Past

The sequence {nα}, where α is an irrational number and {.} denotes fractional part, plays

a fundamental role in probability theory, analysis and number theory. For suitable α, this sequence provides an example for "most uniform" infinite sequences, i.e. sequences whose discrepancy has the

smallest possible order of magnitude. Such 'low discrepancy' sequences have important applications in Monte Carlo integration and other problems of numerical mathematics. For rapidly increasing n_k the behaviour of {n_kα} is similar to that of independent random variables, but its asymptotic properties depend strongly also on the number theoretic properties of n_k, providing a simple example for pseudorandom behaviour. Finally, for periodic f the sequence f(nα) provides a generalization of the trig-onometric system with many interesting  properties. In this lecture, we give a survey of the field  (going back more than 100 years) and formulate new results.

We consider a non linear Schrödinger equation on a compact manifold of dimension d subject to some multiplicative random perturbation. Using some stochastic Strichartz inequality, we prove the existence and uniqueness of a maximal solution in H^1 under some general conditions on the diffusion coefficient. Under stronger conditions on the noise, the nonlinearity and the diffusion coefficient, we deduce the existence of a global solution when d=2. This is a joint work with Z. Brzezniak.

The is the second part of the Analysis and Geometry Seminar today.

The isoperimetric inequality is a fundamental tool in many geometric and analytical issues, beside being the starting point for a great variety of other important inequalities.

We shall present some recent results dealing with the quantitative version of this inequality, an old question raised by Bonnesen at the beginning of last century. Applications of the sharp quantitative isoperimetric inequality to other classic inequalities and to eigenvalue problems will be also discussed.

The theory and computation of convex measures of financial risk has been a very active area of Financial Mathematics, with a rich history in a short number of years. The axioms specify sensible properties that measures of risk should possess (and which the industry's favourite, value-at-risk, does not). The most common example is related to the expectation of an exponential utility function.

A basic application is hedging, that is taking off-setting positions, to optimally reduce the risk measure of a portfolio. In standard continuous-time models with dynamic hedging, this leads to nonlinear PDE problems of HJB type. We discuss so-called static-dynamic hedging of exotic options under convex risk measures, and specifically the existence and uniqueness of an optimal position. We illustrate the computational challenge when we move away from the risk measure associated with exponential utility.

Joint work with Aytac Ilhan (Goldman Sachs) and Mattias Jonsson (University of Michigan).

In 1974 Haim Gaifman conjectured that if a first-order theory T is relatively categorical over T(P) (the theory of the elements satisfying P), then every model of T(P) expands to one of T.

The conjecture has long been known to be true in some special cases, but nothing general is known. I prove it in the case of abelian groups with distinguished subgroups. This is some way outside the previously known cases, but the proof depends so heavily on the Kaplansky-Mackey proof of Ulm's theorem that the jury is out on its generality.

Given a polynomial function f defined on a variety X,

we consider two questions, which are non-commutative analogues

of the Prime Number Theorem and the Linnik Theorem:

- how often the values of f(x) at integral points in X are almost prime?

- can one effectively solve the congruence equation f(x)=b (mod q)

with f(x) being almost prime?

We discuss a solution to these questions when X is a homogeneous

variety (e.g, a quadratic surface).

Whenever we say the words "fluid flows" or "shape changes" we enter the realm of infinite-dimensional geometric mechanics. Water, for example, flows. In fact, Euler's equations tell us that water flows a particular way. Namely, it flows to get out of its own way as adroitly as possible. The shape of water changes by smooth invertible maps called diffeos (short for diffeomorphisms). The flow responsible for this optimal change of shape follows the path of shortest length, the geodesic, defined by the metric of kinetic energy. Not just the flow of water, but the optimal morphing of any shape into another follows one of these optimal paths.

The lecture will be about the commonalities between fluid dynamics and shape changes and will be discussed in the language most suited to fundamental understanding -- the language of geometric mechanics. A common theme will be the use of momentum maps and geometric control for steering along the optimal paths using emergent singular solutions of the initial value problem for a nonlinear partial differential equation called EPDiff, that governs metamorphosis along the geodesic flow of the diffeos. The main application will be in the registration and comparison of Magnetic Resonance Images for clinical diagnosis and medical procedures.

Starting from a Navier-Stokes-Cahn-Hilliard equation for a two-phase flow problem we discuss efficient numerical approaches based on adaptive finite element methods. Various extensions of the model are discussed: a) we consider the model on implicitly described geometries, which is used to simulate the sliding of droplets over nano-patterned surfaces, b) we consider the effect of soluble surfactants and show its influence on tip splitting of droplets under shear flow, and c) we consider bijels as a new class of soft matter materials, in which colloidal particles are jammed on the fluid-fluid interface and effect the motion of the interface due to an elastic force.

The work is based on joint work with Sebastian Aland (TU Dresden), John Lowengrub (UC Irvine) and Knut Erik Teigen (U Trondheim).

A K3 surface of degree two can be seen as a double cover of the complex projective plane, ramified over a nonsingular sextic curve. In this talk we explore two different methods for constructing explicit projective models of threefolds admitting a fibration by such surfaces, and discuss their relative merits.

Singular monopoles are solutions to the Bogomolny equation with prescribed singularities of Dirac monopole type. Previously such monopoles could be constructed only by the Nahm transform, with some difficulty. We therefore formulate a new construction of all singular monopoles. This construction relies on two ideas: Kronheimer's correspondence between singular monopoles on R^3 and self-dual connections on the multi-Taub-NUT space, and Cherkis' recent construction of self-dual connections on curved spaces using bow diagrams. As an example of our method we use it to obtain the explicit solution for a charge one SU(2) singular monopole with an arbitrary number of singularities.

We shall discuss a simple low order numerical integration scheme for ODEs and DAEs. The scheme has a parameter that allows for regularization of Jacobian of stiff problems and for numerically elucidating multi-scale response, if any, in some problems.

We discuss homological finiteness Bredon types FPm with respect to the class of finite subgroups and seperately with respect to the class of virtually cyclic subgroups. We will concentrate to the case of solubles groups and if the time allows to the case of generalized R. Thompson groups of type F. The results announced are joint work with Brita Nucinkis

(Southampton) and Conchita Martinez Perez (Zaragoza) and will appear in papers in Bulletin of LMS and Israel Journal of Mathematics.

I will show that given smooth embedded Lagrangian L in a Calabi-Yau, one can find a perturbation of L which lies in the same hamiltonian isotopy class and such that the correspondent solution to mean curvature flow develops a finite time singularity. This shows in particular that a simplified version of the Thomas-Yau conjecture does not hold.

We revisit the problem of sorting under partial information: sort a finite set given the outcomes of comparisons between some pairs of elements. The input is a partially ordered set P, and solving the problem amounts to discovering an unknown linear extension of P, using pairwise comparisons. The information-theoretic lower bound on the number of comparisons needed in the worst case is log e(P), the binary logarithm of the number of linear extensions of P. In a breakthrough paper, Jeff Kahn and Jeong Han Kim (STOC 1992) showed that there exists a polynomial-time sorting algorithm achieving this bound up to a constant factor. They established a crucial link between the entropy of the input partial order and the information-theoretic lower bound. However, their algorithm invokes the ellipsoid algorithm at each iteration for determining the next comparison, making it unpractical. We develop efficient algorithms for sorting under partial information, derived from approximation and exact algorithms for computing the partial order entropy.

This is joint work with S. Fiorini, G. Joret, R. Jungers, and I. Munro.