Topology of Algebraic Varieties (TAV)

Stable cohomology of compactifications of Ag

Klaus Hulek
Leibniz Universität Hannover
January 14, 2015
A famous result of Borel says that the cohomology of $\mathcal A_g$ stabilizes. This was generalized to the Satake compactification by Charney and Lee. In this talk we will discuss whether the result can also be extended to toroidal compactifictaions. As we shall see this cannot be expected for the second Voronoi compactification, but we shall show that the cohomology of the perfect cone compactification does stabilize. We shall also discuss partial compactifications, in particular the matroidal locus. This is joint work with Sam Grushevsky and Orsola Tommasi.

Normal functions and the geometry of moduli spaces of curves

Richard Hain
Duke University; Member, School of Mathematics
January 13, 2015
In this talk, I will begin by recalling the classification of normal functions over $\mathcal M_{g,n}$, the moduli space of $n$-pointed smooth projective curves of genus $g$. I'll then explain how they can be used to resolve a question of Eliashberg, how they generate the tautological ring of $\mathcal M_{g,n}$, and how they can be used to strengthen slope inequalities of the type proved by Moriwaki.

The law of Aboav--Weaire and extensions

Richard Ehrenborg
University of Kentucky and Princeton University
December 10, 2014
In two-dimensional grain structure, one observes that grains with a small number of sides tend to be surrounded by grains with a large number of sides, and vice-versa. The Law of Aboav--Weaire gives this observation a mathematical formulation, that is, the average number of sides of the neighbors of an $n$-sided grain should be roughly $5 + 6/n$. By introducing the correct error term we prove this law of Material Science. We extend this law to three-dimensional grain structures.

The Andre-Oort conjecture II

Bruno Klingler
Université Paris Diderot; Member, School of Mathematics
December 10, 2014
The Andre-Oort conjecture describes the expected distribution of special points on Shimura varieties (typically: the distribution in the moduli space of principally polarized Abelian varieties of points corresponding to Abelian varieties with complex multiplication). From the point of view of Hodge theory, it completely describes the geometric properties of the Hodge locus in some special instances. In these lectures I will try to introduce Shimura varieties, the Andre-Oort conjecture and recent work on it.

A support theorem for the Hitchin fibration

Pierre-Henri Chaudouard
Université Paris 7; von Neumann Fellow, School of Mathematics
December 9, 2014
The main tool in Ngô's proof of the Langlands-Shelstad fundamental lemma, is a theorem on the support of the relative cohomology of the elliptic part of the Hitchin fibration. For $\mathrm{GL}(n)$ and a divisor of degree $> 2g-2$, the theorem says that the relative cohomology is completely determined by its restriction to any dense open subset of the base of the Hitchin fibration. In this talk, we will explain our extension of that theorem to the whole Hitchin fibration, including the global nilpotent cone (for $\mathrm{GL}(n)$ and a divisor of degree $> 2g-2$).

Minimal log discrepancy of isolated singularities and Reeb orbits

Mark McLean
Stony Brook University
December 3, 2014
Let $A$ be an affine variety inside a complex $N$ dimensional vector space which either has an isolated singularity at the origin or is smooth at the origin. The intersection of $A$ with a very small sphere turns out to be a contact manifold called the link of $A$. If the first Chern class of our link is torsion (I.e. the singularity is numerically $\mathbb Q$ Gorenstein) then we can assign an invariant of our singularity called the minimal discrepancy. We relate the minimal discrepancy with indices of certain Reeb orbits on our link.

Birational geometry of complex hyperbolic manifolds

Gabriele di Cerbo
Columbia University
November 19, 2014
In 1984 Hirzebruch constructed the first examples of smooth toroidal compactifications of ball quotients with non-nef canonical divisor. In this talk, I will show that if the dimension is greater or equal than three then such examples cannot exist. We will use this result to reprove and improve classical theorems, such as boundedness of hyperbolic manifolds, Baily-Borel embeddings and cusps count.

The geometry and topology of rational surfaces with an anticanonical cycle

Robert Friedman
Columbia University
November 18, 2014
Let \(Y\) be a smooth rational surface and let \(D\) be an effective divisor linearly equivalent to \(-K_Y\), such that \(D\) is a cycle of smooth rational curves. Such pairs \((Y,D)\) arise in many contexts, for example in the study of degenerations of \(K3\) surfaces or in the theory of deformations of minimally elliptic singularities. Deformation types of such pairs come with two extra pieces of structure: the “generic” ample cone, i.e.