Gökova Geometry Topology Institute

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===== Silk Road Geometry Conference 2018 (Jun 04 - Jun 08, 2018) =====

{{:events:2018:silkroad18_grouppic.jpg?nolink&800|}}

Turkish Mathematical Society (TMD) supports this event.

=== List of Participants ===
| Gang Tian (BICMR)                   | Yong-Geun Oh (IBS-CGP & POSTECH)           | Eaman Eftekhary (IPM)                                     |
| Weimin Chen (UMass Amherst)         | Cheol Hyun Cho (Seoul National University) | Bohan Fang (BICMR)                                        |
| Jian Ge (BICMR)                     | Shuai Guo (Math School, Peking University) | Ali Kamalinejad (IPM)                                     |
| Bumsig Kim (KIAS)                   | Sang-hyun Kim (Seoul National University)  | Jiayu Li (University of Sciences and Technology of China) |
| Xiaobo Liu (BICMR)                  | Yi Liu (BICMR)                             | Meysam Nassiri (IPM)                                      |
| Reza Seyyedali (IPM)                | Wenyuan Yang (BICMR)                       | Masoumeh Zarei (BICMR)                                    |
| Jian Zhou (Tsinghua University)     | Mehrzad Ajoodanian (IPM)                   | Selman Akbulut (MSU)                                      |
| Mahir Bilen Can (Tulane University) | Craig van Coevering (Boğaziçi University)  | Alex Degtyarev (Bilkent University)                       |
| Şahin Koçak (Anadolu University)    | Muhammed Uludağ (Galatasaray University)   | Yılmaz Akyıldız (emeritus from Boğaziçi University)       |

**Scientific Committee** :  Selman Akbulut, Gang Tian, Yong-Geun Oh\\ 
**Organizing Committee** :  Turgut Önder, Çağrı Karakurt

