# Seminars & Events for 2009-2010

##### The circumference of color-critical graphs

A graph is k-critical if every proper subgraph is (k-1)-colorable, but the graph itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least log n/(100 log k). Examples show the bound cannot be improved to exceed 2(k-1)log n/log(k-2). This is joint work with A. Shapira.

##### Liouville-type theorem for nonnegative Bakry-Emery Ricci tensor

##### Monodromy factorizations and symplectic fillings

By a fundamental result of Giroux, contact structures on 3-manifolds may be described via their open books decomposition. A contact manifold can arise as a boundary of a Stein domain if and only if it has a compatible open book whose monodromy is a product of positive Dehn twists. In principle, one has to examine *all* compatible open books to detect Stein fillings.

##### Toward practical rare event simulation in high dimensions

Prof. Weare will discuss an importance sampling method for certain rare event problems involving small noise diffusions. Standard Monte Carlo schemes for these problems behave exponentially poorly in the small noise limit. Previous work in rare event simulation has focused on developing, in specific situations, estimators with optimal exponential variance decay rates.

##### Transcendence

We all know that e and π are transcendental. How about numbers like $e+π$, $e^π$, or $sqrt(2)^[sqrt(2)]$ If $2n$, $3n$, and $5n$ are all integers, must $n$ be an integer as well? What if only $2n$ and $3n$ are integers? In this talk, I will talk about these and related questions. In particular, I hope to prove the well-known fact above: $e$ and $π$ are transcendental.

##### Quasi-adiabatic continuation and the Topology of Many-body Quantum Systems

Topological arguments play a key role in understanding quantum systems. For example, recently it has been shown that K-theory provides a tool for classifying different phases of non-interacting, or single-particle, systems. However, topological arguments have also been applied to interacting systems.

##### Cohomology groups of structure sheaves

##### Deformation rings of group representations

The motivating open question for this talk is to find for a given prime $p$ all local $Z_p$-algebras which can occur as the deformation ring of some linear representation over $F_p$ of some finite group $G$. We will show for every $p$ that not all of them are complete intersections. This is joint work with Ted Chinburg and Frauke Bleher.

##### Dynamics of bouncing balls

We consider a ball bouncing off infinitely heavy periodically moving wall in the presence of a potential force. We are interested in the question how large is the set of orbits whose energy tends to infinity. Both smooth and piecewise smooth motions of wall will be considered. We also present some related questions about small piecewise smooth perturbations of nearly integrable systems.

##### Complex Monge-Ampere equations on symplectic and hermitian manifolds

We will discuss a program to generalize the complex Monge-Ampere equation to symplectic and hermitian manifolds. We will explain to which extent the classical theory on Kahler manifolds extends to these two cases, and give some applications. This is joint work with B. Weinkove and partly with S.-T. Yau.

##### Large almost monochromatic subsets in hypergraphs

We show that for all $t$ and $\epsilon > 0$ there is a constant $c=c(t,\epsilon)>0$ such that every $t$-coloring of the triples of an $N$-element set contains a subset $S$ of size $c(log N)^{1/2}$ such that at least a $1-\epsilon$ fraction of the triples of $S$ have the same color.

##### Sensor Registration and Synchronisation in Networks

An important problem in distributed and networked sensing is registration of coordinate systems or synchronisation of clocks across the network. The main problem discussed in this talk is as follows. We have a network of sensors each with its own local coordinate system.

##### Regularity of the Hardy Littlewood Function

$Lp$ boundedness of the Hardy-Littlewood function is a classic result in harmonic analysis. But not much is understood about the regularity of it. For instance, if your function $f$ is in the Sobolev W1,1 space, is its maximal function in$L1$? The answer to this question is unknown, but I will discuss our partial understanding.

##### Elliptic curves and Hilbert's 10th problem

In this talk I will introduce elliptic curves and discuss recent work with Barry Mazur on ranks of elliptic curves in families of quadratic twists.

##### Selmer ranks of twists of elliptic curves

In joint work with Barry Mazur, we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions for an elliptic curve to have twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank.

##### Front propagation and phase transitions for fractional diffusion equations

Long range or *anomalous* diffusions, such as diffusions given by the fractional powers $(-\Delta)^s$ of the Laplacian, attract lately interest in Physics, Biology, and Finance. From the mathematical point of view, nonlinear analysis for fractional diffusions is being developed actively in the last years.

##### On the soliton dynamics under a slowly varying medium for generalized KdV equations

We consider the problem of the soliton propagation, in a slowly varying medium, for a generalized Korteweg - de Vries equations (gKdV). We study the effects of inhomogeneities on the dynamics of a standard soliton. We prove that slowly varying media induce on the soliton solution large dispersive effects in large time.

##### Symmetry and regularity of solutions for nonlinear systems of Wolff type

##### Hydraulic Fractures: multiscale phenomena, asymptotic and numerical solutions

Hydraulic fractures (HF) are a class of tensile fractures that propagate in brittle materials by the injection of a pressurized viscous fluid. In this talk I provide examples of natural HF and situations in which HF are used in industrial problems. Natural examples of HF include the formation of dykes by the intrusion of pressurized magma from deep chambers.

##### Stationary Phases and Spherical Averages

In this talk we will give an expository account of the following theorem of Stein about spherical averages, which asserts that if f is a function in Lp on Rn, with n≥3 and p>n/(n-1), then for almost every x in Rn, the average of f over a sphere of radius r centered at x is well-defined, and converges to f(x) as r tends to 0.