University of Warwick

In many applications it is of interest to compute minimizers of

a functional I(u) which is the of the form $J(u)=\Phi(u)+R(u)$,

with $R(u)$ quadratic. We describe a stochastic algorithm for

this problem which avoids explicit computation of gradients of $\Phi$;

it requires only the ability to sample from a Gaussian measure

with Cameron-Martin norm squared equal to $R(u)$, and the ability

to evaluate $\Phi$. We show that, in an appropriate parameter limit,

a piecewise linear interpolant of the algorithm converges weakly to a noisy

gradient flow. \\

Joint work with Natesh Pillai (Harvard) and Alex Thiery (Warwick).

I will describe recent work with Charles Fefferman on a

construction of families of analytic almost-sharp fronts for SQG. These

are special solutions of SQG which have a very sharp transition in a

very thin layer. One of the main difficulties of the construction is the

fact that there is no formal limit for the family of equations. I will

show how to overcome this difficulty, linking the result to joint work

with C. Fefferman and Kevin Luli on the existence of a "spine" for

almost-sharp fronts. This is a curve, defined for every time slice by a

measure-theoretic construction, that describes the evolution of the

almost-sharp front.

The symmetries of a dynamical system have a big effect on its typical behaviour. The most obvious effect is pattern formation - the dynamics itself may be symmetric, though often the symmetry of the system is 'broken', and the state has less symmetry than the system. The resulting phenomena are fairly well understood for steady and time-periodic states, and quite a bit can be said for chaotic dynamics. More recently, the concept of 'symmetry' has been generalised to address applications to physical and biological networks. One consequence is a new approach to patterns of synchrony and phase relations. The lecture will describe some of the high points of the emerging theories, including applications to evolution, locomotion, human balance and fluid dynamics.

Human behaviour can show surprising properties when looked at from a collective point of view. Data on collective behaviour can be gleaned from a number of sources, and mobile phone data are increasingly becoming used. A major challenge is combining behavioural data with health data. In this talk I will describe our approach to understanding behaviour change related to change in health status at a collective level.

I will show that one can (at least in theory) guarantee the "validity" of a numerical approximation of a solution of the 3D Navier-Stokes equations using an explicit a posteriori test, despite the fact that the existence of a unique solution is not known for arbitrary initial data.

The argument relies on the fact that if a regular solution exists for some given initial condition, a regular solution also exists for nearby initial data ("robustness of regularity"); I will outline the proof of robustness of regularity for initial data in $H^{1/2}$.

I will also show how this can be used to prove that one can verify numerically (at least in theory) the following statement, for any fixed R > 0: every initial condition $u_0\in H^1$ with $\|u\|_{H^1}\le R$ gives rise to a solution of the unforced equation that remains regular for all $t\ge 0$.

This is based on joint work with Sergei Chernysehnko (Imperial), Peter Constantin (Chicago), Masoumeh Dashti (Warwick), Pedro Marín-Rubio (Seville), Witold Sadowski (Warsaw/Warwick), and Edriss Titi (UC Irivine/Weizmann).

Tonelli gave the first rigorous treatment of one-dimensional variational problems, providing conditions for existence and regularity of minimizers over the space of absolutely continuous functions.  He also proved a partial regularity theorem, asserting that a minimizer is everywhere differentiable, possible with infinite derivative, and that this derivative is continuous as a map into the extended real line.  Some recent work has lowered the smoothness assumptions on the Lagrangian for this result to various Lispschitz and H\"older conditions.  In this talk we will discuss the partial regularity result, construct examples showing that mere continuity of the Lagrangian is an insufficient condition.

It has long been known that many materials are crystalline when in their energy-minimizing states. Two of the most common crystalline structures are the face-centred cubic (fcc) and hexagonal close-packed (hcp) crystal lattices. Here we introduce the problem of crystallization from a mathematical viewpoint and present an outline of a proof that the ground state of a large system of identical particles, interacting under a suitable potential, behaves asymptotically like fcc or hcp, as the number of particles tends to infinity. An interesting feature of this result is that it holds under no initial assumption on the particle positions. The talk is based upon a joint work in progress with Florian Theil.

We report the numerical realization and demonstration of robustness of certain 2-component structures in Bose-Einstein Condensates in 2 and 3 spatial dimensions with non-trivial topological charge in one of the components. In particular, we identify a stable symbiotic state in which a higher-dimensional bright soliton exists even in a homogeneous setting with defocusing interactions, as a result of the effective potential created by a stable vortex in the other component. The resulting vortex-bright solitary waves, which naturally generalize the recently experimentally observed dark-bright solitons, are examined both in the homogeneous medium and in the presence of parabolic and periodic external confinement and are found to be very robust.

We demonstrate that stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H > 1/2 have similar ergodic properties as SDEs driven by standard Brownian motion. The focus in this article is on hypoelliptic systems satisfying H\"ormander's condition. We show that such systems satisfy a suitable version of the strong Feller property and we conclude that they admit a unique stationary solution that is physical in the sense that it does not "look into the future".

The main technical result required for the analysis is a bound on the moments of the inverse of the Malliavin covariance matrix, conditional on the past of the driving noise.