Mar 28, 2016
Monday

09:45 AM  10:00 AM


Welcome

 Location
 
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10:00 AM  11:00 AM


SYZ mirror symmetry in the complement of a divisor and regular functions on the mirror
Denis Auroux (University of California, Berkeley)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We will give an overview of the StromingerYauZaslow (SYZ) approach to mirror symmetry in the setting of noncompact CalabiYau varieties given by the complement U of an anticanonical divisor D in a projective variety X. Namely, U is expected to carry a Lagrangian torus fibration, and a mirror CalabiYau variety U' can then be
constructed as a (suitably corrected) moduli space of Lagrangian torus fibers equipped with local systems. (Partial) compactifications of U deform the symplectic geometry of these Lagrangian tori by introducing holomorphic discs; counting these discs yields distinguished regular functions on the mirror U'. The goal of the talk will be to illustrate these concepts on simple examples, such as the complement of a conic in C^2.
If time permits we will also try to explain the relation of this story to the symplectic cohomology of U and its product structure
 Supplements


11:00 AM  11:30 AM


Break

 Location
 MSRI: Atrium
 Video


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11:30 AM  12:30 PM


Cluster algebras and cluster varieties
Lauren Williams (University of California, Berkeley)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will give a selfcontained introduction to cluster algebras and cluster varieties.
 Supplements


12:30 PM  02:30 PM


Lunch

 Location
 MSRI: Atrium
 Video


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02:30 PM  03:30 PM


Theta Functions for Log CalabiYau manifolds I II
Sean Keel (University of Texas, Austin)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In my two talks I'll explain the general features of my conjecture, joint with Gross, Hacking and Siebert, that the algebra of regular functions on an affine log CY (with maximal boundary), comes with a natural vector space basis, generalizing the characters of an algebraic torus, for which the structure constants for the multiplication rule are positive integers, counting holomorphic discs (on the mirror).
In particular, I'll explain how using these "theta functions" one can generalize the basic constructions of toric geometry. E.g. a single choice of anticanonical normal crossing divisor on a Fano Y (conjecturally) canonically determines for each line bundle L on Y a basis of sections parameterized by the integer points of a "polytope" (more precisely, a piecewise integer affine manifold with boundary). The talks will be at a very general level  in particular I won't assume any familiarity with cluster varieties or mirror symmetry
 Supplements


03:30 PM  04:00 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
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04:00 PM  05:00 PM


Scattering diagrams, broken lines and theta functions
Mark Gross (University of Cambridge)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
These talks will be part of the continuing exposition of the ideas of GHKK. I will focus in particular on the actual mechanics of how scattering diagrams for cluster algebras are constructed and give rise to theta functions via sums over broken lines.
 Supplements



Mar 29, 2016
Tuesday

10:00 AM  11:00 AM


Examples of cluster varieties and their scattering diagrams.
Paul Hacking (University of Massachusetts, Amherst)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We will describe some basic examples of cluster varieties from the viewpoint of our joint work with Gross, Keel, and Kontsevich
 Supplements


11:00 AM  11:30 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:30 AM  12:30 PM


Scattering diagrams, broken lines and theta functions
Mark Gross (University of Cambridge)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
These talks will be part of the continuing exposition of the ideas of GHKK. I will focus in particular on the actual mechanics of how scattering diagrams for cluster algebras are constructed and give rise to theta functions via sums over broken lines.
 Supplements



12:00 PM  02:30 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:30 PM  03:30 PM


Theta Functions for Log CalabiYau manifolds I II
Sean Keel (University of Texas, Austin)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In my two talks I'll explain the general features of my conjecture, joint with Gross, Hacking and Siebert, that the algebra of regular functions on an affine log CY (with maximal boundary), comes with a natural vector space basis, generalizing the characters of an algebraic torus, for which the structure constants for the multiplication rule are positive integers, counting holomorphic discs (on the mirror).
In particular, I'll explain how using these "theta functions" one can generalize the basic constructions of toric geometry. E.g. a single choice of anticanonical normal crossing divisor on a Fano Y (conjecturally) canonically determines for each line bundle L on Y a basis of sections parameterized by the integer points of a "polytope" (more precisely, a piecewise integer affine manifold with boundary). The talks will be at a very general level  in particular I won't assume any familiarity with cluster varieties or mirror symmetry
 Supplements


03:30 PM  04:00 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



04:00 PM  05:00 PM


Hitchin systems, spectral networks and noncommutative clusters
Andrew Neitzke (University of Texas, Austin)

