Past

In this talk I shall discuss about the classical compressible Euler equations in three

space dimensions for a perfect fluid with an arbitrary equation of state.

We considered initial data which outside a sphere coincide with the data corresponding

to a constant state, we established theorems which gave a complete description of the

maximal development. In particular, we showed that the boundary of the domain of the

maximal development has a singular part where the inverse density of the wave fronts

vanishes, signaling shock formation.

In this talk we present a general method using permanent estimates in order to obtain results about counting and packing Hamilton cycles in dense graphs and oriented graphs. As a warm up we prove that every Dirac graph $G$ contains at least $(reg(G)/e)^n$ many distinct Hamilton cycles, where $reg(G)$ is the maximal degree of a spanning regular subgraph of $G$. We continue with strengthening a result of Cuckler by proving that the number of oriented Hamilton cycles in an almost $cn$-regular oriented graph is $(cn/e)^n(1+o(1))^n$, provided that $c$ is greater than $3/8$. Last, we prove that every graph $G$ of minimum degree at least $n/2+\epsilon n$ contains at least $reg_{even}(G)-\epsilon n$ edge-disjoint Hamilton cycles, where $reg_{even}(G)$ is the maximal even degree of a spanning regular subgraph of $G$. This proves an approximate version of a conjecture made by Osthus and K\"uhn.  Joint work with Michael Krivelevich and Benny Sudakov.

I will present the basics of mathematical finance, and what probabilists do there. More specifically, I will present the basic concepts of replication of a derivative contract by trading, market completeness, arbitrage, and the link with Backward Stochastic Differential Equations (BSDEs).

Crystalline solids are descibed by a material manifold endowed

with a certain structure which we call crystalline. This is characterized by

a canonical 1-form, the integral of which on a closed curve in the material manifold

represents, in the continuum limit, the sum of the Burgers vectors of all the dislocation lines

enclosed by the curve. In the case that the dislocation distribution is uniform, the material manifold

becomes a Lie group upon the choice of an identity element. In this talk crystalline solids

with uniform distributions of the two elementary kinds of dislocations, edge and screw dislocations,

shall be considered. These correspond to the two simplest non-Abelian Lie groups, the affine group

and the Heisenberg group respectively. The statics of a crystalline solid are described in terms of a

mapping from the material domain into Euclidean space. The equilibrium configurations correspond

to mappings which minimize a certain energy integral. The static problem is solved in the case of

a small density of dislocations.

It's well known that the mapping space of two finite dimensional

manifolds can be given the structure of an infinite dimensional manifold

modelled on Frechet spaces (provided the source is compact). However, it is

not that the charts on the original manifolds give the charts on the mapping

space: it is a little bit more complicated than that. These complications

become important when one extends this construction, either to spaces more

general than manifolds or to properties other than being locally linear.

In this talk, I shall show how to describe the type of property needed to

transport local properties of a space to local properties of its mapping

space. As an application, we shall show that applying the mapping

construction to a regular map is again regular.

We study the asymptotic behavior of deterministic dynamics of many interacting particles with random initial data in the limit where the number of particles tends to infinity. A famous example is hard sphere flow, we restrict our attention to the simpler case where particles are removed after the first collision. A fixed number of particles is drawn randomly according to an initial density $f_0(u,v)$ depending on $d$-dimensional position $u$ and velocity $v$. In the Boltzmann Grad scaling, we derive the validity of a Boltzmann equation without gain term for arbitrary long times, when we assume finiteness of moments up to order two and initial data that are $L^\infty$ in space. We characterize the many particle flow by collision trees which encode possible collisions. The convergence of the many-particle dynamics to the Boltzmann dynamics is achieved via the convergence of associated probability measures on collision trees. These probability measures satisfy nonlinear Kolmogorov equations, which are shown to be well-posed by semigroup methods.

 Many geophysical flows over topography can be modeled by two-dimensional
depth-averaged fluid dynamics equations.  The shallow water equations
are the simplest example of this type, and are often sufficiently
accurate for simulating tsunamis and other large-scale flows such
as storm surge.  These hyperbolic partial differential equations
can be modeled using high-resolution finite volume methods.  However,
several features of these flows lead to new algorithmic challenges,
e.g. the need for well-balanced methods to capture small perturbations
to the ocean at rest, the desire to model inundation and flooding,
and that vastly differing spatial scales that must often be modeled,
making adaptive mesh refinement essential. I will discuss some of
the algorithms implemented in the open source software GeoClaw that
is aimed at solving real-world geophysical flow problems over
topography.  I'll also show results of some recent studies of the
11 March 2011 Tohoku Tsunami and discuss the use of tsunami modeling
in probabilistic hazard assessment.

