Past

I plan to discuss some aspects the mysterious relationship between the symmetries of toroidal compactifications of M-theory and helices on del Pezzo surfaces.

We demonstrate a scheme for controlling a large quantum system by acting

on a small subsystem only. The local control is mediated to the larger

system by some fixed coupling Hamiltonian. The scheme allows to transfer

arbitrary and unknown quantum states from a memory to the large system

("upload access") as well as the inverse ("download access").

We give sufficient conditions of the coupling Hamiltonian for the

controllability

of the system which can be checked efficiently by a colour-infection game on

the graph

that describes the couplings.

The B-D equations describe a mean field approximation for a many body system in relaxation to equilibrium. The two B-D equations determine the time evolution of the density c(L,t) of particles with mass L, L=1,2,... One of the equations is a discretized linear diffusion equation for c(L,t), and the other is a non-local constraint equivalent to mass conservation. Existence and uniqueness for the B-D system was established in the 1980's by Ball, Carr and Penrose. Research in the past decade has concentrated on understanding the large time behavior of solutions to the B-D system. This behavior is characterized by the phenomenon of "coarsening", whereby excess density is concentrated in large particles with mass increasing at a definite rate. An important conjecture in the field is that the coarsening rate can be obtained from a particular self- similar solution of the simpler LSW system. In this talk we shall discuss the B-D and LSW equations, and some recent progress by the speaker and others towards the resolution of this conjecture.

Probabilistic methods for the solution of Backward Stochastic Differential Equations (BSDE) provide us with a new approach to the problem of approximating the solution of a semi-linear PDE. Utilizing on the Markovian nature of these BSDE’s we show how one may consider the problem of numerical solutions to BSDEs within the area of weak approximations of diffusions. To emphasize this point, we suggest an algorithm based on the Cubature method on Wiener space of Lyons - Victoir. Instead of using standard discretization techniques of BSDE’s, we choose to work with the actual flow. This allows to take advantage of estimates on the derivatives of the solution of the associated semi-linear PDE and hence, we recover satisfactory convergence estimates.

I will give a few model theoretic properties for fields with a Hasse derivation which are existentially closed. I will explain how some type-definable sets allow us to understand properties of some algebraic varieties, mainly concerning their field of definition.

Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n

has a (generally branched) solution with leading order behaviour

proportional to

(z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic at z_0 and a_N(z_0)\ne 0. Jointly with R.G. Halburd we consider the subclass of equations for which each possible leading order term of

this

form corresponds to a one-parameter family of solutions represented near

z_0

by a Laurent series in fractional powers of z-z_0. For this class of

equations we show that the only movable singularities that can be reached

by

analytic continuation along finite-length curves are of the algebraic type

just described. This work generalizes previous results of S. Shimomura.

The only other possible kind of movable singularity that might occur is an

accumulation point of algebraic singularities that can be reached by

analytic continuation along infinitely long paths ending at a finite point

in the complex plane. This behaviour cannot occur for constant coefficient

equations in the class considered. However, an example of R. A. Smith

shows

that such singularities do occur in solutions of a simple autonomous

second-order differential equation outside the class we consider here.

This talk shall examine a range of problems where nonlinear waves or coherent structures are localised to some portion of a domain. In one spatial dimension, the problem reduces to finding homoclinic connections to equilibria. Two canonical problems emerge when higher-order spatial terms are considered (either via fourth-order operators or discreteness effects). One involves so-called snaking bifurcation diagrams where a fundamental state grows an internal patterned layer via an infinite sequence of fold bifurcations. The other involves the exact vanishing of oscillatory tails as a parameter is varied. It is shown how both problems arise from certain codimension-two limits where they can be captured by beyond-all-orders analysis. Dynamical systems methods can then be used to explain the kind of structures that emerge away from these degenerate points. Applications include moving discrete breathers in atomic lattices, discrete solitons in optical cavities, and theories for two-dimensional localised patterns using Swift-Hohenberg theory.

For a positively graded algebra A we construct a functor from the derived

category of graded A-modules to the derived category of graded modules over

the quadratic dual A^! of A. This functor is an equivalence of certain

bounded subcategories if and only if the algebra A is Koszul. In the latter

case the functor gives the classical Koszul duality. The approach I will

talk about uses the category of linear complexes of projective A-modules.

Its advantage is that the Koszul duality functor is given in a nice and

explicit way for computational applications. The applications I am going to

discuss are Koszul dualities between certain functors on the regular block

of the category O, which lead to connections between different

categorifications of certain knot invariants. (Joint work with S.Ovsienko

and C.Stroppel.)

"Eigenvalue avoidance" or "level repulsion" refers to the tendency of eigenvalues of matrices or operators to be distinct rather than degenerate.

The mathematics goes back to von Neumann and Wigner in 1929 and touches many subjects including numerical linear algebra, random matrix theory, chaotic dynamics, and number theory.

This talk will be an informal illustrated discussion of various aspects of this phenomenon.