 Location
 MSRI:
 Video

 Abstract
Hitchin's integrable system is one of the fundamental examples of mirror symmetry in the sense of StromingerYauZaslow.
Many of the structures described in this workshop can be seen very concretely in this example. This example has moreover some extra structure, coming from the fact that it is actually a hyperkahler space.
I will review an approach to these moduli spaces which arose in my joint work with Davide Gaiotto and Greg Moore. The key player in the story is a set of clustertype coordinate systems, very closely related to those appearing in the work of FockGoncharov. We will also see scattering diagrams very similar to those in the work of GrossHackingKeel.
In the end I will describe a new point: when one tries to extend the cluster description over certain singular loci of the moduli space, the most natural description seems to involve not an ordinary cluster algebra, but rather a noncommutative version thereof. One special case of this story appears closely related to the "noncommutative marked surfaces" recently introduced by BerensteinRetakh; there is also related work of GoncharovKontsevich.
 Supplements


05:00 PM  06:20 PM


Reception

 Location
 MSRI: Atrium
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Mar 30, 2016
Wednesday

10:00 AM  11:00 AM


Scattering diagrams from stability conditions
Tom Sutherland (University of Lisbon)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Stability conditions on triangulated categories allow us to define moduli spaces of objects in the category which undergo wallcrossing as the stability condition varies. We will consider certain CalabiYau3 triangulated categories whose space of stability conditions has a wallandchamber structure which "categorifies" mutation in a corresponding cluster algebra. Following a recent article of Bridgeland I will show how to enhance this wallandchamber structure to a scattering diagram which in certain cases coincides with the scattering diagram associated to the cluster algebra by GrossHackingKeelKontsevich
 Supplements


11:00 AM  11:30 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:30 AM  12:30 PM


DonaldsonThomas transformations for moduli spaces of Glocal systems on surfaces
Linhui Shen (Northwestern University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Kontsevich and Soibelman defined DonaldsonThomas invariants of a 3d CalabiYau category equipped with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in a single formal automorphism of the cluster variety, called the DTtransformation.
Let S be an oriented surface with punctures, and a finite number of special points on the boundary considered modulo isotopy. It give rise to a moduli space X(m, S), closely related to the moduli space of PGL(m)local systems on S, which carries a canonical cluster Poisson variety structure. For each puncture of S, there is a birational Weyl group action on the space X(m, S).
We calculate the DTtransformation of the moduli space X(m,S), with few exceptions.
We discuss the relationship between DTtransformations and duality conjectures in general.
In particular, our main result, combined with the work of Gross, Hacking, Keel and Kontsevich,
gives rise to a canonical basis in the space of regular functions on the cluster variety X(m,S), and in the upper cluster algebra with principal coefficients related to the pair (SL(m), S)
 Supplements



Mar 31, 2016
Thursday

10:00 AM  11:00 AM


DonaldsonThomas transformations for moduli spaces of Glocal systems on surfaces
Linhui Shen (Northwestern University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Kontsevich and Soibelman defined DonaldsonThomas invariants of a 3d CalabiYau category equipped with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in a single formal automorphism of the cluster variety, called the DTtransformation.
Let S be an oriented surface with punctures, and a finite number of special points on the boundary considered modulo isotopy. It give rise to a moduli space X(m, S), closely related to the moduli space of PGL(m)local systems on S, which carries a canonical cluster Poisson variety structure. For each puncture of S, there is a birational Weyl group action on the space X(m, S).
We calculate the DTtransformation of the moduli space X(m,S), with few exceptions.
We discuss the relationship between DTtransformations and duality conjectures in general.
In particular, our main result, combined with the work of Gross, Hacking, Keel and Kontsevich,
gives rise to a canonical basis in the space of regular functions on the cluster variety X(m,S), and in the upper cluster algebra with principal coefficients related to the pair (SL(m), S).
 Supplements


11:00 AM  11:30 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:30 AM  12:30 PM


Mirror symmetry for homogeneous spaces
Clelia Pech (University of Kent at Canterbury)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In this talk I will explain a construction by K. Rietsch of the mirrors of homogeneous spaces. The latter include Grassmannians, quadrics and flag varieties, and their mirrors can be expressed using Lie theory. More precisely the mirrors of homogeneous spaces live on socalled `Richardson varieties', which possess a cluster structure, and the mirror superpotential is defined on these Richardson varieties.
I will start by detailing Rietsch's general construction, then I will present some recent results by MarshRietsch on Grassmannians, as well as joint work with Rietsch (resp. Rietsch and Williams) on Lagrangian Grassmannians (resp. quadrics). In particular I will show how the restriction of the superpotential on each cluster chart is a Laurent polynomial, which changes as we change cluster charts
 Supplements


12:30 PM  02:30 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:30 PM  03:30 PM


