Ben Bakker: Algebraicity of Shafarevich maps

Abstract: For a normal complex algebraic variety X equipped with a complex representation V of its fundamental group, a Shafarevich map f:X->Y is a map which contracts precisely those algebraic subvarieties on which V has finite monodromy.  Such maps were constructed for projective X by Eyssidieux, and recently have been constructed analytically in the quasiprojective case by Brunebarbe and Deng--Yamanoi, in both cases using techniques from non-abelian Hodge theory.  In joint work with Y. Brunebarbe and J. Tsimerman, we show that these maps are algebraic, and that in fact Y is quasiprojective.  This is a generalization of the Griffiths conjecture on the quasiprojectivity of images of period maps, and the proof critically uses o-minimal GAGA.

Luca Battistella: Integral Chow rings from alternative compactifications

Abstract: I will survey some constructions and applications of alternative compactifications of moduli spaces of curves, focussing in particular on the computation of integral Chow rings of moduli spaces of pointed elliptic curves with few markings via "wall-crossing" (joint work in progress with A. Di Lorenzo).

Michael Borinsky: On the top-weight cohomology of the moduli space of curves

Abstract: I will present new results on the asymptotic growth rate of the Euler characteristic of Kontsevich's commutative graph complex. By a recent work of Chan, Galatius, and Payne, these results imply the same asymptotic growth rate for the top-weight Euler characteristic of M_g, the moduli space of curves, and establish the existence of large amounts of unexplained top-weight cohomology in this space.

Gaetan Borot: On Chiodo integral vanishing

Abstract: I will present three types of vanishing of certain integrals involving Chiodo classes C(r,s) on the moduli of r-spin curves. We obtained them with Karev and Lewanski by combining two facts which mainly come from algebraic combinatorics: 1) topological recursion on a spectral curve (r,s) computes intersections of C(r,s) with \psi-classes; 2) the output of topological recursion is analytic along certain deformations of C(r,s) by a complex parameter t (either proved by a combinatorial interpretation in terms of Hurwitz numbers, or by direct analysis); 3) topological recursion on the deformed curves give a Laurent series in t whose coefficients are also integrals of C(r,s). Therefore, the coefficient of negative powers of t must vanish. Some of the vanishing relations obtained in this way generalise a result of Johnson, Pandharipande and Tseng found in their study of double Hurwitz numbers, but some others are of a different kind.

Samir Canning: Motivic structures in the cohomology of the moduli space of curves

Abstract: I will explain how recent progress on computing the Chow rings of moduli spaces of smooth curves combines with older topological results on the cohomology of the mapping class group to verify predictions from the Langlands program on the cohomology of the moduli space of stable curves. This is joint work with Hannah Larson and Sam Payne.

Philip Engel: Compact moduli of K3 and Enriques surfaces

Abstract: Due to Torelli theorems, moduli spaces of surfaces of Kodaira dimension 0 are orthogonal Shimura varieties. In the 60’s-80’s, compactifications of such varieties were constructed by Baily-Borel, Ash-Mumford-Rapaport-Tai, and Looijenga. But are any of these “semitoroidal” compactifications distinguished, in the sense that they parameterize some stable K3 or Enriques surfaces? Work on the Minimal Model Program from the 80’s-00’s by Kollar-Shepherd-Barron-Alexeev proved that an ample divisor on a Calabi-Yau variety defines a notion of stability, leading to compact moduli spaces. I will describe joint work with Alexeev, relating the Hodge-theoretic and MMP approaches to compactification, via the notion of a “recognizable divisor”.

Gavril Farkas: The birational geometry of M_g via tropical geometry and non-abelian Brill-Noether theory.

Abstract: I will discuss how novel ideas from non-abelian  Brill-Noether theory coupled with tropical geometry can be used to prove that the moduli space of  genus 16 is uniruled. This is the highest genus for which the moduli space is known not to be of general type. For the much studied question of  determining the Kodaira dimension of M_g, this case has long been understood to be crucial in order to make further  progress.  This is joint work with Verra. 

Angela Gibney: A reframing of the F-Conjecture

Abstract: The F-Conjecture is an old problem about the Mori Cone of the moduli space of curves. 

 I will describe the conjecture, its origins, and what is known about it.  I’ll also explain how one can

reduce the problem to showing nefness of divisors that contract collections of very particular curves.

Samuel Grushevsky: Compact subvarieties of ${\mathcal A}_g$

Abstract: We study the maximal dimension of compact subvarieties of the moduli space of complex abelian varieties, and of compact subvarieties through a generic point. This is based on joint work in progress with Mondello, Salvati Manni, and Tsimerman.