Robust pricing of an exotic derivative with payoff $\Phi$ can be viewed as the task of estimating its expectation $E_Q \Phi$ with respect to a martingale measure $Q$ satisfying marginal constraints. It has proven fruitful to relate this to the theory of Monge-Kantorovich optimal transport. For instance, the duality theorem from optimal transport leads to new super-replication results. Optimality criteria from the theory of mass transport can be translated to the martingale setup and allow to characterize minimizing/maximizing models in the robust pricing problem. Moreover, the dual viewpoint provides new insights to the classical inequalities of Doob and Burkholder-Davis-Gundy.

• Joseph Parker - Numerical algorithms for the gyrokinetic equations and applications to magnetic confinement fusion
• Rita Schlackow - Global and functional analyses of 3' untranslated regions in fission yeast
• Peter Stewart - Creasing and folding of fibre-reinforced materials

PLEASE NOTE EARLY START TIME TO AVOID CLASH WITH OCCAM GROUP MEETING

The human influenza A virus causes three to five million cases of severe illness and about 250 000 to 500 000 deaths each year. The 1918 Spanish Flu may have killed more than 40 million people. Yet, the underlying cause of the seasonality of the human influenza virus, its preferential transmission in winter in temperate climates, remains controversial. One of the major forms of the human influenza virus is a sphere made up of lipids selectively derived from the host cell along with specialized viral proteins. I have employed molecular dynamics simulations to study the biophysical properties of a single transmissible unit--an approximately spherical influenza A virion in water (i.e., to mimic the water droplets present in normal transmission of the virus). The surface area per lipid can't be calculated as a ratio of the surface area of the sphere to the number of lipids present as there are many different species of lipid for which different surface area values should be calculated. The 'mosaic' of lipid surface areas may be regarded quantitatively as a Voronoi diagram, but construction of a true spherical Voronoi tessellation is more challenging than the well-established methods for planar Voronoi diagrams. I describe my attempt to implement an approach to the spherical Voronoi problem (based on: Hyeon-Suk Na, Chung-Nim Lee, Otfried Cheong. Computational Geometry 23 (2002) 183–194) and the challenges that remain in the implementation of this algorithm.

Newelski suggested that topological dynamics could be used to extend "stable group theory" results outside the stable context. Given a group G, it acts on the left on its type space S_G(M), i.e. (G,S_G(M)) is a G-flow. If every type is definable, S_G(M) can be equipped with a semigroup structure *, and it is isomorphic to the enveloping Ellis semigroup of the flow. The topological dynamics of (G,S_G(M)) is coded in the Ellis semigroup and in its minimal G-invariant subflows, which coincide with the left ideals I of S_G(M). Such ideals contain at least an idempotent r, and r*I forms a group, called "ideal group". Newelski proved that in stable theories and in o-minimal theories r*I is abstractly isomorphic to G/G^{00} as a group. He then asked if this happens for any NIP theory. Pillay recently extended the result to fsg groups; we found instead a counterexample to Newelski`s conjecture in SL(2,R), for which G/G^{00} is trivial but we show r*I has two elements. This is joint work with Jakub Gismatullin and Anand Pillay.

In this talk I shall describe two rather different, but not entirely unrelated,

problems involving thin-film flow of a viscous fluid which I have found of interest

and which may have some application to a number of practical situations,

including condensation in heat exchangers and microfluidics.

The first problem,

which is joint work with Adam Leslie and Brian Duffy at the University of Strathclyde,

concerns the steady three-dimensional flow of a thin, slowly varying ring of fluid

on either the outside or the inside of a uniformly rotating large horizontal cylinder.

Specifically, we study ``full-ring'' solutions, corresponding to a ring of continuous,

finite and non-zero thickness that extends all the way around the cylinder.

These full-ring solutions may be thought of as a three-dimensional generalisation of

the ``full-film'' solutions described by Moffatt (1977) for the corresponding two-dimensional problem.

We describe the behaviour of both the critical and non-critical full-ring solutions.

In particular,

we show that, while for most values of the rotation speed and the load the azimuthal velocity is

in the same direction as the rotation of the cylinder, there is a region of parameter space close

to the critical solution for sufficiently small rotation speed in which backflow occurs in a

small region on the upward-moving side of the cylinder.

