Talks

Dylan Allegretti - Quiver representations, cluster varieties, and categorification of canonical bases

Associated to a compact oriented surface with marked points on its boundary is an interesting class of finite-dimensional algebras. These algebras are examples of gentle algebras, and their representation theory has been studied by many authors in connection with the theory of cluster algebras. An important fact about these algebras is that their indecomposable modules come in two types: string modules, which correspond to arcs connecting marked points on the surface, and band modules, which correspond to closed loops on the surface. Thanks to the work of many mathematicians, the string modules are known to categorify generators of a cluster algebra. In this talk, I will explain how, by including band modules in this story, one can define a family of graded vector spaces which categorify Fock and Goncharov's canonical basis for the algebra of functions on an associated cluster variety. These vector spaces are of interest in mathematical physics, where they are expected to provide a mathematical definition of the space of framed BPS states from the work of Gaiotto, Moore, and Neitzke.

Lara Bossinger - Degenerations of Grassmannians

In the first part of the talk I will focus on the Grassmannian of planes and the corresponding cluster algebra of type A. We show that universal coefficients (as introduced by Fomin-Zelevinsky) give rise to a Gröbner cone in the Gröbner fan of the Plücker ideal. Among the faces of the cone we identify one for every toric variety obtained from a principal coefficients construction.

As a corollary we get a unimodular equivalence between two polytopes defined by tropicalized superpotentials: the Marsh-Rietsch potential and the Gross-Hacking-Keel-Kontsevich potential.

In the second part of the talk we will discuss how the identification of the polytopes can be extended to arbitrary Grassmannians. More details will be discussed in the following talk by Timothy Magee.

The first part of the talk is based on joint work in progress with Fatemeh Mohammadi and Alfredo Nájera Chávez. The second part is based on joint work in progress with Man-Wai Cheung, Timothy Magee and Alfredo Nájera Chávez.

Pierrick Bousseau - Towards DAHA from tropical geometry

I will present a purely tropical construction of a wall structure, which conjecturally produces as output the spherical double affine Hecke algebra for gl(n) (with a canonical basis).

Alfredo Nájera Chávez - Toric degenerations of cluster varieties and cluster duality

In this talk we will introduce cluster Poisson varieties with coefficients. We use this notion to build a flat degeneration of every specially completed cluster Poisson variety (in the sense of Fock-Goncharov) to the toric variety associated to its cluster complex. If time permits we will discuss the following applications:

1) We can define c-vectors of theta functions on cluster Poisson varieties.

2) We can relate cluster duality and Batyrev duality of Gorenstein toric Fanos in the context of mirror symmetry.

This is based on joint work with Lara Bossinger, Bosco Frías-Medina and Timothy Magee.

Man-Wai Cheung - Tropical disks counting, stability conditions in symplectic geometry and quiver representations

Bridgeland developed stability scattering diagrams which then we can discuss scattering diagrams with quiver representations. Scattering diagrams were proposed by Kontsevich-Soibelman, and Gross-Siebert to describe the toric degenerations of Calabi-Yau varieties. The diagrams encode further geometry. Together with Travis Mandel, we relate tropical disks counting with quiver representations by using the stability scattering diagrams. Next, together with Yu-Wei Fan and Yu-Shen Lin, we look at the stable objects for the A2 quiver. It is known that the derived Fukaya-Seidel category of the rational elliptic surface is the derived category of the A2 quiver. Thus we wish to match the special Lagrangian with the stable objects in the derived category of coherent sheaves.

Sara Angela Filippini - Orbital degeneracy loci and applications

We consider a generalization of degeneracy loci of morphisms between vector bundles modelled on orbit closures in representations of algebraic groups. In a preferred class of examples we gain some control over their canonical sheaf. We show how these techniques can be applied to construct Calabi-Yau and Fano varieties of dimension three and four. This is joint work with Vladimiro Benedetti, Laurent Manivel and Fabio Tanturri.

Vladimir Fock - Clusters and plane curve singularities

Tyler Kelly - Open FJRW Theory

I will talk about an open version of FJRW theory which is the enumerative theory for certain gauged Landau-Ginzburg models. I will do this with an eye for explaining what wall crossing looks like for these enumerative invariants. This work is joint with M. Gross and R. Tessler.

Timothy Magee - Grassmannians and mirror symmetry for cluster varieties

At the start of the talk I will discuss a Landau-Ginzburg model for the Grassmannian together with a certain anticanonical divisor, following Gross-Hacking-Keel-Kontsevich. This LG model will lead us in two different directions. First, we will identify the LG model with an LG model of Marsh-Rietsch, and in turn give new interpretations of ideas explored by Rietsch-Williams, as well as new proofs of some of their results. Next, we will draw connections between the LG model and Batyrev-Borisov's construction of (conjecturally) mirror families of Calabi-Yau subvarieties of toric Fanos. These connections occur for other spaces as well, and for the second half of the talk I will discuss our efforts to extend Batyrev-Borisov to the cluster setting. Based on ongoing joint work, primarily a paper with L. Bossinger, M.-W. Cheung, and A. Nájera Chávez and a paper with M.-W. Cheung.

