Rahul Pandharipande (ETH Zürich)
Title: Lehn's conjecture for Segre classes on Hilbert schemes of points of  surfaces and generalizations (I),(II),(III)
Abstract: Let E be a bundle on a nonsingular projective surface S. I will discuss several recent geometric approaches to the question of computing the top Segre (or top Chern) class of the associated tautological bundle E[n] on the Hilbert scheme of points S[n]. Complete results in all ranks are known for K3 surfaces (joint work with Marian and Oprea) where such  classes arise naturally in the virtual geometry of associated Quot schemes (and are related to tautological relations on the moduli of K3 surfaces). An approach to the question using ideas of Reider and Lazarsfeld was found by Voisin which gives complete results for all surfaces in rank 1 (verifying conjectures by Lehn from 1999). For complete results for all surfaces in higher rank, a natural path is the study of the associated localization vertex --- which leads to open questions and conjectures.

E.J.N. Looijenga (Tsinghua University)
Title: Compactification of certain locally symmetric varieties related to Algebraic Geometry

Mihnea Popa (Northwestern University)
Title: Hodge modules in birational and complex geometry
Abstract: I will give an overview of joint work with M. Mustata on Hodge ideals, which are a generalization of multiplier ideals constructed using the theory of mixed Hodge modules, especially the Hodge filtration on the localization along a hypersurface. I will emphasize the case of reduced hypersurfaces, while in a subsequent lecture series Mustata will discuss the case of Q-divisors.  I will explain basic properties they satisfy, like local non-triviality criteria and vanishing theorems, and use these to provide a number of applications regarding the Hodge filtration and the singularities of hypersurfaces in projective space or abelian varieties. 

Burt Totaro (UCLA)
Title: Algebraic cycles and birational geometry
Abstract:
The lectures are centered on the Chow group of algebraic cycles on an algebraic variety. Some introductions to this material are Hartshorne's section II.6 on Divisors and Appendix A on Intersection Theory. We will mostly use what we need from Fulton's Intersection Theory without proof.
One main theme of the lectures is the notion of "decomposition of the diagonal", which describes the geometric consequences that hold for a variety with "small" Chow groups. The basic argument by Bloch and Srinivas is very simple, once you know the formal properties of Chow groups.
The lectures will end with applications of these ideas to birational geometry. In particular, we will use Chow groups to prove a striking recent result: many Fano hypersurfaces X in projective space are not stably rational. (That is, no product of X with projective space is birational to projective space.)

Mircea Mustata (University of Michigan)
Title: Hodge ideals and singularities
Abstract: I will give an introduction to some invariants of singularities of Q-divisors on smooth complex algebraic varieties, that generalize the multiplier ideals from birational geometry, and which come naturally out of Saito’s theory of mixed Hodge modules. This is based on joint work with Mihnea Popa.

Lawrence Ein (UI Chicago)
Title: Measures of irrationality of an algebraic variety
Abstract: We are going to discuss various numerical measure to see how far a variety from a rational variety. In particular, we will discuss jour oint work Bastianelli, De Poi, Lazarsfeld and Ullery on the case the variety is a very general hyper-surface in a projective space.

Yukinobu Toda (Kavli IPMU)
Title: Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds
Abstract: In this talk, I will discuss birational geometry for Joyce’s d-critical loci, by introducing notions such as ‘d-critical flips’, ‘d-critical flops’, etc.  I will show that several wall-crossing phenomena of moduli spaces of stable objects on Calabi-Yau 3-folds are described in terms of d-critical birational geometry,  e.g. certain wall-crossing diagrams of Pandharipande-Thomas stable pair moduli spaces form a d-critical minimal model program. I will also show the existence  of semi-orthogonal decompositions of the derived categories under simple d-critical flips satisfying some conditions. This is motivated by a d-critical  analogue of Bondal-Orlov, Kawamata’s D/K equivalence conjecture, and also gives a categorification of wall-crossing formula of Donaldson-Thomas invariants.

Olivier Debarre (Université Paris Diderot)
Title: Hyperkähler manifolds
Abstract: Hyperkähler manifolds can be considered as higher-dimensional analogs of (complex) K3 surfaces. They are interesting from several points of view: dynamical (some have interesting automorphism groups) and geometric, in particular. It is also one of those rare cases where the Torelli theorem allows for a powerful link between  the  geometry of these manifolds   and lattice theory. Our aim is more to provide, for specific families of  hyperk\"ahler manifolds, a panorama of results about projective embeddings, automorphisms, moduli spaces, period maps and domains.

