#### Gang Tian |
#### Introduction to the Yau-Tian-Donaldson conjecture##### The first lecture is rather general, the second is on K-stability, the last will be more advanced. |

#### Grigory Mikhalkin |
#### Indices of real algebraic curves##### Understanding possible topology of algebraic curves of given degree in the real plane (as well as in the 3-space) is one of the most classical problems in mathematics. It is one of the few unresolved problems in Hilbert’s famous list as of today. I’ll review some of the newest developments in this very old subject. |

#### Sergey Galkin |
#### Cubic forms and related geometries##### To a single homogeneous cubic polynomial one can associate many spaces of different geometric and topological kinds (Fano manifolds, hyper-kähler manifolds, as well as some manifolds of general type). E.g. for a cubic form f(x_{1},...,x_{6}) its zeroes in P^{5} is a Fano fourfold, and there are 3 known constructions of hyper-kähler spaces: 4-folds of Fano-Beauville-Donagi and 8-folds of Lehn-Lehn-Sorger-van Straten are deformations of Hilbert schemes of points on K3 surfaces, and 10-folds of Laza-Sacca-Voisin are deformations of sporadic varieties constructed by O'Grady; much more of similar varieties are yet to be constructed and studied. These spaces hypothetically should be related by concrete geometric constructions, and as a corollary one should obtain algebraic relations between various invariants of this spaces, such as Euler numbers, Chern numbers, Hodge structures, classes in bordisms and other rings of varieties (such as Grothendieck ring), motives, derived categories of coherent sheaves, mirror duals, ... One such geometric relation between variety of lines and symmetric square of a cubic was carefully studied in works by myself and Shinder, Voisin, Laterveer, and its origin traces back to "secant line" construction of Diophantus. The expression for twisted cubics on a cubic hypersurface is not yet known exactly, however we know its approximate form. One talk will cover geometry of cubic hypersurfaces and spaces of its rational curves. In another one I will explain definitions and properties of various groups of varieties, as well as operations on some of them (product, grading, differential, symmetric powers, and other operations); also I will explain "decomplexifications" of such groups to groups of topological manifolds (sometimes with orientations and corners). In third talk I will explain various known and conjectural relations, as well as framework of Herbst-Hori-Page to understand some of these relations as phase transitions and mirror symmetry for the respective spaces. |

#### Weimin Chen |
#### On symplectic Calabi-Yau manifolds##### Symplectic Calabi-Yau surfaces are symplectic 4-manifolds with trivial canonical line bundle, hence are the symplectic analog of Calabi-Yau surfaces. These manifolds are known to be a rather restrictive class of 4-manifolds, and there is a folklore conjecture concerning their smooth classification: a symplectic Calabi-Yau surface is either a K3 surface or an orientable T^{2}-bundle over T^{2}. In this lecture series we shall explain some recent progress towards the above conjecture, namely, under a certain finite symmetry condition, the conjecture is true. In Lecture 1, we will give an overview of symplectic Calabi-Yau surfaces, ending with a statement of our main result. The proof relies on some new construction in symplectic finite group actions on 4-manifolds, which will be explained in Lecture 2. More concretely, let M be a symplectic 4-manifold equipped with a symplectic G-action. We shall associate it with a symplectic 4-manifold M_{G}, which is a certain "symplectic resolution" of the quotient orbifold M/G. Then conjecturally, M_{G} and M obey the following inequality: κ^{s}(M_{G})≤ κ^{s}(M), where κ^{s} is the symplectic Kodaira dimension. In the last lecture, Lecture 3, we shall give an outline and explain the main ingredients of the proof. |

#### Çağrı Karakurt |
#### Heegaard Floer homology of almost rational plumbings##### Heegaard Floer homology, invented by Ozsváth and Szabó, was emerged in early 2000s as a “user friendly” alternative to gauge theoretical invariants of manifolds of dimensions three and four as well as knots. Even though Heegaard Floer invariants are more geometric in nature than their gauge theoretical counterparts, the role of holomorphic disks in their definition make their computation very challenging in general.
In the case that a 3-manifold appears as the boundary of a special type of plumbing Ozsváth and Szabó gave a purely combinatorial method for computing its Heegaard Floer homology. Using algebro-geometric methods, this result was later simplified and extended by Némethi who came up with new invariant of plumbed 3-manifolds so-called Lattice homology which is conjecturally isomorphic to Heegaard Floer homology. For the case of almost rational plumbings this isomor- phism was explicitly established.
The purpose of this lecture series is to understand the aforementioned isomorphism in order to bring together the geometric interpretation of Heegaard Floer invariants and computational convenience of Lattice homology. The lecture series was mainly intended for graduate students in geometry and topology who would like know about the recent progresses in the field. It is assumed that the students are completed the basic course (e.g. algebraic topology, differentiable manifolds etc.). Each lecture will be about three hours long and held in the morning, followed by and informal discussion session in the afternoon.
#### Lecture 1. Heegaard Floer homology and plumbings **Abstract:** We’ll quickly review the definition and basic properties of Heegaard Floer homology and see how it can be computed for those plumbings with at most one bad vertex. #### Lecture 2. (Almost) Rational singularities and graded roots **Abstract:** We’ll relate the Ozsváth-Szabó algorithm in the previous lecture to Artin’s method of finding fundamental cycle of singularities. #### Lecture 3. Applications and further directions **Abstract:** We’ll detect the image of the contact Ozsváth-Szabó under Némethi’s isomorphism. We’ll discuss some applications for the support genus and Stein cobordism problems. In another direction we discuss how to get Hendricks and Manolescu’s involutive Heegaard Floer homology and the corresponding homology cobordism invariants. |