# Abstracts and Materials

### Yamabe Symposium: September 30 - October 2, 2016

#### Title: A Cobordism Outlook on Lagrangian Topology

*Lagrangian Cobordism I**Lagrangian Cobordism and Fukaya Categories**Lagrangian Cobordism in Lefschetz Fibrations*

*Abstract: * Lagrangian cobordism is a natural notion introduced by Arnold in the early days of symplectic topology. In recent years it has been revived and extended to a theory which is relevant in the study of Lagrangian topology and Fukaya categories. In this talk we will survey these developments covering both topological as well as geometric aspects of the story. The talk is based on a series of joint works with Octav Cornea.

#### Title: The Cohomological McKay Correspondence and Symplectic Cohomology

*Mark Mclean (Stony Brook University)*

*Abstract: * Suppose that we have a finite quotient singularity C^n/G admitting a crepant resolution Y (i.e. a resolution with c_1 = 0). The cohomological McKay correspondence says that the cohomology of Y has a basis given by irreducible representations of G (or conjugacy classes of G). Such a result was proven by Batyrev when the coefficient field F of the cohomology group is Q. We give an alternative proof of the cohomological McKay correspondence in some cases by computing symplectic cohomology+ of Y in two different ways. This proof also extends the result to all fields F whose characteristic does not divide |G| and it gives us the corresponding basis of conjugacy classes in H^*(Y). We conjecture that there is an extension to certain non-crepant resolutions. This is joint work with Alex Ritter.

#### Title: Legendrian fronts for affine varieties

*Emmy Murphy (MIT)*

*Abstract: *Given a smooth complex affine variety, we discuss a method for describing its symplectic topology, by representing it as a Weinstein handle body, which in turn is given as an explicitly constructed Legendrian link. After describing this recipe (which is mostly algorithmic), we discuss a number of applications as time allows. Possibilities are: flexibility of Milnor fibers of some non-isolated singularities, mirror symmetry for the Legendrian trefoil, constructions of some new exact Lagrangians, or a proof that the Koras-Russell cubic is symplectomorphic to C^3. This is joint work with Roger Casals.

#### Title: Lagrangian Floer theory - Generating criterion for Fukaya category and related topics

*Kaoru Ono (Kyoto University)*

*Abstract: * I will explain some constructions in Lagrangian Floer theory and present "generating criterion for Fukaya category" and related topics. The talk is based on a joint work with M. Abouzaid, K. Fukaya, Y.-G. Oh and H Ohta and other joint works with K. Fukaya, Y.-G. Oh and H. Ohta.

#### Title: Around Nakamaye's Theorem

*Valentino Tosatti (Northwestern)*

*Abstract: * Nakamaye's Theorem from around 2000 says that the augmented base locus of a nef and big line bundle on a smooth projective variety equals its null locus (the locus of subvarieties where the restriction of the line bundle is not big). I will discuss a transcendental generalization of this theorem to all compact complex manifolds (with line bundles replaced by (1,1) cohomology classes) and some of its applications. I will also discuss a conjectural generalization to big (1,1) classes which are not nef. Joint work with T. Collins.

#### Title: Hyper-Kaehler manifolds

*Claire Voisin (Directrice de recherche, CNRS Institut de Mathmatiques de Jussieu)*

*Abstract: * Hyper-Kaehler manifolds (also called quaternionic or holomorphic symplectic manifolds) are higher dimensional generalizations of K3 surfaces. They are in particular Kaehler but projective ones are dense in the moduli space. The topology of these varieties is rather restricted, although we do not know what are all possible topological types. Their deformation theory is "easy". By Torelli theorem, it is governed by the period map. All known examples have been constructed by algebraic geometry. I will discuss these general facts and describe a few constructions.

#### Title: Transversality and super-rigidity for multiply covered holomorphic curves

*Chris Wendl (UCL/Humboldt University Berlin)*

#### Title: Hyperbolicity of moduli spaces of polarized manifolds

*Sai Kee Yeung (Purdue)*

*Abstract:* In this talk, we would explain some joint work with Wing-Keung To on hyperbolicity of moduli spaces of polarized algebraic manifolds, including families or moduli spaces of Kaehler-Einstein manifolds of negative scalar curvature or trivial scalar curvature, and their quasi- projective versions. Classically, results on moduli space of Riemann surfaces of genus at least two have been obtained by Ahlfors, Royden and Wolpert with Weil-Petersson metric. Study of moduli of higher dimensional manifolds in terms of Weil-Petersson metrics began with a work of Siu thirty years ago. Eventually, we construct a negatively curved Finsler metric on such moduli from which results in hyperbol- icity follow naturally. We also apply the related techniques to study questions related to a conjecture of Viehweg.