Cluster duality and mirror symmetry for the Grassmannian
Lauren Williams (University of California, Berkeley)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In joint work with Konstanze Rietsch, we use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes. From a given plabic graph G we have two coordinate systems: we have a positive chart for our Amodel Grassmannian, and we have a cluster chart for our Bmodel (LandauGinzburg model) Grassmannian. On the Amodel side, we use the positive chart to associate a corresponding NewtonOkounkov (Amodel) polytope. On the Bmodel side, we use the cluster chart to express the superpotential as a Laurent polynomial, and by tropicalizing this expression, we obtain a Bmodel polytope. Our main result is that these two polytopes coincide
 Supplements


03:30 PM  04:00 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



04:00 PM  05:00 PM


Cluster Algebras and Exact Lagrangian Surfaces
Harold Williams (University of Texas, Austin)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We explain a general relationship between cluster theory and the classification of exact Lagrangian surfaces in Weinstein 4manifolds. A key point is the introduction of an operation on singular Lagrangian skeleta which geometrizes the notion of quiver mutation. This lets us produce large classes of exact Lagrangians labeled by clusters in an associated cluster algebra. When the manifold in question is a cotangent bundle and the exact Lagrangians fill a suitable Legendrian knot in its contact boundary, the microlocalization theory of KashiwaraSchapira recovers the cluster structures on positroid strata and moduli spaces of local systems from this symplectic paradigm. This is joint with Vivek Shende and David Treumann, part of which is also joint with Eric Zaslow
 Supplements



Apr 01, 2016
Friday

10:00 AM  11:00 AM


Wall structures in mirror symmetry
Bernd Siebert (Universität Hamburg)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In the talk I will try to explain the appearance and interpretation of wall crossing structures in mirror symmetry
 Supplements


11:00 AM  11:30 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:30 AM  12:30 PM


Mirror symmetry for homogeneous spaces
Clelia Pech (University of Kent at Canterbury)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In this talk I will explain a construction by K. Rietsch of the mirrors of homogeneous spaces. The latter include Grassmannians, quadrics and flag varieties, and their mirrors can be expressed using Lie theory. More precisely the mirrors of homogeneous spaces live on socalled `Richardson varieties', which possess a cluster structure, and the mirror superpotential is defined on these Richardson varieties.
I will start by detailing Rietsch's general construction, then I will present some recent results by MarshRietsch on Grassmannians, as well as joint work with Rietsch (resp. Rietsch and Williams) on Lagrangian Grassmannians (resp. quadrics). In particular I will show how the restriction of the superpotential on each cluster chart is a Laurent polynomial, which changes as we change cluster charts
 Supplements


12:30 PM  02:30 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:30 PM  03:30 PM


Generic bases are dual semicanonical bases for unipotent cells
Christof Geiss (UNAM  Universidad Nacional Autonoma de Mexico)

 Location
 
 Video

 Abstract
This is a report on joint work with B. Leclerc and J. Schröer.
Recall that Derksen, Weyman and Zelvinsky managed to prove many combinatorial conjectures about cluster algebras with symmetric exchange matrix by studying certain representations of the corresponding quiver together with a nondegenerate potential.
We proposed in [GLS2], that in this situation the generic values of Palu's cluster character on the strongly reduced components of the corresponding representation varieties should be a good candidate for a basis of the cluster algebra. It is not hard to see that these elements have the following properties:
(a) contain all cluster monomials,
(b) they are independent of the mutation class,
(c) belong to the upper cluster algebra.
On the other hand, we showed in [GLS2] that the dual of Lusztig's semicanonical basis (which is defined for the enveloping algebra of the positive part of any KacMoody Lie algebra with symmetric Cartan matrix) restricts to a basis of the coordinate ring of corresponding unipotent cell associated to an Weyl group element.
This dual semicanonical basis contains all cluster monomials.
In [GLS2] we showed, that in the above mentioned candidate for a "generic basis" and the dual semicanonical basis coincide.
In this talk I will try to explain this development in the special case of a symmetric Cartan matrix of finite type and the longest element in the Weyl group.
References
[GLS1] Ch. Geiss, B Leclerc, J Schröer:
Kac–Moody groups and cluster algebras
Advances in Mathematics 228 (2011), 329433
[GLS2] Ch. Geiss, B Leclerc, J Schröer:
Generic bases for cluster algebras and the Chamber Ansatz Journal of the American Mathematical Society 25 (2012), 2176
 Supplements


03:30 PM  04:00 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



04:00 PM  05:00 PM


Stability conditions and cluster varieties
Tom Sutherland (University of Lisbon)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We will study the geometry of cluster varieties from the perspective of stability conditions on the associated CalabiYau3 triangulated category. I will focus on ideas introduced by GaiottoMooreNeitzke which suggest how to produce cluster coordinates from stability conditions. In particular we will consider the class of examples associated to triangulations of marked bordered surfaces for which the cluster variety is a moduli space of rank 2 local systems and the space of stability conditions is a space of quadratic differentials with prescribed singularities on an associated closed surface
 Supplements


