Past

The talk will survey old and recent applications of variational techniques in studying the existence, stability and bifurcations of time harmonic, localized in space solutions of the nonlinear Schroedinger equation (NLS). Such solutions are called solitons, when the equation is space invariant, and bound-states, when it is not. Due to the Hamiltonian structure of NLS, solitons/bound-states can be characterized as critical points of the energy functional restricted to sets of functions with fixed $L^2$ norm.

In general, the energy functional is not convex, nor is the set of functions with fixed $L^2$ norm closed under weak convergence. Hence the standard variational arguments fail to imply existence of global minimizers. In addition for ``critical" and ``supercritical" nonlinearities the restricted energy functional is not bounded from below. I will first review the techniques used to overcome these drawbacks.

Then I will discuss recent results in which the characterizations of bound-states as critical points (not necessarily global minima) of the restricted energy functional is used to show their orbital stability/instability with respect to the nonlinear dynamics and symmetry breaking phenomena as the $L^2$ norm of the bound-state is varied.

I will discuss the following

Conjecture B: Finitely generated abstract images of profinite groups are finite.

I will explain how it relates to the width of words and conjugacy classes in finite groups. I will indicate a proof in the special case of 'non-universal' profinite groups and propose several directions for future work.

This conjecture arose in my discussions with various participants of a workshop in Blaubeuren in May 2007 for which I am grateful. (You know who you are!)

In the first part of the talk we will introduce the notion of property testing and briefly discuss some results in testing graph properties in the framework of property testing.

Then, we will discuss a recent result about testing expansion in bounded degree graphs. We focus on the notion of vertex-expansion:   \newline an $a$-expander is a graph $G = (V,E)$ in which every subset $U$ of $V$ of at most $|V|/2$ vertices has a neighborhood of size at least $a|U|$. Our main result is that one can distinguish good expanders from graphs that are far from being weak expanders in time approximately $O(n^{1/2})$.

We design a property testing algorithm that accepts every $a$-expander with probability at least 2/3 and rejects every graph that is $\epsilon$-far from an $a^*$-expander with probability at least 2/3, where $a^* = O(a^2/(d^2 log(n/\epsilon)))$, $d$ is the maximum degree of the graphs, and a graph is called $\epsilon$-far from an $a^*$-expander if one has to modify (add or delete) at least $\epsilon d n$ of its edges to obtain an $a^*$-expander. The algorithm assumes the bounded-degree graphs model with adjacency list graph representation and its running time is $O(d^2 n^{1/2} log(n/\epsilon)/(a^2 \epsilon^3))$.

This is a joint work with Christian Sohler.

In the Gulbenkian Lecture Theatre, St Cross Building, Manor Road.

Tea will be available in the Arumugam Building, St. Catherine's College, from 4.15pm.

We consider transient random walks in random environment on Z with zero asymptotic speed. In a seminal paper, Kesten, Kozlov and Spitzer proved that the hitting time of the level "n" converges in law, after a proper normalization, towards a positive stable law, but the question of the description of its parameter was left open since that time. A new approach to this problem, based on a precise description of Sinai's potential, leads to a complete characterization of this stable law, making a tight link with Kesten's renewal series. The case of Dirichlet environment turns out to be remarkably explicit. Quenched results on this model will be presented if time permits.

We present numerical schemes for nonlinear stochastic differential equations whose solution evolves on a smooth finite dimensional manifold. Given a Lie group action that generates transport along the manifold, we pull back the stochastic flow on the manifold to the Lie group via the action and subsequently to the corresponding Lie algebra.

We construct an approximation to the stochastic flow in the Lie algebra via closed operations and then push back to the manifold, thus ensuring our approximation lies in the manifold. We call such schemes stochastic Munthe-Kaas methods after their deterministic counterparts. We also present stochastic Lie group integration schemes based on Castell--Gaines methods. They become stochastic Lie group integrator schemes if we use Munthe-Kaas methods as the underlying ordinary differential integrator. Lastly, we demonstrate our methods by presenting some numerical examples

We consider a class of energy functionals containing a small parameter ε and a long-range interaction. Such functionals arise from models for phase separation in diblock copolymers and from stationary solutions of FitzHugh–Nagumo type systems.

On an interval of arbitrary length, we show that every global minimizer is periodic, and provide asymptotic expansions for the periods.

In 2D, periodic hexagonal structures are observed in experiments in certain di-block

copolymer melts. Using the modular function and an heuristic reduction of a mathematical model, we present a mathematical account of a hexagonal pattern selection observed in di-block copolymer melts.

We also consider the sharp interface problem arising in the singular limit,

and prove the existence and the nondegeneracy of solutions whose interface is a distorted circle in a two-dimensional bounded domain without any assumption on the symmetry of the domain.

Abstract: I will discuss some recent developments in understanding compactifications of the Heterotic string on Calabi-Yau manifolds. These compactifications are well-described by linear sigma models with (0,2) supersymmetry. I will show how to use these models to compute physical observables, such as genus zero Yukawa couplings, their singularity structure, and dependence on bundle moduli.

In this talk we present a work done with M. Di Giacinto (Università di Cassino - Italy) and Salvatore Federico (Scuola Normale - Pisa - Italy). The subject of the work is a continuous time stochastic model of optimal allocation for a defined contribution pension fund with a minimum guarantee. We adopt the point of view of a fund manager maximizing the expected utility from the fund wealth over an infinite horizon.

The level of wealth is constrained to stay above a "solvency level".

The model is naturally formulated as an optimal control problem of a stochastic delay equation with state constraints and is treated by the dynamic programming approach.

We first present the study in the simplified case of no delay where a satisfactory theory can be built proving the existence of regular feedback control strategies and then go to the more general case showing some first results on the value function and on its properties.

I will prove that certain pairs of ordered structures are dependent. There are basically two cases depending on whether the smaller structure is dense or discrete. I will discuss the proofs of two quite general theorems which construe the dividing line between these cases. Among examples are dense pairs of o-minimal structures in the first case, and tame pairs of o-minimal structures in the latter. This is joint work with P. Hieronymi.

Partial differential equations with a nonlinear pointwise constraint defined through a manifold occur in a variety of applications: The magnetization of a ferromagnet can be described by a unit length vector field and the orientation of the rod-like molecules that constitute a liquid crystal is often modeled by a vector field that attains its values in the real projective plane thus respecting the head-to-tail symmetry of the molecules. Other applications arise in geometric

modeling, quantum mechanics, and general relativity. Simple examples reveal that it is impossible to satisfy pointwise constraints exactly by lowest order finite elements. For two model problems we discuss the practical realization of the constraint, the efficient solution of the resulting nonlinear systems of equations, and weak accumulation of approximations at exact solutions.