=== Talks ===
^ Speaker ^ Title and Abstract ^
| Yong-Geun Oh | **Disjunction energy of compact Lagrangian submanifold from open subset**\\  Study of displacement energy of a subset of symplectic manifold is one important tool for study of symplectic topology, but is a highly nontrivial matter to perform actual measurement. So far such a measurement has been carried out between open subsets and between Lagrangian subamanifolds. In this talk, I will present recent progress of such a measurement for the case of mixture of the two, i.e., between Lagrangian submanifolds and open subsets. |
| Eaman Eftekhary | **Foliations, formal power series and gauge theory**\\ We apply gauge theory to study the space of co-oriented smooth codimension foliations on a smooth manifold M. The quotient of Maurer-Cartan elements by the action of an infinite dimensional non-abelian gauge groupoid forms a moduli space, which contains the space of foliations as a subspace. The quotient of the moduli space under concordance is identified as the space of homotopy classes of maps to the classifying space associated with the groupoid of formal power series (under formal composition). This gives a treatment parallel to study of foliations through Haefliger structures, which may be repeated by replacing real numbers with any commutative algebra of finite rank over reals. In particular, starting from complex numbers we arrive at a residue formula for the Godbillon-Vey invariant. This is joint work with Mehrzad Ajoodanian. |
| Weimin Chen | **Some thoughts on constructing new small symplectic 4-manifolds and related rigidity phenomena in symplectic and algebraic geometry**\\  We discuss a proposal for constructing symplectic exotic $\mathbb{CP}^2$ (though still largely speculative at this point). |
| Cheol Hyun Cho | **Homological mirror functors via Maurer-Cartan formalism**\\ Using formal deformation theory of Lagrangian submanifolds in a symplectic manifold, we can define canonical A-infinity functors from Fukaya category tomatrix factorization category of Landau-Ginzburg models. Different choice of Lagrangians correspond to different charts of the mirror LG model. Building from the basic example of pair of pants, we explain the case of punctured Riemann surfaces via pair of pants decomposition. This is a joint work with Hansol Hong and Siu-Cheong Lau. |
| Bohan Fang | **Crepant resolution conjecture and holomorphic anomaly equation from the remodeling conjecture**\\  I will survey some applications of the remodeling conjecture, such as holomorphic anomaly equations and the crepant resolution conjectures. This talk is based on the joint works with Chiu-Chu Melissa Liu and Zhengyu Zong, as well as on the joint work Yongbin Ruan, Yingchun Zhang and Jie Zhou. |
| Jian Ge | **On Paralel Axiom**\\  We will discuss the rigidity of the Euclidean metric under the assumption of Parallel Axiom and total curvature conditions. This is a joint work with L. Guijarro and P. Solorzano |
| Shuai Guo | **Higher genus mirror symmetry for quintic 3-fold**\\  In this talk. I will try to explain the physics and mathematics that related to a quintic Calabi-Yau hypersurface in the 4-dimensional complex projective space. On the physics side, I will talk about Yamaguchi-Yau's finite generation conjecture, holomorphic anomaly equation and their application in higher genus computation by Huang-Klemm-Quackenbush. On the mathematics side, I will talk about our recent progress on the structures of higher genus Gromov-Witten invariants. This talk is based on the joint works with F. Janda, Y. Ruan and with H-L Chang and J. Li. |
| Bumsig Kim | **Localized Chern characters for 2-periodic complexes**\\ For a two-periodic complex of vector bundles, Polishchuk and Vaintrob have constructed its localized Chern character. I will explore some basic properties of this localized Chern character. In particular, I will show that the cosection localization defined by Kiem and Li is equivalent to a localized Chern character operation for the associated two-periodic Koszul complex, strengthening a work of Chang, Li, and Li. This strong equivalence will be applied to the comparison of virtual classes of the moduli of epsilon-stable quasimaps and the moduli of the corresponding LG epsilon-stable quasimaps, in full generality. The talk is based on joint work with Jeongseok Oh. |
| Sang-hyun Kim | **Diffeomorphism groups of critical regularity**\\  We prove that for each compact connected one-manifold M and for each real number $a\ge 1$ , there exists a finitely generated group G inside the $C^a$ —diffeomorphism group $Diff^a(M)$ such that G admits no injective homomorphisms into the group $\cup_{b>a}Diff^b(M)$. We also prove the dual result for $\cap_{b<a}Diff^b(M)$ . (Joint work with Thomas Koberda) |
| Jiayu Li | **Compactness of Symplectic Critical Surfaces**\\ In this talk, we introduce new functionals to study the existence of holomorphic curves in Kähler surfaces. We study the properties of the critical surfaces of the functionals. We study the compactness of the critical surfaces. |
| Xiaobo Liu | **Connecting Hodge integrals to Gromov-Witten invariants by Virasoro operators**\\  Kontsevich-Witten tau function and Hodge tau functions are important tau functions for KP hierarchy which arise in geometry of moduli space of curves. Alexandrov conjectured that these two functions can be connected by Virasoro operators. In a joint work with Gehao Wang, we have proved Alexandrov's conjecture. In a joint work with Haijiang Yu, we show that this conjecture can also be generalized to Gromov-Witten invariants and Hodge integrals over moduli spaces of stable maps to smooth projective varieties. |
| Yi Liu | **Virtual homological spectral radii for automorphisms of surfaces**\\ A surface automorphism is an orientation-preserving self-homeomorphism of a compact orientable surface. A virtual property for a surface automorphism refers to a property which holds up to lifting to some finite covering space. It has been conjectured by C. T. McMullen that any surface automorphism of positive mapping-class entropy possesses a virtual homological eigenvalue which lies outside the unit circle of the complex plane. In this talk, I will review some background and outline a proof of the conjecture. |
| Meysam Nassiri | **Boundary dynamics for surface homeomorphisms**\\  We discuss some aspects of the topological dynamics of surface homeomorphisms concerning the dynamics on the boundary of invariant domains. In particular, we study the problem of existence of a periodic point on the boundary of an invariant domain for a surface homeomorphism. In the area-preserving setting, a complete classification is given in terms of rationality of Carathéodory’s prime ends rotation number, similar to Poincaré’s theory for circle homeomorphisms. The most important consequences are for the generic area-preserving diffeomorphisms, building on previous work of J. Mather. This talk is based on our joint works with A. Koropecki and P. Le Calvez. |
| Wenyuan Yang | **Martin boundary covers Floyd boundary**\\  In this talk, we discuss a relation between two boundaries for a finitely generated group: Martin boundary associated with a finitely supported symmetric random walk, and Floyd boundary obtained from a conformal scaling of Cayley graphs. We prove that the identity map over the group extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. As applications, we show that harmonic measure is singular with a quasiconformal measure on the limit sets of geometrically finite groups of Gromov hyperbolic spaces. This is joint work with I. Gekhtman, V. Gerasimov and L. Potyagailo. | 
| Selman Akbulut | **Complex $G_2$ manifolds**\\ We will define the notion of complex $G_2$ manifold; as application derive Seiberg-Witten equations as the equation of motion in these manifolds (joint work with Ustun Yildirim). |
| Mahir Bilen Can | **Spherical actions and applications to Schubert calculus**\\ After reviewing the basic theory of spherical varieties, we will focus on Schubert varieties which are spherical under the action of a Levi subgroup. We will present examples of Schubert varieties which are not spherical under any Levi subgroup action. On the bright side, we will show that the nonsingular Schubert varieties are always spherical. By taking this development one step further, we will present some applications to Schubert calculus.  This is a joint work with Reuven Hodges and Venkatramani Lakshmibai. | 
| Craig van Coevering | **A new obstruction for Sasaki-extremal metrics**\\ A Sasaki-extremal metric is a generalization of constant scalar curvaturemetrics on a Sasakian manifold, where the Futaki invariant is not assumed tobe zero. Extremal metrics are unique up to the Hamiltonian holomorphic groupaction, so can be considered a "canonical" metric. On a Sasakian manifold onecan consider the cone of Reeb vector fields C , where each ξ∈C is a polarization. One can define the "extremal cone" to be the open subset E∈C of Reeb vectorfields which admit a compatible Sasaki-extremal metric.  In joint work with Charles P. Boyer we give a generalization of the Lichnerowicz obstruction of Sasaki-Einstein metrics, due to Gauntlett, Martelli, Sparksand Yau, to an obstruction of Sasaki-extremal metrics. Using this obstruction,we will give many examples of Sasakian manifolds for which the extremal coneis empty. In particular, many weighted hypersurface examples are easily shownto have an empty extremal cone. | 
| Alex Degtyarev | **Slopes of colored links**\\  I will discuss various properties of the newly discovered link invariant, which we called slope. In particular, I will address the skein relations for signature in terms of slopes of tangles and the concordance invariance of slopes (as usual, outside of the Knottennullstellen). This is a joint work in progress with Vincent Florens and Ana G. Lecuona. | 
| Muhammed Uludağ | **Mapping Class Groupoids and Thompson’s groups**\\  We concoct a uniform treatment of mapping class groups and Thompson’s groups thereby generalizing them. As a by-product we obtain a description of the outer automorphism group of free groups as the isotropy group of a groupoid, which extends the mapping class groupoid of Mosher and Penner. We illustrate some arithmetic aspects of these groupoids at the end of our talk. |
Gökova Geometry Topology Institute

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