Zoom Cluster Seminar

We shall define a notion of a higher lamination - a graph embedded into a Riemann surface with edges coloured by generators of an affine Weyl group. This notion generalises the notion of the ordinary integral measured lamination and on the other hand of a tropical curve and can be constructed out of a integral Lagrangian submanifold of the cotangent bundle.

I will discuss a positive structure on the varieties of critical points of master functions for the KZ differential equations. The structure is introduced in my recent preprint with V. Schechtman. The positive structure comes as a combination of the ideas from classical works by G. Lusztig and my previous work with E. Mukhin on the relations between these varieties of critical points and the flag varieties.

A number of q-deformations is known in algebra, geometry and combinatorics. However, q-deformations of numbers are well understood, since Euler and Gauss, only for integers. Many classical sequences of integers have interesting q-analogues, the best known one is the notion of q-binomial coefficients. I will talk of a recent attempt, with Sophie Morier-Genoud, to develop a theory of q-deformed rational and real numbers, based on combinatorial properties of continued fractions.

A "q-rational" is a ratio of certain polynomials with positive integer coefficients, I will give a combinatorial interpretation of the coefficients and describe some properties, such as the "total positivity". The construction is similar to that of Gaussian q-binomial coefficients, the Pascal triangle being replaced by the Farey graph; it is related to the cluster algebras and quantum Teichmüller space. A "q-real" is understood as a formal power series with integer coefficients, I will give several concrete examples.

In 1990s John H. Conway proposed "topographic" approach to describe the values of the binary quadratic forms, which can be applied also to the description of the celebrated Markov triples. The growth of the orbits on the corresponding trivalent tree depends on the choice of the branch, which can be labelled by the points of real projective line.

I will show that the function on real projective line, describing the growth of Markov triples and their tropical versions, has some very interesting properties. The classical Markov's results and the link with the hyperbolic geometry, going back to Gorshkov and Cohn, will play a crucial role here.

The talk is based on joint works with K. Spalding.

There is a huge system of parallels between various geometric and topological properties of the moduli space of curves of genus 0 with marked points $M_{0,1+n}$ and the so-called brick manifolds $B_n$, defined a few years ago by Laura Escobar. In fact, it appears that all interesting properties relevant for the topological field theory (considered as general phenomena surrounding $M_{0,1+n}$) find their non-commutative analogs in $B_n$: -- stratification, description of cohomology (that implies the WDVV equations), psi-classes, structure of topological operad, cohomological field theories, Givental group action, surrounding topological operads (little disks, framed little discs, gravity), algebraic structures implied by their representations (hypercommutative algebra, also known in the literature as flat F-manifold, BV/Gerstenhaber algebras, etc.), and so on. (To this end, my favourite alternative title for this talk would be "What is a non-commutative hypercommutative algebra?" ) In addition the brick manifolds have very rich geometric structure: they are toric varieties of Loday's realisations of associahedra and have interpretation as wonderful models of De Concini-Procesi. In the talk, we'll try to show some of the properties mentioned above, step-by-step constructing them both on the side of moduli spaces of curves and on the side of brick manifolds. (along the lines of my joint work with Volodya Dotsenko and Bruno Vallette).

The dilogarithm function was introduced by Euler. It has been known for a long time that the function satisfies various functional identities, which we call dilogarithm identities. I this talk I will explain from scratch how and why a dilogarithm identity is associated with any periodic sequence of mutations in cluster algebras. Part of the talk is based on joint work with M. Gekhtman and D. Rupel.

The associahedron is a famous polytope that is connected to Catalan families. In cluster theory, it can be realized as a polytope whose normal fan in the g-vector fan in type A. Other Dynkin types yield other polytopes, called generalized associahedra. Recently, the space of all polytopal realizations of the g-vector fan of a cluster algebra of Dynkin type has been of interest to physicists. In this talk, I will show how this space is governed by minimal relations between g-vectors. I will then sketch how these minimal relations can be computed using the additive categorification of cluster algebras.
This is a report on a joint work with Arnau Padrol, Yann Palu and Vincent Pilaud.

I want to discuss the set up in which a number of algebraic structures (the ones mentioned in the title) get their natural non-commutative analogs. Surprisingly, these non-commutative analogs (emerged in the literature for completely different reasons) replicate all essential properties of the original structures --- for instance, I can very explicitly show how from nc-BV structure one can get, via the homotopy quotient of the circle action, a new structure that is natural to call the nc-Hycomm algebra / nc-flat F-manifold structure.

