## Invited Lectures :

**Mina Aganagic (Berkeley): Knot Homology from Refined Chern-Simons Theory.**

Abstract: We formulate a refinement of SU(N) Chern-Simons theory via the refined topological string. The refined Chern-Simons theory is defined on any three-manifold with a semi-free circle action. We give an explicit solution of the theory, in terms of a one-parameter refinement of the S and T matrices of Chern-Simons theory, related to the theory of Macdonald polynomials. The ordinary and refined Chern-Simons theory are similar in many ways; for example, the Verlinde formula holds in both. We obtain new topological invariants of Seifert three-manifolds and torus knots inside them. We conjecture that the knot invariants we compute are the Poincare polynomials of the sl(n) knot homology theory. The latter includes the Khovanov-Rozansky knot homology, as a special case. The conjecture passes a number of nontrivial checks. We show that, for a large number of torus knots colored with the fundamental representation of SU(N), our knot invariants agree with the Poincare polynomials of Khovanov-Rozansky homology. As a byproduct, we show that our theory on S^3 has a dual description in term of the refined topological string on X=O(-1)+O(-1)->P^1. This supports the conjecture by Gukov, Schwarz and Vafa relating the spectrum of BPS states on X to sl(n) knot homology.**Fedor Bogomolov (NYU): On rationality of the fields of invariants of linear actions for connected nonsemisimple algebraic groups**(based on my joint work with Christian Boehning and Hans-Christian Graf von Bohtmer).

Abstract: In the talk I want to describe the main steps for the proof of the following general result. Let $AG$ be an affine extension of one of the following complex groups $Sl(n),Sp(2n),SO(n),G_2$ and $V$ be a linear representation of $AG$ with a generically free action. Then the field of invariant $C(V)^{AG}$ is rational for (almost) all $V$. I will explain the meaning of "almost" in my talk.**Fabrizio Catanese (Bayreuth): Special Galois coverings and the singular set of the the moduli space of curves**

Abstract: The Singular locus of the compactified Moduli space of curves $ \overline{\mathfrak M_g}$ corresponds to loci of curves with automorphisms. The description of an irredundant irreducible decomposition for the Singular locus of $ \mathfrak M_g$ was obtained by Cornalba, and I will report on the irredundant irreducible decomposition for the Singular locus of the compactified Moduli space of curves $ \overline{\mathfrak M_g}$.

A special role plays then the proof of irreducibility of the locus of curves with a given topological type for the action of a group G, which follows from Teichmueller theory, as I will explain. The still open and difficult problem is to give simple numerical invariants which determine the topological type.

Special Galois coverings are e.g. cyclic or dihedral coverings, for which I will describe old and new results, and new examples, obtained together with Fabio Perroni and Michael Loenne.

In the case of curves I will show some irreducibility results for coverings of a fixed numerical type: in the cyclic case for smooth curves, and in the cyclic case of prime order for moduli-stable curves. In the dihedral case we have results in work in progress with Michael Loenne and Fabio Perroni: in the case where the genus of the base is 0, or in the case where the covering is \'etale. In this case our work ties in with some general asymptotical study done by Dunfield and Thurston, which show that in the etale case asymptotically (i.e., for genus of the base curve large enough) the main invariant is an element in the second homology group $H_2(G, \Z)$.

We sall try to show also how these results apply also to the locus of real curves.**Alexander Efimov (Steklov Institute): Cohomological Hall algebra and Kac's conjecture.**

Abstract: We will recall Cohomological Hall algebras (COHAs) introduced by Kontsevich and Soibelman and describe the structure of COHA of a symmetric quiver, proving Kontsevich-Soibelman conjecture. This is a recent preprint arXiv:1103.2736. We will also explain how this implies KacĀ“s conjecture for quivers with enough loops, as it was shown by S. Mozgovoy.**Vladimir Fock (Strasbourg): Integrable systems, dimers and cluster varieties. (Joint work with A Marshakov).**

Abstract: We are going to describe a class of discrete algebraically integrable systems with phase having cluster structure and fibred over plane curves of given bidegree, Integrals of motion of these system are given by dimer partition function for a certain graph. In particular cases these systems coincide with the relativistic Toda system, R.Schwarz pentagram system, and gives KdV- like systems in a continuous limit.**Kenji Fukaya (Kyoto): Homological Mirror symmetry of toric manifolds.**

