# Seminars & Events for 2007-2008

##### Star-coloring planar graphs with high girth

A star-coloring is a proper vertex-coloring such that the graph induced by the union of any 2 color classes is a star-forest. Equivalently, it is a proper vertex-coloring with no 2-colored $P_4$. Star-coloring has been studied extensively, even for planar graphs. However, little is known about improved upper bounds for planar graphs with large girth.

##### Real Projective Structures and Non-standard analysis

We investigate the analog of the Thurston boundary of Teichmüller space in the context of convex real projective structures on closed manifolds. In particular we give a new interpretation of measured laminations in terms of non-standard hyperbolic structures over the hyper-reals.

##### Counting rational points on a cubic surface

A conjecture of Manin predicts precise asymptotic for the density of rational points on del Pezzo surfaces. This has been satisfactorily settled for del Pezzo surfaces of higher degree. But for lower degree not much is known.

##### No mass drop for mean curvature flow of mean convex hypersurfaces

A possible evolution of a compact hypersurface in $R^{n+1}$ by mean curvature past singularities is defined via the level set flow. In the case that the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. We obtain as a consequence that no mass drop can occur along such a flow.

##### Branched Polymers

A branched polymer is a finite, connected set of non-overlapping unit balls in space. The powerful "dimension reduction" theorem of Brydges and Imbrie permits computation of the volume of the space of branched polymers of size N in dimensions 2 or 3. We will show how these and some related computations can be done using elementary calculus and combinatorics.

##### Maps between moduli spaces of curves and Gieseker-Petri divisors

We study contractions of the moduli space of stable curves beyond the minimal model of M_g by resolving and giving a complete enumerative description of the rational map between moduli spaces of curves Mg --> Mh which associates to a curve C of genus g, the Brill-Noether locus of special divisors in the case this locus is a curve.

##### Arithmetic invariants of discrete Langlands parameters

Let $G$ be a reductive algebraic group over a local field $k$. Hiraga, Ichino and Ikeda have recently proposed a general conjecture for the formal degree of a discrete series representation of $G(k)$, using special values of the adjoint L-function and $\epsilon$ factor of its (conjectural) Langlands parameter.

##### The Resolvent Algebra: A Novel Approach to Canonical Quantum Systems

The standard C*-algebraic version of the algebra of canonical commutation relations, the Weyl algebra, frequently causes difficulties in applications since it neither admits the automorphic action of physically interesting dynamics nor does it incorporate pertinent physical observables such as (bounded functions of) the Hamiltonian.

##### Chaoticity of the Teichmüller flow

A non-zero Abelian differential on a compact Riemann surface determines an atlas, outside the singularities, whose coordinate changes are translations. The vertical flow with respect to this translation structure generalizes the genus one notion of rational and irrational flows on tori.

##### Spherical billiards with many 3-periodic orbits

It is known that the Lebesgue measure of 3-periodic trajectories in a planar (Birkhoff) billiards is zero (and a well-known conjecture states that the same is true for any period). On the sphere, however, it is easy to construct a billiard domain with 2-dimensional family of 3-periodic orbits (take the intersection of the sphere with the positive octant).

##### Sumsets in finite fields and Cayley sum graphs

I will sketch a proof of the fact that for a prime $p$, every complement of a set of roughly $\sqrt{p}$ elements of the finite field $Z_p$ is a sumset, that is, is of the form $A+A$, whereas there are complements of sets of size roughly $p^{2/3}$ which are not sumsets. This improves estimates of Green and Gowers, and can also be used to settle a recent problem of Nathanson.

##### Variational principles on triangulated surfaces

We will discuss various applications of recently discovered 2-dimensional counterparts of the Schlaefli formula.

##### Iwasawa theory of elliptic curves for supersingular primes

Studying the Selmer groups of elliptic curves for a supersingular prime is difficult. It turned out we should instead use the plus/minus Selmer groups defined by Kobayashi.

##### Special Lagrangian fibrations, instanton corrections and mirror symmetry

We study the extension of mirror symmetry to the case of Kahler manifolds which are not Calabi-Yau: the mirror is then a Landau-Ginzburg model, i.e. a noncompact manifold equipped with a holomorphic function called superpotential.

##### The abstract concept of Duality and some related facts (part of a joint project with Shiri Artstein-Avidan)

We discuss in the talk an unexpected observation that very minimal basic properties essentially uniquely define some classical transforms which traditionally are defined in a concrete and quite involved form. We start with a characterization of a very basic concept in Convexity: Duality and the Legendre transform.

##### A worldwide web of images

In this talk we'll explore the emerging potential of computer vision to transform the way we think about the interconnectedness of digital imagery and the Web, and how these relate to our physical environment.

##### Mass inequalities for Cauchy data in general relativity

We will survey a number of inequalities involving the total mass and other invariants for initial data for the Einstein equations in general relativity.

##### On dimensionality of mean structure from a single data matrix

We consider inference from data matrices that have low dimensional mean structures. In educational testing and in probe-level microarray data, estimation and inference are often made from a single data matrix believed to have a uni-dimensional mean structure.

##### Minimizing the ground state energy of an electron in a randomly deformed lattice

We provide a characterization of the spectral minimum for a random Schrödinger operator of the form $H=-\Delta + \sum_i {\in Z^d} q(x-i-\omega_i)$ in $L^2(R^d)$, where the single site potential $q$ is reflection symmetric, compactly supported in the unit cube centered at $0$, and the displacement parameters $\omega_i$ are restricted so that adjacent single site potentials do not overlap.

##### The birational geometry of Kontsevich moduli spaces

I will describe the stable base loci of linear systems on the Kontsevich moduli spaces of maps to projective spaces and Grassmannians. This description allows us to run the log minimal model program for these moduli spaces in small degree. I will give some examples where interesting classical moduli spaces occur.