and the "new realization" of the Yangian in types B, C and D. We give a solution which is based on the Gauss decomposition of the generator matrix in the R-matrix presentation. The proof relies on some properties of the Gelfand-Retakh quasideterminants which allow one to embed the Yangian of rank n-1 into the Yangian of rank n of the same type.

This is joint work with Naihuan Jing and Ming Liu.

Valentin Ovsienko (CNRS, University of Reims)

Discrete projective geometry and combinatorics

This talk is a variation on the subject closely related to combinatorics of cluster algebras: triangulations of n-gons and coordinates on Grassmannians. I will introduce a more general notion of "3d-dissection of n-gons" that arose from the Conway and Coxeter theorem with the total positivity condition relaxed.

Gus Schrader (Columbia University)

Quantum total positivity

The notion of total positivity plays an important role in Lie theory and geometry, and led to the development of the theory of cluster algebras by Fomin and Zelevinsky. In this talk I'll describe a quantum counterpart of the totally positive part of a cluster variety, and outline some applications of this quantum total positivity to the quantization of higher Teichmüller theory, and to integrable systems.

Vera Serganova (UC Berkeley)

Universal tensor categories arising from classical supergroups

For all classical supergroups series GL, OSP, P and Q we provide a construction of certain abelian tensor categories. Those categories are abelian envelopes of Karoubian categories defined by Deligne and satisfy nice universal properties.

Alexander Shapiro (University of Toronto)

On parallel transports in quantum Teichmüller theory

Two approaches to quantizing higher Teichmüller theory were suggested in the works of Fock-Rosly and Fock-Goncharov. Although, the two constructions might seem different at the first sight, their equivalence has always been a part of the mathematical folklore. In this talk, I will suggest a way to see the former construction from the latter one. Although, making folklore statements into theorems is rarely a rewarding activity, in this case we will arrive at two rather interesting results along the way: first, we will obtain cluster structure on quantum groups and their doubles, second, we will see that the so-called "RLL-type" relations follow from relations in the modular groupoid. Contents of this talk follow works (most of which are still in progress) with Michael Gekhtman, Ivan Ip, David Jordan, Ian Le, Gus Schrader, and Michael Shapiro.

Linhui Shen (Michigan State University)

Cluster Duality of Grassmannian

The Grassmannian Gr(k,n) parametrizes k-dimensional subspaces in C^n. We introduce decorated configuration space X(k,n) equipped with a natural potential function W. We prove that the tropicalization of (X(k,n), W) canonically parametrizes a linear basis of C[Gr(k,n)], as expected by the Duality Conjecture of Fock-Goncharov. We identify the tropical set of (X(k,n), W) with the set of plane partitions. As an application, we show a cyclic sieving phenomenon for plane partitions. This is joint work with Daping Weng.

Sergei Tabachnikov (Penn State)

Cross-ratio dynamics on ideal polygons

Define a relation between labeled ideal polygons in the hyperbolic space by requiring that the complex distance (a combination of the distance and the angle) between the respective sides of the first and second polygons equals c; the complex number c is a parameter of the relation. This defines a 1-parameter family of maps on the moduli space of ideal polygons in the hyperbolic space (or, in its real version, in the hyperbolic plane). I shall discuss the completely integrable dynamics of this family of maps.

Arkady Vaintrob (University of Oregon)

Deformations and normal forms of dg-manifolds

A dg-manifold is a manifold with a sheaf of quasi-free differential graded algebras. (They are also sometimes called Q-manifolds or supermanifolds with homological vector fields.) Many geometric and algebraic structures can be described in terms of dg-manifolds. I will discuss general results about deformations and local normal forms of dg-manifolds and related objects (such as dg-submanifolds and bundles) and will show how they can be used to study Lie (bi)algebroids, Poisson manifolds, homotopy actions, generalized complex manifolds, etc.

Doron Zeilberger (Rutgers University)

Experimenting with the Berenstein-Retakh Intriguing Non-Commutative Catalan "Numbers"

My (and most combinatorial enumerators') favorite numbers are the Catalan numbers, that have at least five hundred different non-trivially equivalent definitions. Recently Arkady Berenstein and the birthday boy, Volodia Retakh, came up with an amazing generalization, making them non-commutative polynomials. It is fun to experiment with these new toys.