Victoria Hoskins: Motivic mirror symmetry and chi-independence for Higgs bundles

Abstract: Moduli spaces of Higgs bundles on a curve for Langlands dual groups are conjecturally related by a form of mirror symmetry. For SLn and PGLn, Hausel and Thaddeus conjectured a topological mirror symmetry given by an equality of (twisted orbifold) Hodge numbers, which was proven by Groechenig-Wyss-Ziegler and later by Maulik-Shen. We lift this to an isomorphism of Voevodsky motives, and thus in particular an equality of (twisted orbifold) rational Chow groups. Our method is based on Maulik and Shen's approach to the conjecture of Hausel-Thaddeus, as well as showing certain motives are abelian, in order to use conservativity of the Betti realisation on abelian motives. The same idea also enables us to prove a motivic chi-independence result. If there is time, I will explain how motivic nearby cycles can be used to specialise these results to positive characteristic. This is joint work with Simon Pepin Lehalleur.

Daniel Huybrechts: Moduli spaces of twisted line bundles and splitting Brauer classes.

Abstract: I report on joint work with Dominique Mattei in which we show that every Brauer class (or Azumaya algebra) splits after base change to a torsor for an abelian variety of bounded dimension. The construction uses the notion of moduli space of twisted line bundles and in particular leads to a uniform bound for the index. The latter is seen as a step towards the solution of the period-index conjecture. We will highlight the case of elliptic K3 surfaces where the situation is more transparent.

Bruno Klingler: Recent progress on Hodge loci

Abstract: Given a quasi projective family S of complex algebraic varieties, its Hodge locus is the locus of points of S where the corresponding variety admits exceptional Hodge classes (conjecturally: exceptional algebraic cycles). In this talk I will survey the many recent advances in our understanding of such loci, as well as the remaining open questions. 

Rahul Pandharipande: Cycles on the moduli space of abelian varieties

Abstract: I will discuss tautological projections of various cycle classes on the moduli space of PPAVs. The geometry is connected to many different directions: Pixton's relations for the moduli of curves, the Schubert calculus of the Lagrangian Grassmannian, and the quantum cohomology of the Hilbert scheme of points of the plane. The talk represents work (joint and disjoint) with S. Canning, A. Iribar Lopez, H. Larson, S. Molcho, D. Oprea, A. Pixton, and J. Schmitt.

Beatrice Pozzetti: The real spectrum compactification of character varieties

Abstract: I will discuss joint work with Burger, Iozzi and Parreau in which we investigate properties of a natural (real) algebraic compactification of character varieties of finitely generated groups in semisimple Lie groups. After describing how points at infinity in such compactification can be interpreted as equivalence classes of actions on buildings, I will explain the relation with the Thurston-Parreau marked lenght spectrum compactification and mention applications to Hitchin and maximal character varieties.

Dhruv Ranganathan: A tour of logarithmic moduli theory

Abstract: Logarithmic structures arise naturally from simple questions about compactifications and degenerations of algebraic varieties, and so are heavily intertwined with the geometry of moduli spaces. The typical logarithmic space is partly geometric and partly combinatorial, and the two pieces are beautifully tied together by the theory of algebraic stacks. Logarithmic geometry has undergone significant development in the last few decades, particularly through its interactions with enumerative geometry, tropical geometry, and mirror symmetry. I will aim to share some of the highlights, focusing in on the geometry of some of these moduli spaces and some of the lessons about logarithmic geometry that we have learned along the way. Some of the newer aspects of the story are based on joint work with Davesh Maulik and with Patrick Kennedy-Hunt. 

David Rydh: Non-reductive and topological moduli spaces

Abstract: I will present a notion of moduli spaces that singles out the topological features of reductive (good & adequate, GIT) as well as non-reductive moduli spaces (NRGIT). In particular, I'll report on an existence theorem for topological moduli spaces generalizing the recent result by Alper–Halpern-Leistner–Heinloth for reductive moduli spaces.

Adrian Sauvaget: Volumes of moduli spaces of cone surfaces

Abstract: I will give an overview of the computation of various volume functions associated with moduli spaces of surfaces with cone singularities, including Weil-Petersson polynomials and Masur-Veech volumes. I will emphasize the problems associated with cone singularities of integral angles. For these specific values, the moduli spaces carry a foliation defined via isomonodromic deformations of cone surfaces. Isomonodromic foliations can be used to give an alternative proof of Virasoro constraints of the point (Witten-Kontsevich theorem).