The phase transition is a phenomenon that appears in natural sciences in various contexts. In the random graph theory, the phase transition refers to a dramatic change in the number of vertices in the largest components by addition of a few edges around a critical value, which was first discussed on the standard random graphs in the seminal paper by Erdos and Renyi. Since then, the phase transition has been a central theme of the random graph theory. In this talk we discuss the phase transition in random graphs with a given degree sequence and random graph processes with degree constraints.

Carlos and Benson will give an update on their research.

Current numerical relativity codes model neutron star matter as a perfect fluid, with an unphysical "atmosphere" surrounding the star to avoid the breakdown of the equations at the fluid-vacuum interface at the surface of the star. To design numerical methods that do not require an unphysical atmosphere, it is useful to know what a generic sound wave looks near the surface. After a review of relevant mathematical methods, I will present results for low (finite) amplitude waves that remain smooth and, perhaps, for high amplitude waves that form a shock.

we present some sharp regularity results for the stationary and the evolution Navier-Stokes equations with shear dependent viscosity, under the no-slip boundary condition. This is a classical turbulence model, considered by von Neumann and Richtmeyer in the 50's, and by Smagorinski in the beginning of the 60's (for p= 3). The model was extended to other physical situations, and deeply studied from a mathematical point of view, by Ladyzhenskaya in the second half of the 60's. We consider the shear thickening case p>2. We are interested in regularity results in Sobolev spaces, up to the boundary, in dimension n=3, for the second order derivatives of the velocity and the first order derivatives of the pressure. In spite of the very rich literature on the subject, sharp regularity results up to the boundary are quite new.

An important and challenging problem in mathematical finance is how to choose a pricing measure in an incomplete market, i.e. how to find a probability measure under which expected payoffs are calculated and fair option prices are derived under some notion of optimality.

The notion of q-optimality is linked to the unique equivalent martingale measure (EMM) with minimal q-moment (if q > 1) or minimal relative entropy (if q=1). Hobson's (2004) approach to identifying the q-optimal measure (through a so-called fundamental equation) suggests a relaxation of an essential condition appearing in Delbaen & Schachermayer (1996). This condition states that for the case q=2, the Radon-Nikodym process, whose last element is the density of the candidate measure, is a uniformly integrable martingale with respect to any EMM with a bounded second moment. Hobson (2004) alleges that it suffices to show that the above is true only with respect to the candidate measure itself and extrapolates for the case q>1. Cerny & Kallsen (2008) however presented a counterexample (for q=2) which demonstrates that the above relaxation does not hold in general.

The speaker will present the general form of the q-optimal measure following the approach of Delbaen & Schachermayer (1994) and prove its existence under mild conditions. Moreover, in the light of the counterexample in Cerny & Kallsen (2008) concerning Hobson's (2004) approach, necessary and sufficient conditions will be presented in order to determine when a candidate measure is the q-optimal measure.

We present the ideas of Malliavin calculus in the context of rough differential equations (RDEs) driven by Gaussian signals. We then prove an analogue of Hörmander's theorem for this set-up, finishing with the conclusion that, for positive times, a solution to an RDE driven by Gaussian noise will have a density with respect to Lebesgue measure under Hörmander's conditions on the vector fields.

Efficient numerical solutions of several important partial-differential equation based models in mathematical finance are impeded by the fact that they contain operators which are Lipschitz continuous but not continuously differentiable. As a consequence, Newton methods are not directly applicable and, more importantly, do not provide their typical fast convergence properties.

In this talk semi-smooth Newton methods are presented as a remedy to the the above-mentioned difficulties. We also discuss algorithmic issues including the primal-dual active set strategy and path following techniques.

Finally we will discuss some recent developments in attacking the shock reflection-diffraction problems, including the existence, stability, and regularity of global regular configurations of shock reflection-diffraction by wedges. The approach includes techniques to handle free boundary problems, degenerate elliptic equations, and corner singularities, which is highly motivated by experimental, computational, and asymptotic results. Further trends and open problems in this direction will be also addressed. This talk will be mainly based on joint work with M. Feldman.

Cover a plane with curves, one curve through each point

in each direction. How can you tell whether these curves are

the geodesics of some metric?

This problem gives rise to a certain closed system of partial

differential equations and hence to obstructions to finding such a

metric. It has been an open problem for at least 80 years. Surprisingly

it is harder in two dimensions than in higher dimensions. I shall present

a solution obtained jointly with Robert Bryant and Mike Eastwood.

"Time-periodic shocks in systems of viscous conservation laws are shown to be nonlinearly stable. The result is obtained by representing the evolution associated to the linearized, time-periodic operator using a contour integral, similar to that of strongly continuous semigroups. This yields detailed pointwise estimates on the Green's function for the time-periodic operator. The evolution associated to the embedded zero eigenvalues is then extracted.

Stability follows from a Gronwall-type estimate, proving algebraic decay of perturbations."