Abstract: In this talk I will explain a joint work with M. Abouzaid, Y.G. Oh, H. Ohta and K. Ono. We show the homological Mirror symmetry between toric A model and Landau-Ginzburg B model. More precisely we show that for any Lagrangian submanifold on a toric manifold with nontrivial Floer homology there is an idenpotent of the direct sum of toric fibers whoes image is Floer theoretically equivalent. Also each of the toric fiber corresponds to an appropriate skyscraper sheaf of the Landau-Ginzburg B model.**Alexander Goncharov (Yale): Dimers and cluster integrability.****Mark Gross (UCSD): Examples of stable log maps and tropical geometry.**

Abstract: Expanding on Bernd Siebert's talk, I will discuss some examples of moduli spaces of stable log maps and show the connection between these moduli spaces and tropical geometry, along with some sample calculations.**Sergei Gukov (Caltech): From hyperholomorphic sheaves to quantum group invariants via Langlands duality.**

**Ilia Itenberg (Strasbourg): Topology of real tropical hypersurfaces**

Abstract: We present several restrictions on the Betti numbers of the real part of a nonsingular real tropical hypersurface in a tropical projective space ${\Bbb T}P^n$. The restrictions are formulated in terms of Hodge numbers of complex hypersurfaces in ${\Bbb P}^n$.**Mikhail Kapranov (Yale): Arithmetic Hall algebras.**

**Ludmil Katzarkov (Miami & Vienna): Degenerations and wall crossings.**

Abstract: In this talk we will explain a connection between classical technique of degenerations and some category invariants.**Viatcheslav Kharlamov (Strasbourg): Anti-symplectic involutions on rational symplectic 4-manifolds.****Maxim Kontsevich (IHES): Integrable systems and canonical bases.**

**Andrei Losev (ITEP): Homotopical beta-function of the instantonic sigma-model and bosonic string Einstein equation on schemes.**

**Diego Matessi (Piemonte): Conifold transitions and tropical geometry.**

Abstract: The process of degenerating a complex variety X to a singular variety X_0 with double points and then resolving to obtain X' is called a conifold transition. Simultaneously resolving a set of double points is obstructed in the symplectic world (Smith-Thomas-Yau), while simultaneously smoothing is obstructed in the complex world (Tian-Friedman). In this talk, on a joint work with R. Castano-Bernard, I will explain how the two obstructions are unified by tropical geometry. For instance, I will show how good relations among vanishing cycles can be seen inside the tropical manifold and the corresponding nodes resolved on one side, or smoothed on the mirror side.**David Morrison (UCSB): Mirror symmetry and non-complete-intersection Calabi-Yau manifolds.**

Abstract: The elegant combinatorial mirror construction of Batyrev and Borisov for Calabi-Yau complete intersections in toric varieties has received much study, in part due to the availability of a tool -- Witten's gauged linear sigma model with an abelian gauge group -- to analyze the corresponding physical theories. There has been significant recent progress, starting with work of Hori and Tong in 2006, in understanding gauged linear sigma models with nonabelian gauge groups. In this talk we will explain some of this progress, and draw some implications for the study of mirror symmetry.**Andy Neitzke (UT Austin): A 2d-4d wall-crossing formula.**

Abstract: I will describe a wall-crossing formula which combines and extends formulas previously considered by Cecotti-Vafa and by Kontsevich-Soibelman. This formula is expected to be relevant to an "open-closed" version of the theory of generalized Donaldson-Thomas invariants. It is also the key ingredient in a new construction of hyperholomorphic vector bundles. This is joint work with Davide Gaiotto and Greg Moore.**Nikita Nekrasov (IHES): Surprises with four dimensional N=2 gauge theories.**

**Dimitri Orlov (Steklov): Mirror symmetry, B-branes and strange Arnold duality.**

**Tony Pantev (UPenn): Mirror symmetry and mixed Hodge structures.**

Abstract: I will explain how essential information about the structure of symplectic manifolds is captured by algebraic data, and specifically by the non-commutative mixed Hodge structure on the cohomology of the Fukaya category. I will discuss computable Hodge theoretic invariants arising from twist functors, and from geometric extensions. This is a joint work with L. Katzarkov and M. Kontsevich.**Pierre Schapira (Paris VI): Microlocal theory of sheaves and symplectic topology: results and open problems.**

