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.

Malliavin calculus provides a framework to differentiate functionals defined on a Gaussian probability space with respect to the underlying noise. This allows to develop analysis on path space with infinite-dimensional generalisations of Fourier analysis, Sobolev spaces, etc from R^d. In this talk, we attempt to build a Lipschitz à la E. M. Stein (as opposed to Sobolev) function theory on rough path space. This framework allows to pathwise differentiate functionals on rough paths with respect to the underlying rough path. Time permitting, we show how to obtain Feynman-Kac-type representations for solutions to some high-order (>2) linear parabolic equations on R^d.

The stability of structures continues to be scientifically fascinating and technically important.  Shell buckling emerged as one of the most challenging nonlinear problems in mechanics more than fifty years ago when it was intensively studied.  It has returned to life with new challenges motivated not only by structural applications but also by developments in the life sciences and in soft materials.  It is not at all uncommon for slightly imperfect thin cylindrical shells under axial compression or spherical shells under external pressure to buckle at 20% of the buckling load of the perfect shell.  A historical overview of shell buckling will be presented followed by a discussion of recent work by the speaker and his collaborators on the buckling of spherical shells.  Experimental and theoretical work will be described with a focus on imperfection-sensitivity and on viewing the phenomena within the larger context of nonlinear stability.