# Seminars & Events for 2012-2013

##### TBA

##### Regular del Pezzo surfaces with irregularity

Over perfect fields, the geometry of regular del Pezzo surfaces has been classified, but over imperfect fields, the problem remains largely open. We construct the first examples of regular del Pezzo surfaces X that have positive irregularity h^1(X, O_X ) > 0. Our construction is by quotienting a regular, quasi-linear surface (i.e.

##### Derangements

A fixed point free permutation of a set is called a derangement. It is an old result of Jordan that if G is a transitive permutation group on a finite set, then derangements exist. In fact, some anaylsis of this problem predates group theory going back to the early 1700's. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Singularity Theory

Pioneered by Whitney in the 1950's, singularity theory deals with the local behavior of smooth mappings between manifolds. We will discuss some basics of singularity theory, including what a "generic" smooth map looks like, and hopefully hint at some applications in topology as well as the real world.

##### An introduction to motives

**PLEASE NOTE SPECIAL TIME AND LOCATION.** ** This is in addition to the regular PU/IAS Number Theory Seminar**. We review the construction of the triangulated categories of motives over a base scheme (following the method of Morel and Voevodsky).

##### Relative Artin motives and the reductive Borel-Serre compactification of a locally symmetric variety

Let $X$ be a locally symmetric variety, $\bar{X}$ its Baily-Borel compactification, $\bar{X}^{rbs}$ its reductive Borel-Serre compactification and $p:\bar{X}^{rbs} \to \bar{X}$ the canonical map. We prove that the derived direct image sheaf $Rp_*\mathbb{Q}$ is the realization of a canonical motive associated to the variety $\bar{X}$.

##### Structure and rigidity of totally periodic pseudo-Anosov flows in graph manifolds

This is joint work with Thierry Barbot. A graph manifold is an irreducible manifold so that all pieces of the torus decomposition are Seifert fibered. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Toric b-symplectic and origami manifolds

Origami manifolds and b-symplectic manifolds are examples of manifolds which are symplectic except on a hypersurface Z: in the origami case, the symplectic form vanishes at Z, in the b-case, it explodes to infinity at Z.

##### Regularity theory for area-minimizing currents

It was established by Almgren at the beginning of the eighties that area-minimizing $n$-dimensional currents in Riemannian manifolds are regular up to a singular set of dimension at most $n-2$. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Approximation Bounds for Sparse Principal Component Analysis

We produce approximation bounds on a semidefinite programming relaxation for sparse principal component analysis. These bounds control approximation ratios for tractable statistics in hypothesis testing problems where data points are sampled from Gaussian models with a single sparse leading component.

##### Analysis and Partial Differential Equations on Polyhedral Domains

The classical theory of partial differential equations (PDEs) on smooth, bounded domains--a great achievement of modern mathematics--is well understood and has many applications in both pure and applied mathematics. Many domains that arise in applications are, however, not smooth (that is, they do not have a smooth boundary). PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Minerva Lecture I: Sets with few ordinary lines

Given n points in the plane, an _ordinary line_ is a line that contains exactly two of these points, and a _3-rich line_ is a line that contains exactly three of these points. CLICK ON LECTURE TITLE FOR COMPLETE ABSTRACT.

##### Motivic Galois groups and periods

We explain how to associate a universal pro-algebraic group to the Betti realization functor from the triangulated category of motives over a subfield of $\mathbb{C}$. We then give a concrete description of the torsor of isomorphisms between the Betti realization and the de Rham realization. If time permits, some applications to periods will be sketched.

##### Minerva Lecture II: Polynomial expanders and an algebraic regularity lemma

The _sum-product phenomenon_ is the observation that given a finite subset A in a ring, at least one of the sumset A+A or the product set A.A is typically significantly larger than A itself, except when A is "very close" to a field in some sense. CLICK ON LECTURE TITLE FOR COMPLETE ABSTRACT.

##### Topological Invariants in Knot Theory

The primary motivation for studying knot theory is the immediate applications to 3 and 4 dimensional topology. This has led to the development of homology theories and other invariants meant to extract information in this context. However, there are much simpler topological invariants that arise naturally and, despite their simplicity, are incredibly difficult to calculate.

##### Entropy and the localization of eigenfunctions on negatively curved manifolds - I

We are interested in the behaviour of laplacian eigenfunctions on negatively curved manifolds, in the high frequency limit. The Quantum Unique Ergodicity conjecture predicts that they should become uniformly distributed over phase space, and the Shnirelman theorem states that this is true if we allow ourselves to possibly drop a ``negligible'' family of eigenfunctions.

##### On the conservation of (equivariant) homeomorphism classes of M(J)-manifolds

A /toric manifold / is a closed manifold of dimension 2n which admits a locally standard half dimensional torus action whose orbit space can be identified with an n-dimensional simple polytope. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Regularized periods of automorphic forms

Following Jacquet, Lapid and Rogawski, we define a regularized period of an automorphic form on GL(n+1) x GL(n) along the diagonal subgroup GL(n) and express it in terms of the Rankin-Selberg integral of

Jacquet, Piatetski-Shapiro and Shalika. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Polar codes and randomness extraction for structured sources

Polar codes have recently emerged as a new class of low-complexity codes achieving Shannon capacity. This talk introduces polar codes with emphasis on the probabilistic phenomenon underlying the code construction. New results and connections to matroid theory and randomness extraction are discussed.

##### Minerva Lecture III: Universality for Wigner random matrices

Wigner random matrices are a basic example of a Hermitian random matrix model, in which the upper-triangular entries are jointly independent. CLICK ON LECTURE TITLE FOR COMPLETE ABSTRACT.