Past

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.

Abstract: Compact $G_2$ manifolds with isolated conical singularities arise naturally in M-theory. I will discuss such manifolds, and explain a method to ``desingularize'' them by glueing in pieces of asymptotically conical $G_2$ manifolds. There are topological obstructions to such desingularizations that depend on the rate of convergence to the cone at the singularities, and on the geometry of the links of the cones. If time permits, I will also briefly discuss a new related project with Dominic Joyce which could provide the first examples of such manifolds, as well as a possible new construction of smooth compact $G_2$ manifolds.

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.

In this talk we will start with various shock reflection-diffraction phenomena, their fundamental scientific issues, and their theoretical roles in the mathematical theory of multidimensional hyperbolic systems of conservation laws. Then we will describe how the global shock reflection-diffraction problems can be formulated as free boundary problems for nonlinear conservation laws of mixed-composite hyperbolic-elliptic type.

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."

It is a wide open question whether the set of rational points on a smooth quartic surface in projective three-space can be nonempty, yet finite. In this talk I will treat the case of diagonal quartics V, which are given by: a x^4 + b y^4 + c z^4 + d w^4 = 0 for some nonzero rational a,b,c,d. I will assume that the product abcd is a square and that V contains at least one rational point P. I will prove that if none of the coordinates of P is zero, and P is not contained in one of the 48 lines on V, then the set of rational points on V is dense. This is based on joint work with Adam Logan and David McKinnon.

Lusztig discover an integral lift of the Frobenius morphism for algebraic groups in positive characteristic to quantum groups at a root of unity. We will describe how this map may be constructed via the Hall algebra realization of a quantum group.

(based on joint work with Helen Haworth, William Shaw, and Ben Hambly)

The simulation of multi-name credit derivatives raises significant challenges, among others from the perspective of dependence modelling, calibration, and computational complexity. Structural models are based on the evolution of firm values, often modelled by market and idiosyncratic factors, to create a rich correlation structure. In addition to this, we will allow for contagious effects, to account for the important scenarios where the default of a number of companies has a time-decaying impact on the credit quality of others. If any further evidence for the importance of this was needed, the recent developments in the credit markets have furnished it. We will give illustrations for small n-th-to-default baskets, and propose extensions to large basket credit derivatives by exploring the limit for an increasing number of firms

It is well known that the three dimensional, incompressible Navier-Stokes equations have a unique, global solution provided the initial data is small enough in a scale invariant space (say L3 for instance). We are interested in finding examples for which no smallness condition is imposed, but nevertheless the associate solution is global and unique. The examples we will present are due to collaborations with Jean-Yves Chemin, and with Marius Paicu.

As a small step towards an understanding of the relationship of the two fields in the title, I will present a uniformness result for embeddings of finitely generated, virtually free groups as cocompact, discrete subgroups in totally disconnected, locally compact groups.

Modern molecular biology research produces data on a massive scale. This

data

is predominantly high-dimensional, consisting of genome-wide measurements of

the transcriptome, proteome and metabalome. Analysis of these data sets

often

face the additional problem of having small sample sizes, as experimental

data

points may be difficult and expensive to come by. Many analysis algorithms

are

based upon estimating the covariance structure from this high-dimensional

small sample size data, with the consequence that the eigenvalues and eigenvectors

of

the estimated covariance matrix are markedly different from the true values.

Techniques from statistical physics and Random Matrix Theory allow us to

understand how these discrepancies in the eigenstructure arise, and in

particular locate the phase transition points where the eigenvalues and

eigenvectors of the estimated covariance matrix begin to genuinely reflect

the

underlying biological signals present in the data. In this talk I will give

a

brief non-specialist introduction to the biological background motivating

the

work and highlight some recent results obtained within the statistical

physics

approach.