Abstract: PDF**Bernd Siebert (Hamburg): Logarithmic Gromov-Witten invariants.**

Abstract: So far Gromov-Witten invariants relative to a divisor D could only be defined under the assumption that D is smooth. In the talk I will survey recent joint work with Mark Gross (1102.4322v1) establishing Gromov-Witten theory in log smooth situations. In particular, this includes the cases that D has (simple) normal crossings, or of fibres of semi-stable families. There is an interesting appearance of tropical geometry in the story.**Yan Soibelman (Kansas State): Integrable systems and wall-crossing formulas**

Abstract: One can use the ideas of Mirror Symmetry and Tropical Geometry in order to construct one-parameter family of complex symplectic manifolds starting with a complex integrable system with exact holomorphic symplectic form. This class of integrable systems includes Hitchin systems or more generally those coming from local Calabi-Yau 3-folds. If time permits I will discuss the relationship with the theory of (generalized) Donaldson-Thomas invariants.-

**Jake Solomon (Hebrew University): Entropy of Lagrangian submanifolds.**

Abstract: I'll discuss an invariant of a pair of Maslov zero Lagrangian submanifolds in a Calabi-Yau manifold with a given Hamiltonian isotopy between them. The invariant is constructed using moduli of holomorphic disks. It depends only on the homotopy class of the Hamiltonian isotopy. The invariant is analogous under mirror symmetry to the Donaldson-Bott-Chern functional important in the study of stable vector bundles. It suggests a modification of the special Lagrangian condition for stability of Lagrangian submanifolds. This is joint work with G. Tian.**Piotr Sulkowski (Caltech): Quantum curves and topological recursion.**

Abstract: Classical solutions to various exactly solvable mathematical and physical models, or theories, are encoded in complex, algebraic curves. These models/theories include: random matrix theory, Gromov-Witten theory on toric threefolds, some issues in knot theory (physically: Chern-Simons theory), Seiberg-Witten theory, intersection theory on the moduli space of complex curves, and many others... Apart from the classical solution, in all these models one constructs infinite class of "quantum" invariants. It turns out that these invariants can be determined from the geometry of the underlying classical complex curve, in terms of the so-called topological recursion. I will present this topological recursion, discuss its properties, and explain how it can be used to obtain, in a universal way, information of various quantum invariants in various, otherwise completely distinct, theories.**Valerio Toledano Laredo (Northeastern): Yangians, quantum loop algebras and trigonometric connections.**

Abstract: I will describe monodromy representations of affine braid groups arising from a flat connection with values in the Yangian of a simple Lie algebra g. These representations are related to those arising from the quantum Weyl group operators of the quantum loop algebra of g. Matching these two classes of representations involves in particular the construction of a functor relating finite-dimensional modules of those two quantum groups. This is based on joint work with Sachin Gautam.**Ilia Zharkov (Kansas State): Tropical Homology**

Abstract: Tropical varieties arising as degenerations of complex algebraic one parameter families still carry a lot of information inherited from the complex varieties. I will define tropical homology (p,q)-groups and show that in some cases they recover the classical Hodge structures.**Anton Zorich (Rennes): Degeneration of flat versus hyperbolic metric on Riemann surfaces, determinant of Laplacian, and Lyapunov exponents of the Hodge bundle.**(in collaboration with A. Eskin and M. Kontsevich)

Abstract: A holomorphic 1-form on a Riemann surface defines flat metric with conical singularities. We compare the flat metric with the canonical hyperbolic metric when a Riemann surface tends to a stable curve. Such a comparison allows us to compare determinants of the Laplacian in the flat and hyperbolic metrics and to describe the "mean monodromy" of the Hodge bundle along the Teichmuller geodesic flow.

