# Seminars & Events for 2010-2011

##### Serre´s conjectures on the number of rational points of bounded height

We give a survey of recent results on conjectures of Heath-Brown and Serre on the asymptotic density of rational points of bounded height. The main tool in the proofs is a new global determinant method inspired by the local real and p-adic determinant methods of Bombieri-Pila and Heath-Brown.

##### Computational aspects of bordered Floer homology

I will give a brief outline of bordered Floer homology, and explain how it can be used to describe aspects of Heegaard Floer homology. I will pay special emphasis on the branched double-covers of links in $S3$. This is joint work with Robert Lipshitz and Dylan Thurston.

##### Parabolic equations and the Ricci flow on manifolds with boundary

In the first part of the talk, we will focus on a second-order quasilinear parabolic equation in a vector bundle over a compact manifold~$M$ with boundary. Our goal is to explore the short-time existence of solutions to this equation. In the second part, we will discuss the Ricci flow on~$M$.

##### Relative Entropy Indicators and applications to Authorship Attribution

Estimating the entropy of an unknown (ergodic, stationary) stochastic source with unknown memory, only relying on finite samples, poses interesting and challenging mathematical questions. In this talk we aim first to review some recent and less recent relative entropy indicators, such as the zipper's based methods.

##### On the three compactifications of Siegel space

The moduli space of $A_g$ of abelian varieties has three classical toroidal compactifications: (1) perfect, (2) 2nd Voronoi, and (3) Igusa blowup, each with its own distinct geometric meaning. It is an interesting problem to understand exactly how these compactifications are related.

##### L-spaces and left-orderability

Various families of examples suggest an interesting correspondence between L-spaces and 3-manifolds with non-left-orderable fundamental group. This motivates the study of left-orderability in the context of Dehn surgery. In particular, since every knot group is left-orderable, we study the phenomenon of when a left-order descends to the quotient group associated with the surgery.

##### Geometry of level sets of elliptic equations and a conjecture of De Giorgi

I will describe recent results on the rigidity of level sets of solutions of local and non local elliptic equations on the euclidean space and on riemannian manifolds in connection with a conjecture by De Giorgi.

##### Higher order Fourier analysis

In a famous paper Timothy Gowers introduced a sequence of norms $U(k)$ defined for functions on abelian groups. He used these norms to give quantitative bounds for Szemeredi's theorem on arithmetic progressions. The behavior of the $U(2)$ norm is closely tied to Fourier analysis.

##### Relative Homotopy type and obstructions to the existence of rational points

In 1969 Artin and Mazur defined the etale homotopy type $Et(X)$ of scheme $X$, as a way to homotopically realize the etale topos of a $X$. In the talk I shall present for a map of schemes $X\rightarrow S$ a relative version of this notion. We denoted this construction by $Et(X/S)$ and call it the homotopy type of $X$ over $S$.

##### A Heegaard Floer characterization of Borromean knots

If the total rank of the knot Floer homology of a knot is equal to the total rank of the Heegaard Floer (hat) homology of the ambient 3-manifold, we say that the knot has simple knot Floer homology (or the knot is Floer simple). It is known that the unknot is the only Floer simple knot in $S3$.

##### Theta Functions

In this talk we consider Weierstrass points for the linear space of meromorphic functions on a compact Riemann surface whose divisors are multiples of $\frac{1}{P_0^{\alpha}P_1..P_{g-1}}$ where the points $P_i$ are points on the surface and $\alpha$ is a positive integer for which there is no holomorphic differential whose divisor is a multiple of $P_0^{\alpha)P_1..P_{g-1}}$.

##### Finding (dis)similarities between surfaces using conformal geometry

##### Forward model coding in compressive optical imagers

##### Geometry and combinatorics for revolute-jointed robot arms

(This is joint work with I. Streinu) We present a complete theoretical characterization and a method for calculating the reachable workspace boundary for all serial manipulators with revolute joints having any pair of consecutive joint axes coplanar. The number of joints is arbitrary.

##### On spaces of homomorphisms and spaces of representations

The subject of this talk is the structure of the space of homomorphisms from a group $\pi$ to a Lie group $G$ denoted $Hom(\pi,G)$. The space of representations $Hom(\pi,G)/G$ obtained from the adjoint action of $G$ will be considered. In special cases, these spaces can be assembled into a single space analogous to the classifying space of the group $G$.

##### The Unstable Chromatic Spectral Sequence

The chromatic spectral sequence has had a profound influence towards calculation of stable homotopy groups of the spheres. I will outline the changes necessary to set up an unstable chromatic spectral sequence. there will be applications to Hopf invariants.