Institute for Geometry and
Physics

Organizers: Denis Auroux, Ludmil Katzarkov, Maxim Kontsevich, Elizabeth Gasparim, Ernesto Lupercio
This event will be held at the University of Miami (Coral Gables, Florida).
The workshop will start on Monday morning (January 26) and will end on Saturday (January 31) midday. There will be four minicourses of three lectures each, individual research talks, as well as time for informal discussions.
The workshop is partially supported by the NSF FRG ``Wallcrossings in Geometry and Physics'' (grants DMS1265228, 1265196, 1264662, 1265230, 1262531).
Venue: The conference will be held in the BankUnited Center Field House on all days (including Wednesday contrary to what was announced earlier). See this campus map (or that one).
Accommodation: Most participants will be staying at the Holiday Inn Coral Gables/University of Miami, located right next to the University of Miami campus. The hotel address is: 1350 South Dixie Highway, Coral Gables FL 33146; phone: 13056675611.
Invited/registered participants: please send your arrival and departure dates to Dania Puerto, dania@math.miami.edu, by December 1, in order to secure a room reservation. Students will be expected to share rooms; if you have a preferred roommate please indicate their name in your message to Dania, otherwise we will assign one for you.
Registration fee: there will be a nominal registration fee to cover the cost of refreshments.
Airport: Miami International Airport is about 7 miles from campus. The most convenient way to reach the hotel or the campus is to take a taxi. Invited participants: When booking flights, please keep in mind NSF and University of Miami rules. Only USbased airlines are permitted (more specifically your plane ticket mush be issued by a US air carrier and show US carrier flight numbers). Please keep all your original boarding passes since they will be needed for reimbursement.
9:3010:30:  Kontsevich I 
11:0012:00:  Moore I 
Lunch break  
2:003:00:  Voisin I 
3:304:30:  Abouzaid 
5:006:00:  Soibelman 
9:3010:30:  Seidel I 
11:0012:00:  Moore II 
Lunch break  
2:003:00:  Kontsevich II 
3:304:30:  Ganatra 
5:006:00:  Keating 
9:0010:00:  Seidel II 
10:1511:15:  Voisin II 
11:4512:45:  Perutz 
Free afternoon 
9:3010:30:  Kontsevich III 
11:0012:00:  Moore III 
Lunch break  
2:003:00:  Goncharov 
3:304:30:  Toda 
5:006:00:  Corti 
9:3010:30:  Seidel III 
11:0012:00:  Voisin III 
Lunch break  
2:003:00:  Fukaya 
3:304:30:  Lunts 
5:006:00:  Kapranov 
9:0010:00:  Pantev 
10:1511:15:  Kaledin 
11:4512:45:  Efimov 
Lectures 1 and 2: Holomorphic Floer cohomology
Abstract: For a complex symplectic manifold M and two holomorphic Lagrangian submanifolds L_{0}, L_{1} I define holomorphic Floer cohomology HF(L_{0},L_{1}) as a holomorphic bundle of the complex line of Planck constants. In the complement to Stokes rays this bundle is identified with global vanishing cycles of the intersection locus, and pseudoholomorphic discs controls the jumps along Stokes rays. In the case of M being cotangent bundle to a curve and of vertical L_{0}, this construction gives a new interpretation of GaiottoMooreNeitzke solution of Hitchin equations via wallcrossing. This is a joint work in progress with Y. Soibelman.
Lecture 3: Iterated stability
Abstract: I'll give an update on the project from last year on defining Fukaya categories with coefficients and constructing stability conditions. In particular, I'll describe explicitly all semistable objects in the derived category of the product of elliptic curves defined over a multidimensional local field, for a large family of stability conditions which are far from the degenerate ones. All "special Lagrangians" are described as multidimensional piecewise linear objects. This is a joint work in progress with F. Haiden, L. Katzarkov and P. Pandit.
Abstract: In their study of mirror symmetry for del Pezzo surfaces, Auroux, Katzarkov and Orlov encountered the following phenomenon: fibrewise compactification (properification) of the LandauGinzburg model leaves its Amodel (Fukaya) category unchanged. By now, this is an accepted part of the ideas surrounding mirror symmetry, and has explanations from various viewpoints. In the lectures, we will focus purely on the symplectic geometry aspect.
Abstract: The lectures will discuss cohomological and Chowtheoretic obstructions to rationality or stable rationality of complex projective varieties. Of course, there are many such geometric obstructions, like the plurigenera, and we will rather focus on the case of rationally connected varieties, where these obvious obstructions do not appear. We will discuss 1) unramified cohomology and its link to the integral Hodge conjecture, and 2) various notions of decompositions of the diagonal.
Abstract: We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We summarise the available evidence. The conjectures relate mutationequivalence classes of Fano polygons with QGorenstein deformation classes of del Pezzo surfaces.
Abstract: We will define a certain class of smooth and proper DG categories, which we call weakly geometric.These are just retracts of derived categories of smooth projective varieties (in the Morita homotopy category of DG categories). We will give a criterion of weak geometricity in terms of noncommutative K_0motives. As an application, we will show for example that any phantom DG category, as well as the derived category of determinantal Barlow surface, can be represented as a retract of some iterated projectivization of vector bundles.
Abstract: To prove various properties of GromovWitten invariants, Floer homology etc. we need to use a map between moduli spaces which forget certain data such as marked points or maps. We need to establish consistency of perturbation and forgetful map. This point looks so much technical but it actually becomes most delicate point in order to work out the whole story. To do it in complete generality is in general impossible. I will explain what kinds of properties we can establish using forgetful map, the method to realize it, and what are the difficulty to realize it.