## Other Contributions :

**Oren Ben-Bassat (University of Haifa): Deformations of Open Surfaces and their Stacks of Bundles.**

Abstract: We give a concrete description of some moduli stacks of vector bundles on formal neighborhoods of curves in surfaces. Some outcomes of this description are (1) an understanding of the deformations of the moduli stacks as well as (2) a description of the topology of the moduli stacks. This is work in progress with E. Gasparim.**Colin Diemer (U of Miami): Circuit Relations and the Secondary Stack.**

Abstract: We discuss the framed secondary stack of a polytope and its relevance for studying monodromy of toric hypersurfaces. The one dimensional strata (circuits) can be understood concretely, and allow one to see more refined information than just passing to a single tropical limit. Applications to A-hypergeometric series will be discussed. This is part of a joint work with Gabriel Kerr and Ludmil Katzarkov.**David Favero (UPenn): Graded matrix factorizations, functor categories, and orbit categories.**

Abstract: The category of dg functors between categories of graded matrix factorizations, can be described by a different category of matrix factorizations whose grading is a bit more subtle. This gives rise to some nice "geometric" behavior between various categories, perhaps more easily seen through the lens of homological mirror symmetry. I will give an overview of recent results obtained in joint work with Matthew Ballard and Ludmil Katzarkov on this subject and provide examples of some of the geometry alluded to. If time permits I will very briefly describe some applications to algebraic cohomology classes and Rouquier dimension for derived categories of coherent sheaves.**Gabriel Kerr (U of Miami): Circuit Relation in the Symplectic Mapping Class Group.**

Abstract: I will discuss some generators and relations for the symplectic mapping class group $Symp(Y)$ of a toric hypersurface $Y \subset X$. The generators are those coming from spherical Dehn twists and large complex limit monodromy. The relations can be read directly from the combinatorics of the secondary polytope of X. The subgroup they generate will be discussed and examples in dimension one will be illustrated.**Helge Ruddat (UC Berkeley): Mirror Symmetry partners via vanishing cycles.**

Abstract: In a joint work with Mark Gross and Ludmil Katzarkov, we propose a construction for mirror symmetry partners which works for any Kodaira dimension, e.g. this allows us to produce the mirror of a complex curve of any genus. In general, the mirror will be very singular and equipped with a sheaf of vanishing cycles. We associate Hodge numbers to the singular space with this sheaf and show that these fulfil the mirror duality h^{p,q}(X) = h^{d-p,q}(Y) for X,Y a mirror pair of our construction. In this talk, I intend to introduce the construction and demonstrate a few examples.**Nick Sheridan (MIT): On the Homological Mirror Symmetry conjecture for pairs of pants.**

Abstract: The n-dimensional pair of pants is CP^n with n+2 generic hyperplanes removed. We construct an immersed Lagrangian sphere in the pair of pants and compute its A_{\infty} endomorphism algebra in the Fukaya category. On the level of cohomology, the algebra is an exterior algebra with n+2 generators. It is not formal, and we must compute certain higher products to determine it up to quasi-isomorphism. This allows us to give some evidence for the Homological Mirror Symmetry conjecture: the pair of pants is conjectured to be mirror to the Landau-Ginzburg model (C^{n+2},W), where W = z_1 ... z_{n+2}. We show that the endomorphism algebra of our Lagrangian is quasi-isomorphic to the endomorphism (dg) algebra of the structure sheaf of the origin in the mirror. This implies similar results for finite covers of the pair of pants, in particular for certain affine Fermat hypersurfaces. Preprint: arXiv:1012.3238.**Nicolo Sibilla (Northwestern):**

Abstract: In this talk I will explain how to construct a model for the Fukaya category of punctured Riemann surfaces in terms of a sheaf of dg-categories over a suitable category of ribbon graphs. The main ingredients will be Nadler and Zaslow's work on cotangent bundles, and Kontsevich' recent ideas on the locality of the Fukaya category of a Stein manifold. Further, I will explain applications to homological mirror symmetry for degenerate elliptic curves. This work is joint with David Treumann and Eric Zaslow.**Alexander Soibelman (UNC Chapel Hill):**

Abstract: In their paper on the quantization of Hitchin's integrable system, Beilinson and Drinfeld introduce and prove the "very goodness" property of the moduli space Bun_G of G-bundles over a curve. We use certain quiver representation methods, developed by Crawley-Boevey, to establish this property for parabolic bundles of degree zero over the projective line. As an application, we use "very goodness" to demonstrate that the space of solutions to the so-called additive Deligne-Simpson problem is nonempty.

Last update: June 30, 2011