Valery Alexeev

Title: Generalizations of Losev-Manin moduli spaces
Abstract: Motivated by the study of degenerations of K3 surfaces, we construct moduli spaces associated with many reflections groups, including the Weyl groups of ADE type. They generalize Losev-Manin moduli spaces of pointed curves, which is the $A_n$ case. This is a joint work with Alan Thompson.


Benjamin Bakker

Title: The birational geometry of complex ball quotients
Abstract: Quotients of the complex ball by discrete groups of holomorphic isometries naturally arise in many moduli problems---for instance, those of low genus curves, del Pezzo surfaces, certain K3 surfaces, and cubic threefolds. The fact that the complex ball admits nonarithmetic lattices means new techniques are required to understand the birational geometry of such quotients. In joint work with J. Tsimerman, we show that in dimension $n\geq4$ every smooth complex ball quotient is of general type, and further that the canonical bundle $K_{\overline{X}}$ of the toroidal compactification $\overline{X}$ is ample for $n\geq 6$. The proof uses a hybrid technique employing both the hyperbolic geometry of the uniformizing group and the algebraic geometry of the toroidal compactification. We will also discuss applications to bounding the number of cusps and the Green--Griffiths conjecture.


Michel Brion

Title: Commutative algebraic groups up to isogeny
Abstract: The commutative algebraic groups over a field k are the objects of an abelian category C, with morphisms being the homomorphisms of algebraic k-groups. If k is algebraically closed, then by work of Serre and Oort, the homological dimension of C is 1 in characteristic 0 and 2 in positive characteristics. The talk will address the isogeny category of commutative algebraic groups, defined as the quotient of C by the Serre subcategory F of finite algebraic groups. In particular, we will show that the homological dimension of the isogeny category is 1 for any field k, and we will discuss analogies with some other categories of homological dimension 1 arising in algebraic geometry.


Ana-Maria Castravet

Title: Derived category of moduli spaces of pointed stable rational curves
Abstract: I will report on joint work with Jenia Tevelev on Orlov's question on exceptional collections on moduli of pointed stable rational curves and related spaces.


Daniel Chan

Title: Moduli stacks of Serre stable representations
Abstract: In 1987, Geigle and Lenzing introduced the study of 1-dimensional smooth rational Deligne-Mumford stacks with trivial generic stabiliser into representation theory, to elucidate the module category of Ringel's canonical algebras. In particular, they constructed a derived equivalence between the stack and the corresponding algebra. In this talk, we will introduce a new moduli stack called the Serre stable moduli stack which roughly corresponds to studying families of point objects in the sense of Bondal and Orlov. This gives a moduli-theoretic interpretation of Geigle and Lenzing's derived equivalence.


Dawei Chen

Title: Compactification of strata of abelian differentials
Abstract: Many questions about Riemann surfaces boil down to study their flat structures that are induced from abelian differentials. On the other hand, loci of abelian differentials with prescribed number and multiplicities of zeros form a natural stratification of the Hodge bundle. The geometry of these strata has interesting properties and applications to moduli of complex curves. In this talk we focus on the question of compactifying the strata of abelian differentials, from the viewpoints of algebraic geometry, complex analytic geometry, and flat geometry. In particular, we provide a complete description of the strata compactification in the Hodge bundle over the Deligne-Mumford moduli space of stable pointed curves. The upshot is a global residue condition compatible with a full order on the dual graph of a stable curve. This is joint work with Bainbridge, Gendron, Grushevsky, and Moeller, based on arXiv:1604.08834.


François Charles

Title: Bertini irreducibility theorems in arithmetic
Abstract: The classical Bertini irreducibility theorem states that if X is an irreducible projective variety of dimension at least 2 over an infinite field, then X has an irreducible hyperplane section. The proof does not apply in a situation where X has only finitely many hyperplane sections, be it over finite fields or if X is an arithmetic variety endowed with an ample hermitian line bundle. I will discuss how to amend the theorem in these cases (joint with Bjorn Poonen over finite fields).


