Workshop

Parabolic skein modules implement parabolic reduction along a Borel subgroup B of a reductive group G using codimension-1 defects in skein theory. These parabolic defects appear naturally when defining quantum cluster coordinates on skein categories and modules, and when quantising the celebrated knot invariant known as the A-polynomial.

We'll introduce defect skein theory, focusing on it's relevance to cluster coordinates and the quantum A-polynomial, With that done, we will see how the former construction helps us compute a localisation of the later.

We reformulate the q-difference linear system corresponding to the q-Painlevé equation of type A₇⁽¹⁾' as a Riemann-Hilbert problem on a circle. Jump of this Riemann-Hilbert problem can be used to build a Fredholm determinant that gives the tau function of our q-Painlevé equation. Minor expansion of this determinant coincides with the known expansion in terms of Nekrasov functions. We also use this determinant representation to study the global properties of the tau function and find that it has a non-trivial z->1/z involution with generating function given by elliptic dilogarithms and connection constant given by elliptic Gamma functions. The talk will be based on the paper https://arxiv.org/abs/2501.01419

In this talk I will introduce C*-categories as the dimension 2 component of a 4 dimensional fully extended topological field theory. Factorization Homology will then be introduced as the principle by which one associates C*-cateogories to surfaces, enforcing the strong locality feature that characterizes fully extended theories. As a main source of examples and inspiration, I will say a few words about my ongoing project with  non-compact topological gauge theories, where the theory of locally compact quantum groups naturally enters the picture.

This talk is based on an on-going project with Subri Murugesan, in which we aim to find a deeper understanding of Fenchel-Nielsen type spectral networks in quantum Toda theory. One of our motivations is to understand the relation between the SL(3) parallel transports computed in the free field formalism by Coman, Pomoni and Teschner to the SL(3) parallel transports computed with respect to Fenchel-Nielsen type spectral network by Hollands-Neitzke. Another is to extend the concept of the "non-perturbative" topological string partition function to define conformal blocks with respect to different types of spectral networks. In this talk I will review the computation of semi-classical Toda blocks through spectral networks techniques, and make comments on its quantisation.

Linear incidence theorems are statements about points/lines/subspaces and their incidences. Examples include classical theorems of Desargues and Pappus. In this talk, we'll see how cluster integrable systems of Goncharov and Kenyon lead to a new type of results in linear incidence geometry: dynamical incidence theorems.

Based on joint work with Pavlo Pylyavskyy (Minnesota).

In this talk, I will revisit the derivation of q-Painlevé equations from cluster algebras and present a proposal that their solutions can be described by partition functions of dimer models on the infinite plane, subject to suitable boundary conditions. The proposal rests on two key observations. First, solutions to q-Painlevé equations are expressible in terms of partition functions of topological string theory, which involve counts of coupled 3d–2d–1d Young diagram configurations. Second, these configurations naturally emerge from dimer models on infinite planar graphs, obtained as lifts from toroidal graphs associated with Goncharov–Kenyon cluster integrable systems. Within this framework, the q-Painlevé equation should emerge as a bilinear relation among minors of the Kasteleyn matrix of this bipartite graph. I will illustrate the proposal using the q-Painlevé III example and explain why it doesn't work naively.

Let G be a semisimple group over C. Let \beta be a positive braid whose Demazure product is the longest Weyl group element. The braid variety M(\beta) generalizes many well known varieties, including positroid cells, open Richardson varieties, and double Bruhat cells. We provide a concrete construction of the cluster structures on M(\beta), using the weaves of Casals and Zaslow and a new combinatorial construction called Lusztig cycles. We show that the coordinate ring of M(\beta) is a cluster algebra, which confirms a conjecture of Leclerc as special cases. As an application, we show that M(\beta) admits a natural Poisson structure and can be further quantized. This talk is based on joint work with Roger Casals, Eugene Gorsky, Mikhail Gorsky, Ian Le, and Jose Simental.

Quantisation of complex Chern-Simons Theory for G=SL(2) reduces to the  quantisation of the moduli space of flat SL(2)-connections. The framework of Schur quantisation introduced in work of  D. Gaiotto and myself offers a new approach to this old problem, the main new feature being the existence of spherical cyclic vector generating a representation of the skein algebra. The aim of my talk will be to review some aspects of this approach with focus on the cluster algebra aspects.

Weaves were first introduced by Casals and Zaslow as a graphical tool to describe a family of Legendrian surfaces living inside the 1-jet space of a base surface. Casals, Gorsky, Gorsky, Le, Shen, and Simental later generalized weaves to all Dynkin types such that the original weaves for Legendrian surfaces belong to Dynkin type A, and they use weaves of general Dynkin types to describe the cluster structure on braid varieties. In my previous joint work with Casals, we gave a topological interpretation of the cluster structures associated with weaves of Dynkin type A by associating the quiver with intersections of certain 1-cycles on surfaces and associating cluster variables with merodromies (parallel transports) along dual relative 1-cycles. In this talk, I will generalize this topological interpretation to all general Dynkin types by introducing a new diagrammatic object called “weighted cycles” and constructing an intersection pairing between them. I will define the merodromy along a weighted cycle and explain how to describe cluster variables using merodromies. If time allows, I will also mention a connection to quantum groups and skein algebras.

Formal state integrals were first introduced by Hikami giving an algebraic description of perturbative invariants of knots. Andersen-Kashaev introduced an analytic description giving a precise convergent expression for an invariant of links. We will discuss the appropriate homology and cohomology in which the Andersen-Kashaev invariant arises as a natural period pairing. This will prove various conjectures of Garoufalidis-Gu-Mariño; notably that the 3d index of Dimofte-Giaotto-Gukov is given by an entry of a Stokes automorphism. This entry will be computed by intersections in this homology and be manifestly integral. This is based on joint work with Fantini and Andersen-Fantini-Kontsevich.