Institute for Geometry and Physics Miami-Cinvestav-Campinas Collaboration in Geometry & Physics in the Americas

Conference on Homological Mirror Symmetry – January 28-February 1, 2013

Organizers: Denis Auroux, Ludmil Katzarkov, Maxim Kontsevich, Elizabeth Gasparim, Ernesto Lupercio, Tony Pantev

This event will be held at the University of Miami (Coral Gables, Florida), and will be followed by the Third Latin Congress on Symmetries in Geometry and Physics (Maranhão, Brazil, February 1-9, 2013).

New: A very incomplete (so far) set of notes from the talks is now available here.

The workshop will start on Monday morning (January 28) and will end on Friday afternoon (February 1). There will be three mini-courses of three lectures each, individual research talks, as well as time for informal discussions.

Talks: The conference will be held in the University Center, Flamingo Ballroom C & D (2nd Floor). See the campus map for directions (the arrows show the path from the hotel); there's also a newer campus map.

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: 1-305-667-5611.

Invited/registered participants: please send your arrival and departure dates to Dania Puerto, dania@math.miami.edu, as soon as possible, in order to secure a room reservation. We will soon have to release the block of rooms reserved for the conference and rates will go up.

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 US-based 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.

Monday January 28

Tuesday January 29

Wednesday January 30

Thursday January 31

Friday February 1

Minicourses:

• Mohammed Abouzaid (Columbia and SCGP): Dilations, equivariance, and formality in symplectic topology

  Abstract: Mirror symmetry provides a dictionary which can be used to pass from algebraic geometry to symplectic topology. An instance of such a passage is the notion of a Floer-theoretic dilation was introduced by Seidel and Solomon as the mirror of an infinitesimal automorphism of a Calabi-Yau manifold which dilates the holomorphic volume form. The lectures will (1) introduce the Floer theoretic and mirror symmetry background required to define these structures, (2) survey applications to symplectic topology due to Seidel and A-Smith, and (3) discuss the available geometric methods for constructing dilations.

• Sean Keel (UT Austin): Mirror symmetry for CYs with maximal boundary

  Abstract: I will explain my conjecture, joint with Hacking and Gross, which gives the mirror of an open affine CY manifold with maximal boundary. The mirror is given as a vector space with a canonical basis, a natural Mori theoretic generalisation of Thurston's space of laminations on a punctured Riemann surface, with a multiplication rule determined by counts of rational curves. I'll explain how the classical theta function for Abelian varieties, the natural trace functions on the Hitchin space of SL₂ local systems on a punctured Riemann surface, and the Fock-Goncharov dual basis conjecture for cluster varieties are very special cases. I will explain our main results -- a proof of the conjecture in dimension two, and partial results, e.g. the acyclic case, for cluster varieties.

• Maxim Kontsevich (IHES):
  I. On complex moduli for Landau-Ginzburg models

  Abstract: (joint work with L.Katzarkov and T.Pantev) I'll formulate and prove a Tian-Todorov type result for LG models, and a conjectural description of Hodge numbers for mirror dual Fano varieties.

  II. Weak Calabi-Yau algebras

  Abstract: (joint work with Y.Vlassopoulos) I will discuss the notion of weak Calabi-Yau algebra, combining finite-dimensional and homologically smooth cases. This is a generalization of a smooth variety endowed with a section of the anticanonical bundle.

  III. Algebraicity and integrality for DT-series.

  Abstract: In many circumstances the generating series of DT-invariants are algebraic. Moreover, the corresponding canonical transformations preserve not only symplectic form but also a class in K₂ (like cluster mutations). This constraint implies integrality of DT-invariants.

Research talks:

• Mina Aganagic (Berkeley): Knots and Mirror Symmetry

  Abstract: I will describe a recent conjecture relating knot theory and mirror symmetry. To every knot K, one can associate a Calabi-Yau manifold YK. The conjecture is that YK arises by a generalization of SYZ mirror symmetry, as mirror to the conifold, O(-1)+O(-1) -> P¹. The conifold has infinitely many mirrors, one for each knot. Moreover, by studying B-model on YK with a collection of B-branes, one should recover quantum invariants of the knot, colored by arbitrary representations. The talk is based on joint work with C. Vafa, and followup work together with T. Ekholm and L. Ng.

• Rina Anno (Pittsburgh): DG approach to Fourier-Mukai transforms

• Alexander Efimov (RAS): Homotopy finiteness of DG categories from algebraic geometry

  Abstract: We will explain that for any separated scheme of finite type over a field of characteristic zero, the DG category D^bcoh(X) is homotopically finitely presented. This uses categorical resolution of singularities by Kuznetsov and Lunts, and confirms a conjecture of Kontsevich. We will also explain how to show the analogous result for coherent matrix factorizations on such schemes.

