Abstracts and Materials

Title: Resolving singularities in families

Dan Abramovich (Brown University)

Abstract: Semistable reduction is often the first step in constructing compactified moduli spaces, and can be used to discover their properties. I will describe work-in-progress with Michael Temkin and Jaroslaw Wlodarczyk in which we prove functorial semistable reduction for families of varieties in characteristic 0, refining work with Karu from 2000. Techniques developed for moduli spaces enter in unexpected ways.

Title: "Smoothings of singular Calabi-Yau threefolds"

Robert Friedman (Columbia University)

Abstract: Calabi-Yau threefolds can degenerate in many interesting ways. In this talk, which is a report on joint work with Radu Laza, we survey some results on Calabi-Yau threefolds acquiring canonical singularities and describe, under suitable hypotheses, how the geometry of smoothings is related to the topology of an appropriate resolution. 

Title: Mixed Spin P field theory and Gromov-Witten invariants of quintic Calabi-Yau threefold

Jun Li (Stanford)

Abstract: Mixed Spin P field is a mathematical realization of Witten’s vision to study Gromov Witten invariants of quintic Calabi-Yau threefolds. In this talk, I will report the current progress toward solving all genus generating functions of the Gromov-Witten invariants of quintic Calabi-Yau threefolds.

Title: Segre series for Hilbert schemes of points on surfaces. 

Alina Marian (Northeastern University)

Abstact: The calculation of intersection-theoretic invariants for Hilbert schemes of points on a smooth surface is a basic problem in enumerative geometry. An interesting series of invariants is given by the top Segre classes of tautological vector bundles over the Hilbert scheme. In the case of tautological bundles arising from line bundles over the surface, conjectural formulas for the Segre series were proposed by M. Lehn in 1999, with a  proof emerging recently in work of C. Voisin and a joint project with D. Oprea and R. Pandharipande. I will explain this circle of ideas, and will describe progress in the higher-rank case. The talk is based on ongoing work  with Dragos Oprea and Rahul Pandharipande. 

Title: Umemura 3-folds as moduli of abelian surfaces, and the Horrocks-Mumford bundle

Shigeru Mukai (RIMS Kyoto)

Abstract: The (compactified) moduli of elliptic curves of full d-level structure is the projective line P1 with a polyhedral group action for d=2,3,4,5.  For these value of d, the moduli of (1,d)-polarized abelian surfaces with suitable level structure is P3 (d=2,3,4) or has the P3 blown-up at 60 points as its small resolution (d=5). After a quick review of these result together with principally polarized case (d=1), I discuss the reconstruction of abelian surfaces via the Horrocks-Mumford vector bundle on P4 in the case d=5.

Title: Gromov-Witten theory of elliptic fibrations

Aaron Pixton (MIT)

Abstract: Let X be a variety. Gromov-Witten invariants of X are defined by integrals on the moduli space of stable maps from a genus g curve to X. I will outline a conjecture interpreting these invariants as coefficients of certain automorphic forms if X is the total space of an elliptic fibration X -> B. I will also discuss a relationship between this conjecture and Givental's R-matrix action on cohomological field theories. This is joint work with Georg Oberdieck.

Title: Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds

Yukinobu Toda (Kavli IPMU and University of Tokyo)

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.

Title: On the construction of moduli of Fano varieties with K-stability

Chenyiang Xu (BICMR Beijing)

Abstract: Higher dimensional geometry naturally leads to the  fundamental problem of parametrizing Fano varieties. However, it seems in general a nice parametrization doesn't exist, unless we post some stability condition. The notion of K-stability, which was first formulated by differential geometers like Tian and Donaldson, has a deep connection with Kahler-Einstein metric problem and suggests a good candidate. But to carry out it in algebraic geometry is a challenging problem, because of the 'infinite' nature involved in the original definition. In the recent years, there is a lot of progress in the algebraic geometry theory of K-stability, by the contribution of many people. In this talk, I will survey the progress, with a concentration on new techniques from birational geometry, e.g. the minimal model program.