Multi-well relaxation problem emerges e.g. in characterising effective properties of composites and in phase transformations. This is a nonlinear problem and one approach uses its reformulation in Fourier space, known in the theory of composites as Hashin-Shtrikman approach, adapted to nonlinear composites by Talbot and Willis. Characterisation of admissible mixtures, subjected to appropriate differential constraints, leads to a quasiconvexification problem. The latter is equivalently reformulated in the Fourier space as minimisation with respect to (extremal points of) H-measures of characteristic functions (Kohn), which in a sense separates the microgeometry of mixing from the differential constraints. For three-phase mixtures in 3D we obtain a full characterisation of certain extremal H-measures. This employs Muller's Haar wavelet expansion estimates in terms of Riesz transform to establish via the tools of harmonic analysis weak lower semicontinuity of certain functionals with rank-2 convex integrands. As a by-product, this allows to fully solve the problem of characterisation of quasiconvex hulls for three arbitrary divergence-free wells. We discuss the applicability of the results to problems with other kinematic constraints, and other generalisations. Joint work with Mariapia Palombaro, Leipzig.

Let u be a vector field on a bounded domain in R^3. The absolute boundary condition states that both the normal part of u and the tangential part of curl(u) vanish on the boundary. After motivating the use of this condition in the context of the Navier Stokes equation, we prove local (in time) existence with this boundary behaviour. This work is together with Dr. Z. Qian and Prof. G. Q. Chen, Northwestern University.

We will analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. These results will be applied to graphs converging locally to trees and derive a new formula for the Stieljes transform of the spectral measure of such graphs. We illustrate our results on the uniform regular graphs, Erdos-Renyi graphs and graphs with prescribed degree distribution. We will sketch examples of application for weighted graphs, bipartite graphs and the uniform spanning tree of n vertices. If time allows, we will discuss related open problems. This is a joint work with Marc Lelarge (INRIA & Ecole Normale Supérieure).

A tree-rooted map is a planar map together with a

distinguished spanning tree. In the sixties, Mullin proved that the

number of tree-rooted maps with $n$ edges is the product $C_n C_{n+1}$

of two consecutive Catalan numbers. We will present a bijection

between tree-rooted maps (of size $n$) and pairs made of two trees (of

size $n$ and $n+1$ respectively) explaining this result.

Then, we will show that our bijection generalizes a correspondence by

Schaeffer between quandrangulations and so-called \emph{well labelled

trees}. We will also explain how this bijection can be used in order

to count bijectively several classes of planar maps

In many dispersed multiphase flows droplets, bubbles and particles move and disappear due to a phase change. Practical examples include fuel droplets evaporating in a hot gas, vapour bubbles condensing in subcooled liquids and ice crystals melting in water. After these `bodies' have disappeared, they leave behind a remnant `ghost' vortex as an expression of momentum conservation.

A general framework is developed to analyse how a ghost vortex is generated. A study of these processes is incomplete without a detailed discussion of the concept of momentum for unbounded flows. We show how momentum can be defined unambiguously for unbounded flows and show its connection with other expressions, particularly that of Lighthill (1986). We apply our analysis to interpret new observations of condensing vapour bubble and discuss droplet evaporation. We show that the use of integral invariants, widely applied in turbulence, introduces a new perspective to dispersed multiphase flows

Abstract: We consider deformations of torsion-free $ G_2 $ structures, defined by the $ G_2 $-invariant 3-form $ \phi $ and compute the expansion of the Hodge star of $ \phi $ to fourth order in the deformations of $ \phi $. By considering M-theory compactified on a $ G_2 $ manifold, the $ G_2 $ moduli space is naturally complexified, and we get a Kahler metric on it. Using the expansion of the Hodge star of $ \phi $ we work out the full curvature of this metric and relate it to the Yukawa coupling.

Fracture mechanics is a significant scientific field of great practical importance. Recently the subject has been invigorated by a number of important accomplishments. From the viewpoint of fundamental science there have been interesting new developments aimed at understanding fracture at the atomic scale; simultaneously, active research programmes have focussed on mathematical modelling, experimentation and computation at macroscopic scales. The workshop aims to examine various different approaches to the modelling, analysis and computation of fracture. The programme will allow time for discussion.

Invited speakers include:

Andrea Braides (Università di Roma II, Italy)

Adriana Garroni (Università di Roma, “La Sapienza”, Italy)

Christopher Larsen (Worcester Polytechnic Institute, USA)

Matteo Negri (Università di Pavia, Italy)

Robert Rudd (Lawrence Livermore National Laboratory, USA)