• David Favero (Vienna): Homological Mirror Symmetry for non-Fano toric varieties, Part I

  Abstract: Following the work of Auroux, Katzarkov, and Orlov, I will describe a framework for Homological Mirror Symmetry for (non-Fano) toric varieties. This is obtained by pairing semi-orthogonal decompositions of both the bounded derived category of coherent sheaves on a toric variety and the Fukaya-Seidel category of its Landau-Ginzburg mirror coming from toric minimal model runs. This is joint work with M. Ballard, C. Diemer, L. Katzarkov, and G. Kerr.

• Sheel Ganatra (Stanford): Symplectic cohomology and duality for the wrapped Fukaya category

  Abstract: To an exact symplectic manifold M, one can associate two important A-model invariants, symplectic cohomology SH*(M) and the wrapped Fukaya category W(M). We will explain how, when M contains enough Lagrangians, the natural geometric open-closed string maps between the Hochschild homology of W(M), symplectic cohomology, and the Hochschild cohomology of W(M) are all isomorphisms. The induced isomorphism between Hochschild homology and cohomology is an instance of a new self-duality for the wrapped Fukaya category, a non-compact version of a Calabi-Yau structure.

• Alexander Goncharov (Yale): G-laminations and canonical bases

  Abstract: A pair (G; S), where G is a split reductive group and S a decorated surface, gives rise to a moduli space A(G;S), a positive space related to the moduli space of G-local systems on S. We introduce a rational function W on A(G;S), the potential. Its tropicalisation determines a subset of integral tropical points of A(G;S), which we call a set of G-laminations on S. We prove that G-laminations parametrise top components of an infinite dimensional stack, the surface affine Grassmannian. These cycles materialise canonical bases. For example, when S is a disc with special points on the boundary, we get canonical basis in invariants of tensor products of representation of the Langlands dual group. We suggest that the potential W itself, not only its tropicalization, is important, providing the potential for a Landau-Ginzburg model on A(G;S). We conjecture that the mirror dual to the pair (A(G,S); W) is the moduli space of local systems on S for the Langlands dual group. This is a joint work with Linhui Shen (Yale).

• Mark Gross (UCSD): Gluing for log GW invariants

• Fabian Haiden (Vienna): Quadratic differentials of exponential type and stability

  Abstract: This is a report on work in progress relating spaces of stability conditions of some triangulated categories of tame representation type to moduli of flat surfaces with infinite-angle singularities. Joint with L. Katzarkov and M. Kontsevich.

• Paul Horja (Vienna): Derived categories of coherent sheaves on toric stacks

  Abstract: I will discuss an approach towards the understanding of the categories of coherent sheaves on toric stacks using a version of the grade restriction rule of Herbst-Hori-Page and Ballard-Favero-Katzarkov for transport across moduli spaces. The construction can be interpreted as a categorification of the Gelfand-Kapranov-Zelevinsky combinatorial framework.

• Dmitry Kaledin (RAS): Beilinson-Kato conjecture for finite-dimensional algebras

  Abstract: I am going to discuss the first of Beilinson conjectures on L-functions -- namely, the conjecture relating K-theory and Hodge-Deligne cohomology of varieties over Q. I am going to explain that the conjecture, in an elegant reformulation due to Kato, makes sense for non-commutative algebras, and moreover, is almost obvious when the algebra is finite-dimensional.

• Mikhail Kapranov (Yale): Derived schemes beyond [0,1]

  Abstract: The most familiar type of derived schemes is that of [0,1]-schemes (also known as quasi-smooth derived schemes): those for which the tangent dg-space at any point has cohomology in degrees 0 and 1 only. They give rise to perfect obstruction theories.

• Gabriel Kerr (U. Miami): Homological Mirror Symmetry for non-Fano toric varieties, Part 2

  Abstract: I will discuss a toric compactification of LG models that are mirror to smooth projective nef Fano toric stacks. The fixed points of this compactification are in bijective correspondence with straight line minimal model runs of the mirror toric stack. These points also parameterize maximal degenerations of the LG model and have a tropical interpretation. I will illustrate a new example of the HMS conjecture when the nef Fano condition fails and discuss a general strategy to prove HMS for general smooth projective toric stacks.

• Bernd Siebert (Hamburg): On standard constructions in mirror symmetry

  Abstract: In the degeneration approach to mirror symmetry often much can be gained by considering higher dimensional parameter spaces. In the talk I will illustrate this claim by several examples including type II degenerations, conifold transitions and non-toric surface singularities.

• Yan Soibelman (Kansas State): Wall-crossing structures

  Abstract: I am going to discuss mathematical structures which unify wall-crossing formulas appearing in several areas of mathematics (theory of Donaldson-Thomas invariants, cluster transformations, complex integrable systems of Hitchin type, count of geodesics of quadratic differentials, Mirror Symmetry, etc.) and physics (2d and 4d gauge theories, BPS count for supersymmetric black holes, etc.).