What is also very interesting --- while the geometric models for these structures (nc-Gerst and nc-BV) are quite simple, they are related to Brick manifolds (that I discussed a few weeks ago, but now I'll give a quite different independent definition) in exactly the same way as (framed) little disks are related to the moduli spaces of curves of genus $0$ --- via the real orientable blow-up along the boundary.

I intend to make this talk completely independent from the previous one.

Along the lines of my joint work with Volodya Dotsenko and Bruno Vallette.

I plan to talk about a topological aspect of the double affine Hecke algebra.

I will discuss a relationship with the skein algebra, the Dehn twists, and quantum knot invariants.

The goal of the talk is to express the extended Goldman symplectic structure on the $SL(n)$ character variety of a punctured Riemann surface in terms of Fock-Goncharov coordinates; the associated symplectic form has integer coefficients expressed via the inverse of the Cartan matrix. The main technical tool is a canonical two-form associated to a flat graph connection. We discuss the relationship between the extension of the Goldman Poisson structure and the Poisson structure defined by Fock and Goncharov. We elucidate the role of the Rogers' dilogarithm as generating function of the symplectomorpism defined by a graph transformation. The talk is based on joint work with Dmitri Korotkin.

The Grassmannian Gr(k,n) admits an action by a finite cyclic group of order n via the cyclic shift automorphism. The combinatorics underlying both total positivity and clusters for Gr(k,n) are cyclically equivariant, which is one explanation for the particular elegance of these structures in the case of Gr(k,n). We will explore the L-shift locus in Gr(k,n), i.e. the subvariety of points fixed by the Lth power of the cyclic shift. Steven Karp recently showed that the 1-shift locus consists of finitely many points. On the other hand the n-shift locus is Gr(k,n) itself. Our theorems interpolate between these extremes: we give a geometric description of the L-shift locus for any L, describe its total nonnegativity locus as a stratified space, and propose an atlas of generalized cluster charts (in the sense of Chekhov-Shapiro) whose clusters are total positivity tests.

Consider the standard contact structure on $\mathbb{R}^3_{xyz}$ with contact 1-form $\alpha=dz-ydx$. A Legendrian link $\Lambda$ is a link in $\mathbb{R}^3$ along which $\alpha$ vanishes. Chekanov associated a dga (also known as the Chekanov-Eliashberg dga) to every Legendrian link $\Lambda$ and proved that the stable-tame equivalence class of such dga is invariant under Legendrian isotopy of Legendrian links. The augmentation variety of $\Lambda$ is defined to be the moduli space of augmentations of its CE dga.

Let $\mathbb{R}^4_{xyzt}$ be the symplectization of the standard contact $\mathbb{R}^3_{xyz}$ and consider the symplectic field theory of exact Lagrangian cobordisms between Legendrian links placed at different constant $t$ slices. Ekholm, Honda, Kalman constructed a contravariant functor that maps an exact Lagrangian cobordism to a dga homomorphism between the corresponding CE dga’s associated to the Legendrian links at the two ends. This induces a covariant functor that maps exact Lagrangian cobordisms to morphisms between augmentation varieties. In the special case of an exact Lagrangian filling of a Legendrian link $\Lambda$, i.e., an exact Lagrangian cobordism from the empty link to $\Lambda$, such covariant functor defines an algebraic torus inside the augmentation variety of $\Lambda$.

In this talk, I will focus on the cases of rainbow closures for positive braids, which are naturally Legendrian links. By using an enhanced version of the CE dga, my collaborators and I construct a cluster $K_2$ structure on the augmentation variety of the rainbow closure $\Lambda_\beta$ for any positive braid $\beta$, and prove that the algebraic tori arisen from exact Lagrangian fillings constructed by pinching crossings are cluster charts in this cluster structure. We also relate the Kalman automorphism to the cluster Donaldson-Thomas transformation on these augmentation varieties. As an application of this new cluster structure, we give a sufficient condition on which $\Lambda_\beta$ admits infinitely many non-Hamiltonian-isotopic exact Lagrangian fillings based on the order of DT, solving a conjecture on the existence of Legendrian links that admit infinitely many fillings. This is joint work in progress with Honghao Gao and Linhui Shen.