The second problem,

which is joint work with Phil Trinh and Howard Stone at Princeton University,

concerns a rigid plate moving steadily on the free surface of a thin film of fluid.

Specifically, we study two problems

involving a rigid flat (but not, in general, horizontal) plate:

the pinned problem, in which the upstream end of plate is pinned at a fixed position,

the fluid pressure at the upstream end of the plate takes a prescribed value and there is a free surface downstream of the plate, and

the free problem, in which the plate is freely floating and there are free surfaces both upstream and downstream of the plate.

For both problems, the motion of the fluid and the position of the plate

(and, in particular, its angle of tilt to the horizontal) depend in a non-trivial manner on the

competing effects of the relative motion of the plate and the substrate,

the surface tension of the free surface, and of the viscosity of the fluid,

together with the value of the prescribed pressure in the pinned case.

Specifically, for the pinned problem we show that,

depending on the value of an appropriately defined capillary number and on the value of the

prescribed fluid pressure, there can be either none, one, two or three equilibrium solutions

with non-zero tilt angle.

Furthermore, for the free problem we show that the solutions

with a horizontal plate (i.e.\ zero tilt angle) conjectured by Moriarty and Terrill (1996)

do not, in general, exist, and in fact there is a unique equilibrium solution with,

in general, a non-zero tilt angle for all values of the capillary number.

Finally, if time permits some preliminary results for an elastic plate will be presented.

Part of this work was undertaken while I was a

Visiting Fellow in the Department of Mechanical and Aerospace Engineering

in the School of Engineering and Applied Science at Princeton University, Princeton, USA.

Another part of this work was undertaken while I was a

Visiting Fellow in the Oxford Centre for Collaborative Applied Mathematics (OCCAM),

University of Oxford, United Kingdom.

This publication was based on work supported in part by Award No KUK-C1-013-04,

made by King Abdullah University of Science and Technology (KAUST).

We will discuss the Littlewood conjecture from Diophantine approximation, and recent variants of the conjecture in which one of the real components is replaced by a p-adic absolute value (or more generally a "pseudo-absolute value''). The Littlewood conjecture has a dynamical formulation in terms of orbits of the action of the diagonal subgroup on SL_3(R)/SL_3(Z). It turns out that the mixed version of the conjecture has a similar formulation in terms of homogeneous dynamics, as well as meaningful connections to several other dynamical systems. This allows us to apply tools from combinatorics and ergodic theory, as well as estimates for linear forms in logarithms, to obtain new results.

Fix a connected manifold-with-boundary M and a closed, connected submanifold P of its boundary. The set of all possible submanifolds of M whose components are pairwise unlinked and each isotopic to P can be given a natural topology, and splits into a disjoint union depending on the number of components of the submanifold. When P is a point this is just the usual (unordered) configuration space on M. It is a classical result, going back to Segal and McDuff, that for these spaces their homology in any fixed degree is eventually independent of the number of points of the configuration (as the number of points goes to infinity). I will talk about some very recent work on extending this result to higher-dimensional submanifolds: in the above setup, as long as P is of sufficiently large codimension in M, the homology in any fixed degree is eventually independent of the number of components. In particular I will try to give an idea of how the codimension restriction arises, and how it can be improved in some special cases.

ALE formulations are useful when approximating solutions of problems in deformable domains, such as fluid-structure interactions. For realistic simulations involving fluids in 3d, it is important that the ALE method is at least of second order of accuracy. Second order ALE methods in time, without any constraint on the time step, do not exist in the literature and the role of the so-called geometric conservation law (GCL) for stability and accuracy is not clear. We propose discontinuous Galerkin (dG) methods of any order in time for a time dependent advection-diffusion model problem in moving domains. We prove that our proposed schemes are unconditionally stable and that the conservative and non conservative formulations are equivalent. The same results remain true when appropriate quadrature is used for the approximation of the involved integrals in time. The analysis hinges on the validity of a discrete Reynolds' identity and generalises the GCL to higher order methods. We also prove that the computationally less intensive Runge-Kutta-Radau (RKR) methods of any order are stable, subject to a mild ALE constraint. A priori and a posteriori error analysis is provided. The final estimates are of optimal order of accuracy. Numerical experiments confirm and complement our theoretical results.

This is joint work with Andrea Bonito and Ricardo H. Nochetto.