Navid Nabijou - Gromov-Witten theory in genus one and the joy of desingularisation

The toroidal approach to logarithmic geometry views a logarithmic scheme as a scheme equipped with a “locally toric structure.” Viewed through this lens, various standard constructions in toric geometry (for instance: toric blowups, line bundles associated to piecewise-linear functions, etc.) can be extended to the logarithmic setting. This has myriad applications, one of the most exciting of which is to the study of moduli spaces and enumerative invariants.

In this talk I will explore this circle of ideas, basing the discussion around the interplay with polyhedral geometry. I will conclude by discussing joint work with L. Battistella and D. Ranganathan, in which we use log geometry to produce a desingularisation of the moduli space of relative stable maps in genus one, to carefully study the boundary strata of this space and to derive recursion formulae for Gromov-Witten invariants of hypersurfaces.

Clelia Pech - Mirror symmetry for (cominuscule) homogeneous spaces

In this talk, based on joint work with K. Rietsch and L. Williams, I will explain some constructions of Landau-Ginzburg models for some cominuscule homogeneous spaces. The origin of these constructions is the Lie-theoretic mirror of K. Rietsch. More precisely, the mirror of a homogeneous space lives on a Langlands dual Richardson variety, which possesses a cluster structure, and the mirror superpotential is defined on this variety. I will start by recalling Rietsch's construction, then I will present some results on Lagrangian Grassmannians and quadrics.

Thomas Prince - New Calabi-Yau threefolds via the Gross-Siebert algorithm

We describe how to use to Gross--Siebert algorithm to construct three dimensional polarised tropical manifolds - and hence Calabi-Yau threefolds - from a class of four dimensional reflexive polytopes. These (simply connected) Calabi-Yau threefolds are expected to be anti-canonical sections of Fano fourfolds with isolated Gorenstein singularites. We explain how to compute topological invariants of these spaces using a constructible sheaf on the 1-skeleton of the polytope, and provide an example of a single polytope from which 91 such tropical manifolds can be constructed, providing 22 topological types of Calabi--Yau threefolds; 9 of which are new topological types with b_2=1.

Sibylle Schroll - A geometric model for gentle algebras

Gentle algebras are a class of tame algebras which naturally arise in various different contexts such as categorifications of cluster algebras, N=2 gauge theories and Fukaya categories of surfaces. In this talk we will describe a geometric model of the bounded derived category of a gentle algebra and we will show how this model relates to the Fukaya category of a surface with boundary and stops. We will use the geometric model to give a complete derived invariant for gentle algebras.

Harold Williams - Kasteleyn operators from mirror symmetry

In this talk we explain an interpretation of the Kasteleyn operator of a doubly-periodic bipartite graph from the perspective of homological mirror symmetry. Specifically, given a consistent bipartite graph $\Gamma$ in $T^2$ with a complex-valued edge weighting $\mathcal{E}$ we show the following two constructions are the same. The first is to form the Kasteleyn operator of $(\Gamma, \mathcal{E})$ and pass to its spectral transform, a coherent sheaf supported on a spectral curve in $(\mathbb{C}^\times)^2$. The second is to take a certain Lagrangian surface $L \subset T^* T^2$ canonically associated to $\Gamma$, equip it with a brane structure prescribed by $\mathcal{E}$, and pass to its mirror coherent sheaf. This lives on a toric compactification of $(\mathbb{C}^\times)^2$ determined by the Legendrian link which lifts the zig-zag paths of $\Gamma$ (and to which the noncompact Lagrangian $L$ is asymptotic). We work in the setting of the coherent-constructible correspondence, a sheaf-theoretic model of toric mirror symmetry. This is joint work with David Treumann and Eric Zaslow.

Tony Yue Yu - Cluster algebra via non-archimedean geometry

I will explain the enumeration of non-archimedean curves in cluster varieties. We can construct a scattering diagram via the enumeration of infinitesimal non-archimedean cylinders and prove its consistency. Then we prove a comparison theorem with the combinatorial constructions of Gross-Hacking-Keel-Kontsevich. This has several very nice implications, such as the broken-line convexity conjecture, a geometric proof of the positivity in the Laurent phenomenon, and the independence of the mirror algebra on the choice of cluster structure, as conjectured by GHKK. This is part of my joint work with S. Keel.