Ngaiming Mok (HKU)
Title: Curvature and Uniformization on Quotient Spaces of Bounded Symmetric Domains of Finite Volume
Abstract: By the Uniformization Theorem a compact Riemann surface other than the Riemann Sphere or an elliptic curve is uniformized by the unit disk and equivalently by the upper half plane.  The upper half plane is also the universal covering space of the moduli space of elliptic curves equipped with a suitable level structure.  In Several Complex Variables, the Siegel upper half plane is an analogue of the upper half plane, and it is the universal covering space of moduli spaces of polarized Abelian varieties with level structures. The Siegel upper half plane belongs, up to biholomorphic equivalence, to the set of bounded symmetric domains, on which a great deal of mathematical research is taking place. Especially, finite-volume quotients of bounded symmetric domains, which are naturally quasi-projective varieties, are objects of immense interest to Several Complex Variables, Algebraic Geometry and Number Theory, and an important topic is the study of uniformizations of algebraic subsets of such quasi-projective varieties. While a lot has already been achieved from methods of Diophantine Geometry, Model Theory, Hodge Theory and Algebraic Geometry for Shimura varieties, techniques for the general case of not necessarily arithmetic quotients have just begun to be developed. We will explain a differential-geometric approach to the study of such algebraic subsets revolving around the notion of asymptotic curvature behavior and the use of rescaling arguments, and illustrate how this approach using transcendental techniques leads to various characterization results for totally geodesic subvarieties of finite-volume quotients without the assumption of arithmeticity. Especially, we will explain how the study of holomorphic isometric embeddings of the Poincar\'e disk and more generally complex unit balls into bounded symmetric domains can be further developed to derive uniformization theorems for bi-algebraic varieties and more generally for the Zariski closure of images of algebraic sets under the universal covering map.

Zhiwei Yun (MIT)
Title: An Introduction to the Moduli of Shtukas

Junyan Cao (IMJ-PRG)
Title: Singular hermitian metrics and some applications in complex geometry
Abstract: The positivity of direct images of relative pluricanonical bundles is a powerful tool in complex geometry, due to the pioneering works of Fujita, Kawamata, Kollár, Viehweg among many others. In the lectures, we will first explain some metric properties of these sheaves established in the works of Berndtsson, Păun and Takayama. Then we will explain some applications: (1) some particular case of the Iitaka conjecture (2) a structure theorem of projective manifolds with nef anticanonical bundles.

Jun-Muk Hwang (Korea Institute for Advanced Study)
Title: Minimal rational curves on Fano manifolds of Picard number 1
Abstract: The overall theme of the lectures is the question, to what extent minimal rational curves on a Fano manifold of Picard number 1 determine the Fano manifold. A key result is Cartan-Fubini type extension theorem which has two different aspects, the theory of VMRT-structures and the procedure of analytic continuation. We explain the differential-geometric idea of VMRT-structures and see how it is used in concrete geometric problems. We also examine the analytic continuation procedure and discuss the problem of weakening the holomorphic condition to a formal condition.

Radu Laza (Stony Brook University)
Title: Birational geometry of the moduli of K3 surfaces, and applications

Zsolt Patakfalvi (EPFL)
Title: Projectivity of moduli spaces of K-semi-stable varieties
Abstract: The talk will be about how one can show that the moduli spaces of K-semi-stable Fano and general type varieties are (quasi-)projective schemes assuming that they are already constructed as proper algebraic spaces. I will very briefly explain how the above (quasi-)projectivity translates to semi-positivty and positvity questions of certain functorial bundles on base-spaces of families. Then the main focus will be on explaining how these semi-positivity and positivity questions can be shown. The emphasis will be on the Fano case (joint work with Codogni). However, as time permits I might be able to also mention a few details on the general type case  in characteristic zero (joint with Kovács and Xu), as well as on the general type surface case in positive/mixed characteristic.

Hiromu Tanaka (The University of Tokyo)
Title: On varieties of Fano type in positive characteristic
Abstract: The notion of varieties of Fano type is a generalisation of smooth Fano varieties, which is known as one of important classes in minimal model program. However, varieties of Fano type behave pathologically in characteristic p. In this series of talk, we first overview the fundamental properties of varieties of Fano type. We then study some examples that appear only in characteristic p.