Abstract: The Fukaya category of a LandauGinzburg model (M,W) enlarges the Fukaya category of M by including certain noncompact Lagrangians with asymmetric perturbations at infinity involving W. I will explain work in progress with Mohammed Abouzaid giving a criterion, in the spirit of work of Abouzaid/AbouzaidFukayaOhOhtaOno, for when a finite collection of Lagrangians splitgenerates this Fukaya category. As first applications we show the derived Fukaya category is preserved by quadratic stabilization and give a new proof of exact triangles associated to (fibered) Dehn twists. The new ingredients include Serre bimodules and a closed string Floer homology ring associated to (M,W), conjecturally mirror to polyvector fields.
Abstract: Given a split reductive group G and a surface with n
punctures S,
there is a dual pair of spaces closely related to the spaces of G / G^{L}
local systems on S: A(G,S) and X(G^{L},S).
Here G^{L} is the Langlands dual group.
Duality conjectures predict a canonical
pairing between integral tropical points of one space and the other space.
It provides a collection of functions on one of the spaces
parametrised by the tropical points of the other.
These functions on the space A(G,S) can not span
the space of all regular functions. To describe the subspace they should
span, we define a birational action of the group W^{n},
where W the Weyl group of G, on each of the spaces.
The image should consist of all functions which remain regular under the
W^{n} action.
The longest element of the W^{n} should give the DTtransformation.
We expect a similar picture in the cluster setup with DTtransformation
replacing the Waction.
We will also discuss the mirror symmetry incarnation of this picture,
and most general dual pairs for decorated surfaces.
This is a joint work with Linhui Shen.
Abstract: Periodic cyclic homology is defined by taking the producttotal complex of a certain bicomplex. For algebras over Q, taking the sumtotal complex of the same bicomplex gives 0. It has been suggested by Kontsevich some years ago that in characteristic p, the sumtotal complex is a nontrivial and interesting invariant. At the time, the suggestion was not pursued seriously; however, recently a very similar phenomenon appeared in the work of Beilinson and Bhatt on padic Hodge theory. I want to revisit the subject and follow through on Kontsevich's idea, both for algebras and DG algebras (where there are even more possibilities for interesting theories, five in total). I will also explain how this is related to the noncommutative Hodgetode Rham degeneration.
Abstract: Many explicit descriptions of perverse sheaves admit natural categorifications. We call the resulting categorified objects ("perverse sheaves of triangulated categories") perverse Schobers. When the base manifold is a (topological, or Riemann) surface, perverse Schobers can be defined explicitly, in terms of certain diagrams of categories and spherical functors. This suggests a natural approach to defining "Fukaya category with coefficients in a perverse Schober", for the case of a punctured surface. The talk is based on joint work and projects with V. Schechtman and Y. Soibelman.
Abstract: We consider Milnor fibres of nonADE (i.e., positive modality) isolated hypersurface singularities, focusing on the threevariable case, and modalityone singularities of type T_{p,q,r}. We will explain features which do not arise from PicardLefschetz theory, including existence of exact Lagrangian tori, and properties of the symplectic mapping class group. Time allowing, we will sketch workinprogress regarding mirror symmetry for T_{p,q,r}, and candidate mirrors to the symplectic phenomena observed.
Abstract: This will be a survey of some interesting motivic measures which have connections to birational geometry, derived categories of coherent sheaves, categories of matrix factorizations and motivic vanishing fibers. I will also say about a work in progress on motivic McKay correspondence.
Abstract: I will report on joint work with Nick Sheridan and, in part, Sheel Ganatra, concerning mirror symmetry for CalabiYau manifolds. I will explain how to use the Mukai pairing to show that a hypothesis of "core homological mirror symmetry" (a relatively tractable fragment of Kontsevich's HMS conjecture) implies that the closedopen string map is an isomorphism. In the last part of the talk I'll outline a scheme to show how HMS determines the normalization of the holomorphic volume form and the mirror map, whereby Sheridan's work on HMS for the quintic 3fold implies the formula predicted by Candelas et al. for rational curvecounts on the quintic.
Abstract: I am going to explain a general approach to algebraicity of certain generating series which appear in DonaldsonThomas theory and cluster algebras. It is based on the techniques of nonarchimedean and tropical geometry. This is a joint project with Maxim Kontsevich.
Abstract: About four years ago, together with Bayer and Macri, I proposed a conjectural BogomolovGieseker type inequality for Chern characters of certain two term complexes in the derived category of coherent sheaves on projective 3folds. I will explain the current status of the above conjecture, and show that it implies the existence of algebraic moduli stacks of Bridgeland semistable objects in the derived category. This result is used to construct DonaldsonThomas invariants counting Bridgeland semistable objects on CalabiYau 3folds which are deformation invariant. This is a work in progress with Dulip Piyaratne.
Mohammed Abouzaid 
Heather Lee Yanki Lekili Timothy Logvinenko Valery Lunts Ernesto Lupercio Andrew MacPherson Greg Moore Alexander Noll Ignacio Otero Pranav Pandit Tony Pantev SeoRee Park James Pascaleff Tim Perutz Victor Przyjalkowski Larry Richardson Helge Ruddat Olaf Schnürer Paul Seidel Nick Sheridan Artan Sheshmani Yan Soibelman Zack Sylvan Yukinobu Toda Umut Varolgunes Sara Venkatesh Michael Viscardi Claire Voisin Miguel Xicotencatl Tony Yue Yu Jingyu Zhao Ilia Zharkov Peng Zhou 