# Seminars & Events for 2007-2008

##### Metric and kernel learning

Many problems in data mining and machine learning, both in supervised and unsupervised learning, depend crucially on the choice of an appropriate distance or similarity measure. The appropriateness of such a measure can ultimately dictate the success or failure of the learning algorithm, but its choice is highly problem and application dependent.

##### Robust statistical techniques for financial modeling

A Series of Eight Lectures Mondays & Wednesdays at the Bendheim Center for Finance (BCF)

##### Integral Apollonian circle packings

Apollonian circle packings are infinite packings of circles, constructed recursively from an initial configuration of four mutually touching circles by adding circles externally tangent to triples of such circles. Configurations of four mutually touching circles were studied by Descartes in 1643. If the initial four circles have integer curvatures, so do all the circles in the packing.

##### Hyperbolic 3-Manifolds and Arithmetic

Thurston's geometrisation program tells us that we can understand all closed 3-manifolds if we can understand those with a finite volume hyperbolic structure. This is not easy, however, and it is not even clear at the outset that there exist more than a few such manifolds.

##### The bilinear Hardy-Littlewood function for the tail

##### Run time bounds for a model related to sieving

The following problem is very close to one arising in factorization algorithms.

Generate random numbers in the interval $[1,N]$ until some subset has a product which is a square. How long does this take (and how can you tell when you're done)?

##### Invariants of Legendrian knots in Heegaard Floer homology

A new invariant of Legendrian knots will be defined, taking values in the knot Floer homology of the underlying null-homologous knot. With the aid of this invariant we find transversely non-simple knots in many overtwisted contact structures, and show that the Eliashberg-Chekanov twist knots (in particular the $7_2$ knot in Rolfsen's table) are not transversely simple.

##### Stochastic convex optimization using mirror averaging algorithms

Several statistical problems where the goal is to minimize an unknown convex risk function, can be formulated in the general framework of stochastic convex optimization.

##### Hilbert Spaces of Entire Functions and Automorphic L-Functions

We review the de Branges theory of Hilbert spaces of entire functions. This theory gives a canonical form for a class of operators as multiplication operator together with a generalized Fourier transform taking such an operator to a generalized differential operator.

##### Noncommutative differential operators, unparametrized paths and Hodge structures

Joint Columbia-Courant-Princeton Algebraic Geometry Seminar at Columbia

##### (Conjectural) triply graded link homology groups of the Hopf link and Hilbert schemes of points on the plane

Gukov et al. suggested triply graded link homology groups via refined BPS counting on the deformed conifold. Through large N duality they identify their Poincaré polynomials as refined topological vertices. I further apply the geometric engineering to interpret them as holomorphic Euler characteristics of natural vector bundles over Hilbert schemes of points on the affine plane.

##### The cone theorem revisited

Joint Columbia-Courant-Princeton Algebraic Geometry Seminar at Columbia

##### The Structure of Shrinking Solitons

We discuss a handful of structure theorems for shrinking solitons with bounded curvature. In particular we prove a priori injectivity radi.

##### Measuring wild ramification using rigid geometry

Joint Columbia-Courant-Princeton Algebraic Geometry Seminar at Columbia

##### Closing the optimality gap using affinity propagation

An important problem in science and engineering is how to find and associate constituent patterns or motifs in large amounts of high-dimensional data. Examples include the identification and modeling of object parts in images, and the detection and association of RNA motifs that regulate tissue-dependent gene splicing in mammals.

##### Null structure and almost optimal local well-posedness of the Maxwell-Dirac system

In this talk I will present recent joint work with P. D'Ancona and D. Foschi on the classical Maxwell-Dirac system, which is the fundamental PDE in quantum electrodynamics. We show that the system has some special structural properties ("null" structure) which improve the regularity of solutions.

##### Generalized eigenfunctions and spectrum for Dirichlet forms

How existence of certain solutions determines the spectrum is a classical issue for Schrödinger operators. We will discuss such results in the context of Dirichlet forms. The framework of Dirichlet forms covers in particular rather general elliptic operators on manifolds as well as (suitable) quantum graphs.

##### Moduli of polarized symplectic manifolds

In many ways irreducible symplectic manifolds behave similar to K3-surfaces, although it is known that the global Torelli theorem fails in general. Nevertheless, it is possible to relate moduli spaces of polarized irreducible symplectic manifolds to quotients of type IV domains by an arithmetic group.

##### The harmonic mean curvature flow of a 2-dimensional hypersurface

The harmonic mean curvature flow is the flow that moves a hypersurface embedded in $R^3$ by the speed given by a ratio of the Gauss and the mean curvature of the given surface in the direction of its normal. It is a fully nonlinear, weakly parabolic equation, degenerate at the points at which our hypersurface changes its convexity and fast diffusion when the mean curvature tends to zero.

##### Prime Splitting Laws

Algebraic number theory seeks to understand and somehow classify all Galois extensions of a number field $K$. One perspective on this basic classification problem emerges from the Cebotarev density theorem, which implies that such extensions $L/K$ are determined by the primes of $K$ that split completely in $L$.