Izzet Coskun

Title: The birational geometry of moduli spaces of sheaves on surfaces
Abstract: In this talk, I will describe recent progress in computing ample and effective cones of moduli spaces of Gieseker semistable sheaves on surfaces. The main tool is Bayer and Macri's construction of a nef divisor on moduli spaces of Bridgeland semistable objects. After briefly reviewing the developments for P^2 and P^1 x P^1, I will describe how Bridgeland stability allows us to carry out these computations on a large range of surfaces including surfaces of general type such as very general hypersurfaces in P^3 and very general cyclic covers of the plane. This is joint work with Jack Huizenga and builds on joint work with Arcara, Bertram and Woolf.


Eduardo Esteves

Title: Toric tilings, compactified Jacobians and limit linear series
Abstract: This is a report on joint work with Omid Amini (ENS Paris). The goal is to construct a new compactification of the Jacobian over reducible nodal curves, by using what we may call toric tilings. These are unions of toric varieties described by combinatorial data which serve as bases for families of line bundles and their degenerations over reducible nodal curves. While the approach taken to date considers as objects of the moduli problem line bundles, our objects are rather these families over toric tilings. I will contend in this talk that this new compactification is suitable for the study of limit linear series on all reducible nodal curves.


Jochen Heinloth

Title: A stack theoretic GIT criterion to construct separated coarse moduli
Abstract: One of the applications of GIT is that it allows to single out parts of moduli problems that admit separated coarse moduli spaces. Numerical stability criteria can be formulated purely in terms of moduli problems (this was formulated independently by Daniel Halpern-Leistner). One can also give a stack theoretic criterion showing that this often defines separated substacks of moduli stacks admitting coarse moduli spaces. Sometimes this allows to avoid difficult GIT calculations. We apply this to parahoric group schemes on curves.


Klaus Hulek

Title: Complete moduli of cubic threefolds
Abstract: Cubic threefolds have played an important role in algebraic geometry, ever since Clemens and Griffiths showed that these are unirational but not rational varieties. The crucial ingredient in their proof is a careful analysis of the intermediate Jacobian. The map which associates to a cubic threefold $X$ its intermediate Jacobian $IJ(X)$ defines an injective map from the GIT moduli space of smooth cubics to the moduli space $A_5$ of principally polarized abelian $5$-folds. Here we shall discuss the behaviour of this map when the cubic threefold acquires singularities. Our main result is that the intermediate Jacobian map extends to a regular morphism $\overline{IJ}: \widetilde M \to A_5^{\mathrm{Vor}}$ from the wonderful blow-up of the GIT moduli space of cubic threefolds to the second Voronoi compactification. This enables us to understand the degenerations of intermediate Jacobians geometrically. This is joint work with S. Casalaina-Martin, S. Grushevsky and R. Laza.


David Jensen

Title: A nonarchimedean analytic approach to the geometry of general curves
Abstract: A standard approach to the study of general curves is via degeneration arguments. One considers a family of smooth curves degenerating to a singular curve, and attempts to deduce geometric properties of the general fiber using information about the singular fiber. Recently, it has been observed that this theory fits within a broader framework of nonarchimedean analytic techniques originally due to Berkovich and further developed by several other authors. We will discuss recent applications of nonarchimedean analytic geometry to problems concerning the geometry of general curves, including the Maximal Rank Conjecture and properties of the Gaussian-Wahl map.


Michael Kemeny

Title: Betti numbers of canonical curves and Hurwitz spaces
Abstract: The coordinate ring of a canonical curve and its asociated Betti numbers have been studied since the very beginning of algebraic geometry, in works of Petri, Hilbert and others. More recently, Mark Green offered a conjectural description of precisely which of the Betti numbers vanish in the 80s which was later resolved for general curves by Vosin. This talk will concern a conjecture of Schreyer which goes beyond Green's conjecture to predict the values of the extremal nonzero Betti numbers. The geometry of Hurwitz spaces plays a crucial role. This is joint work with Gabi Farkas.


Sándor Kovács

Title: On the boundedness of slc surfaces of general type
Abstract: This is a report on joint work with Christopher Hacon. We give a new proof of Alexeev's boundedness result for stable surfaces which is independent of the base field. We also highlight some important consequences of this result.


Emanuele Macri

Title: Bridgeland stability for semiorthogonal decompositions
Abstract: I will present a general method to induce Bridgeland stability conditions on semiorthogonal decompositions. We will show, in particular, the existence of Bridgeland stability conditions on the Kuznetsov component of the derived category of (some) Fano 3folds and of cubic fourfolds. This is joint work in progress with Arend Bayer, Martí Lahoz, and Paolo Stellari.


