In this talk, we consider a split connected semisimple group G defined over a global field F. Let A denote the ring of adèles of F and K a maximal compact subgroup of G(A) with the property that the local factors of K are hyperspecial at every non-archimedian place. Our interest is to study a certain subspace of the space of square-integrable functions on the adelic quotient G(F)\G(A). Namely, we want to study functions coming from induced representations from an unramified character of a Borel subgroup and which are K-invariant.

Our goal is to describe how the decomposition of such space can be related with the Plancherel decomposition of a graded affine Hecke algebra (GAHA).

The main ingredients are standard analytic properties of the Dedekind zeta-function as well as known properties of the so-called residue distributions, introduced by Heckman-Opdam in their study of the Plancherel decomposition of a GAHA and a result by M. Reeder on the support of the weight spaces of
the anti-spherical  discrete series representations of affine Hecke algebras. These last ingredients are of a purely local nature.

This talk is based on joint work with V. Heiermann and E. Opdam.

Commitment schemes are a fundamental primitive in cryptography. Their security (more precisely the computational binding property) is closely tied to the notion of collision-resistance of hash functions. Classical definitions of binding and collision-resistance turn out too be weaker than expected when used in the quantum setting. We present strengthened notions (collapse-binding commitments and collapsing hash functions), explain why they are "better", and show how they be realized under standard assumptions.

Ousman Kodio

Lubricated wrinkles: imposed constraints affect the dynamics of wrinkle coarsening

We investigate the problem of an elastic beam above a thin viscous layer. The beam is subjected to
a fixed end-to-end displacement, which will ultimately cause it to adopt the Euler-buckled
state. However, additional liquid must be drawn in to allow this buckling. In the interim, the beam
forms a wrinkled state with wrinkles coarsening over time. This problem has been studied
experimentally by Vandeparre \textit{et al.~Soft Matter} (2010), who provides a scaling argument
suggesting that the wavelength, $\lambda$, of the wrinkles grows according to $\lambda\sim t^{1/6}$.
However, a more detailed theoretical analysis shows that, in fact, $\lambda\sim(t/\log t)^{1/6}$.
We present numerical results to confirm this and show that this result provides a better account of
previous experiments.

Edward Rolls

Multiscale modelling of polymer dynamics: applications to DNA

We are interested in generalising existing polymer dynamics models which are applicable to DNA into multiscale models. We do this by simulating localized regions of a polymer chain with high spatial and temporal resolution, while using a coarser modelling approach to describe the rest of the polymer chain in order to increase computational speeds. The simulation maintains key macroscale properties for the entire polymer. We study the Rouse model, which describes a polymer chain of beads connected by springs by developing a numerical scheme which considers the a filament with varying spring constants as well as different timesteps to advance the positions of different beads, in order to extend the Rouse model to a multiscale model. This is applied directly to a binding model of a protein to a DNA filament. We will also discuss other polymer models and how it might be possible to introduce multiscale modelling to them.

In this talk I will present a recent result on the characterisation of the equilibrium measure for a nonlocal and non-radial energy arising as the Gamma-limit of discrete interacting dislocations.

It is well-known that the stochastic heat equation on R^n has a Hölder continuous function-valued solution in the case n=1, and that in dimensions 2 and above the solution is not function-valued but is forced to take values in some wider space of distributions. So what happens if the space has, in some sense, a dimension in between 1 and 2? We turn to the theory of fractals in order to answer this question. It has been shown (Kigami, 2001) that there exists a class of self-similar sets on which natural Laplacians can be defined, and so an analogue to the stochastic heat equation can be posed. In this talk we cover the following questions: Is the solution to this equation function-valued? If so, is it Hölder continuous? To answer the latter we must first prove an analogue of Kolmogorov's celebrated continuity theorem for the self-similar sets that we are working on. Joint work with Ben Hambly.

We present global rates of convergence for a general class of methods for nonconvex smooth optimization that include linesearch, trust-region and regularisation strategies, but that allow inaccurate problem information. Namely, we assume the local (first- or second-order) models of our function are only sufficiently accurate with a certain probability, and they can be arbitrarily poor otherwise. This framework subsumes certain stochastic gradient analyses and derivative-free techniques based on random sampling of function values. It can also be viewed as a robustness
assessment of deterministic methods and their resilience to inaccurate derivative computation such as due to processor failure in a distribute framework. We show that in terms of the order of the accuracy, the evaluation complexity of such methods is the same as their counterparts that use deterministic accurate models; the use of probabilistic models only increases the complexity by a constant, which depends on the probability of the models being good. Time permitting, we also discuss the case of inaccurate, probabilistic function value information, that arises in stochastic optimization. This work is joint with Katya Scheinberg (Lehigh University, USA).
 

We give an explicit scheme to reconstruct any C^1 curve from its signature. It is implementable and comes with detailed stability properties. The key of the inversion scheme is the use of a symmetrisation procedure that separates the behaviour of the path at small and large scales. Joint work with Terry Lyons.