Topological similarity of random cell complexes, and applications

Benjamin Schweinhart
Princeton University
December 10, 2014
Although random cell complexes occur throughout the physical sciences, there does not appear to be a standard way to quantify their statistical similarities and differences. I'll introduce the notions of a 'swatch' and a 'cloth', which provide a description of the local topology of cell complexes which is both general (any physical system that may be represented as a regular cell complex is admissible) and complete (any statistical question about the local topology may be answered from the cloth).

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.

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.

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.

Small value parallel repetition for general games

Ankit Garg
Princeton University
January 20, 2015
We prove a parallel repetition theorem for general games with value tending to 0. Previously Dinur and Steurer proved such a theorem for the special case of projection games. Our proofs also extend to the high value regime (value close to 1) and provide alternate proofs for the parallel repetition theorems of Holenstein and Rao for general and projection games respectively. Our techniques are elementary in that we only need to employ basic information theory and discrete probability in the small-value parallel repetition proof.

On descending cohomology geometrically

Sebastian Casalaina-Martin
University of Colorado at Boulder
January 20, 2015
In this talk I will present some joint work with Jeff Achter concerning the problem of determining when the cohomology of a smooth projective variety over the rational numbers can be modeled by an abelian variety. The primary motivation is a problem posed by Barry Mazur. We provide an answer to Mazur's question in two situations. First, we show that the third cohomology group can be modeled by the cohomology of an abelian variety over the rationals provided the Chow group of points is supported on a curve.

A birational model of the Cartwright-Steger surface

Igor Dolgachev
University of Michigan
January 21, 2015
A Cartwright-Steger surface is a complex ball quotient by a certain arithmetic cocompact group associated to the cyclotomic field $Q(e^{2\pi i/12})$, its numerical invariants are with $c_1^2 = 3c_2 = 9, p_g = q = 1$. It is a cyclic degree 3 cover of a simply connected surface of general type with $c_1^2 = 2, p_g = 1$. A similar construction in the case of the cyclotomic fields $Q(e^{2\pi i/5})$ (resp. $Q(e^{2\pi i/7})$) leads to the beautiful geometry of a del Pezzo surface of degree 5 and its K3 double cover branched along the union of lines (resp.

Publicly-verifiable non-interactive arguments for delegating computation

Guy Rothblum
Stanford University
January 26, 2015
We construct publicly verifiable non-interactive arguments that can be used to delegate polynomial time computations. These computationally sound proof systems are completely non-interactive in the common reference string model. The verifier’s running time is nearly-linear in the input length, and poly-logarithmic in the complexity of the delegated computation. Our protocol is based on graded encoding schemes, introduced by Garg, Gentry and Halevi (Eurocrypt 2012). Security is proved under a falsifiable and arguably simple cryptographic assumption about graded encodings.