Daniel Murfet

Title: Generalised orbifolding of simple singularities
Abstract: The simple hypersurface singularities have a famous ADE classification due to Arnold. Surprisingly, there are still new things to say about these basic objects of singularity theory: inspired by orbifolding relations among minimal conformal field theories, which also have an ADE classification, a trio of mathematical physicists Ros Camacho-Carqueville-Runkel showed that all simple singularities are “generalised orbifolds” of A-type singularities. I will explain their theorem, and how it uses results on a bicategory of isolated hypersurface singularities and matrix factorisations, developed in my own joint work with Carqueville.


Peter O'Sullivan

Title: Universal principal bundles over a scheme
Abstract: Let X be a scheme over a field k and x be a k-point of X. Consider the functor on affine k-groups up to conjugacy that assigns to G the set of isomorphism classes of principal G-bundles trivial above x. In general, this functor is not representable. However its restriction to an appropriate full subcategory may be representable, under suitable conditions on X. We discuss some cases where this is so.


Brett Parker

Title: Tropical enumeration of curves in blowups of the projective plane
Abstract: The degeneration formula for Gromov-Witten invariants in normal crossings or log smooth degenerations involves a sum over tropical curves. The contribution of each tropical curve is combination of relative invariants corresponding to each vertex. These relative invariants are often readily computable from consistency constraints. In this talk, I will explain how this works with the example of counting holomorphic curves in blowups of the projective plane. I plan to make most aspects of the talk accessible to non-experts. There will be lots of pictures and fun zooming around.


Jason Starr

Title: Spaces of Rational Curves and Varieties in Positive Characteristic
Abstract: I will survey techniques for proving results about varieties in positive characteristic, mostly about Fano manifolds, using spaces of rational curves: Serre's Conjecture II, Period-Index Theorems, existence of rational surfaces on 2-Fano varieties, results about fundamental groups and Picard groups, and results about specializations (ala the Ax conjecture). The emphasis will be on extending to char p the well-known techniques from characteristic 0.


Burt Totaro

Title: Rationality does not specialize among terminal varieties
Abstract: Hassett, Pirutka, and Tschinkel showed that rationality is not an open condition among smooth complex projective 4-folds. One remaining question is whether rationality is a closed condition in a certain sense. Namely: given a family of smooth projective varieties for which very general fibers are rational, is every fiber rational? We discuss the positive and negative results on this problem if we allow mildly singular (terminal) varieties.


Tuyen Truong

Title: Some etale ideas in algebraic and topological dynamical systems
Abstract: I will present the definition and some main properties of dynamical degrees, which are birational invariants, and explain their relation to topological entropy, an important invariant of dynamical systems. Then I will describe some conjectures, using etale analogs of dynamical systems, aiming to extend results from holomorphic dynamics to dynamics over fields different from C. One conjecture among these aims to extend a recent result by Esnault and Srinivas that dynamical degrees of an automorphism of surfaces can be computed in terms of l-adic cohomology groups. Thus this conjecture can be viewed as an extension of Weil’s conjecture to the dynamical setting.


Claire Voisin

Title: Cubic fourfolds and O'Grady 10-dimensional hyper-Kaehler manifolds
Abstract: This is joint work with R. Laza and G. Saccà. We construct a smooth hyper-Kaehler compactification of the intermediate Jacobian fibration associated to a general cubic fourfold and show that this variety is a deformation of the 10-dimensional O'Grady hyper-Kaehler manifold. This realizes a project that had been initiated by D. Markushevich.


Ruibin Zhang

Title: Invariant theory of classical supergroups
Abstract: We establish first and second fundamental theorems (FFT and SFT) of invariant theory for the group super schemes $\mathrm{GL}(V)$ and $\mathrm{OSp}(V)$, where $V$ is a vector superspace. The theorems reduce respectively to FFT and SFT of invariant theory for the ordinary general linear group, orthogonal group and symplectic group when $V$ is purely even or purely odd. The proof of the SFT for the orthosymplectic supergroup $\mathrm{OSp}(V)$ yields a new proof of the SFTs for orthogonal and symplectic groups. This is joint work with Deligne and Lehrer.