Abstract: A central problem in algebraic geometry is to construct and study moduli spaces of objects of interesting geometric objects. The classical tool for this is geometric invariant theory, which requires you to approximate a moduli problem by an orbit space X/G for some reductive group G acting on a variety X. In recent years, I and others have developed a set of tools for proving structural results about a moduli problem, such as the existence of a moduli space, without approximating the moduli problem by an orbit space. I will discuss one example in which these tools work very cleanly: the moduli space of maps from a smooth curve to a quotient stack V/G, where V is a linear representation of G. I will also discuss the one major thing that the intrinsic approach does not get you: quasi-projectivity of the moduli space. Recently I have been exploring ways to fix that using, of all things, geometric invariant theory.

Abstract: We will discuss a natural perspective on stable reduction that extends Deligne--Mumford's stable reduction for curves to higher dimensions. From this perspective, we will outline a proof of stable reduction for surfaces in large characteristic.

Abstract: The Du Bois invariants are natural invariants for singularities. In this talk, I will explain what they are, why they are interesting, and report on recent and ongoing work studying their properties.

Abstract: We adapt Bost's algebraicity characterization to the situation of a germ in a compact Kähler manifold. As a consequence, we extend the algebraic integrability criteria of Campana-Paun and of Druel to foliations on compact Kähler manifolds. As an application, we prove that a compact Kähler manifold is uniruled if and only if its canonical line bundle is not pseudoeffective.

Abstract: We show that the minimal log discrepancy of any isolated Fano cone singularity is at most the dimension of the variety. This is based on its relation with dimensions of moduli spaces of orbifold rational curves. We also propose a conjectural characterization of weighted projective spaces as Fano orbifolds in terms of orbifold rational curves.

Abstract: Hassett showed that there are natural reduction morphisms between moduli spaces of weighted pointed stable curves when we reduce weights. I will discuss some joint work with Ziquan Zhuang which constructs similar morphisms between moduli of stable pairs in higher dimension.

Abstract: The monodromy conjecture of Denef—Loeser is a conjecture in singularity theory that predicts that given a complex polynomial f, and any pole s of its motivic zeta function, exp(2πis) is a "monodromy eigenvalue" associated to f. I will formulate a "birational geometric" version of the conjecture, and briefly sketch ongoing work to reduce the conjecture to the case of Newton non-degenerate hypersurfaces. These are hypersurface singularities whose singularities are governed, up to a certain extent, by faces of their Newton polyhedra. The extent to which the former is governed by the latter is a key aspect of the conjecture.

Abstract: The Hodge diamond of a smooth projective complex variety exhibits fundamental symmetries, arising from Poincaré duality and the purity of Hodge structures. In the case of a singular projective variety, the complexity of the singularities is closely related to the symmetries of the analogous Hodge–Du Bois diamond. For example, the failure of the first nontrivial Poincaré duality is reflected in the defect of factoriality. Based on joint work with Mihnea Popa, I will discuss how local and global conditions on singularities influence the topology of algebraic varieties.

Abstract: I will present a compactification of the moduli space of maps from families of curves, to certain moduli spaces M, via the example of M being the GIT moduli space of binary forms of degree 2n. The main application of our results is the construction of certain moduli of fibered log Calabi-Yau pairs. This is a joint work with Andrea Di Lorenzo.

Abstract: The enumerative geometry of the moduli spaces of curves, which began in the 1980s with a famous paper of Mumford, is now an extremely developed field, with perhaps thousands of papers dedicated to it. It's high time to do the higher dimensional case! The analogues of M_{g,n} in higher dimensions are the KSBA spaces. I will introduce some characteristic classes on the KSBA spaces that are analogous to the kappa, lambda and psi classes on the moduli of curves, and present some results and speculations.