University of Warwick

A familiar technique in PDE theory is to use mollification to adjust a function controlled in some weak norm into a smooth function with corresponding control on its $C^k$ norm. It would be extremely useful to be able to do the same sort of regularisation for Riemannian metrics, and one might hope to use Ricci flow to do this. However, attempting to do so throws up some fundamental problems concerning the well-posedness of Ricci flow. I will explain some recent developments that allow us to use Ricci flow in this way in certain important cases. In particular, the Ricci flow will now allow us to adjust a `noncollapsed’ 3-manifold with a lower bound on its Ricci curvature through a family of such manifolds, without disturbing the Riemannian distance function too much, and so that we instantly obtain uniform bounds on the full curvature tensor and all its derivatives. These ideas lead to the resolution of some long-standing open problems in geometry.

No previous knowledge of Ricci flow will be assumed, and differential geometry prerequisites will be kept to a minimum.

Joint work with Miles Simon.
 

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.

Ambitious mathematical models of highly complex natural phenomena are challenging to analyse, and more and more computationally expensive to evaluate. This is a particularly acute problem for many tasks of interest and numerical methods will tend to be slow, due to the complexity of the models, and potentially lead to sub-optimal solutions with high levels of uncertainty which needs to be accounted for and subsequently propagated in the statistical reasoning process. This talk will introduce our contributions to an emerging area of research defining a nexus of applied mathematics, statistical science and computer science, called "probabilistic numerics". The aim is to consider numerical problems from a statistical viewpoint, and as such provide numerical methods for which numerical error can be quantified and controlled in a probabilistic manner. This philosophy will be illustrated on problems ranging from predictive policing via crime modelling to computer vision, where probabilistic numerical methods provide a rich and essential quantification of the uncertainty associated with such models and their computation. 

The cover of the December 2016 AMS Notices shows an eye-like region picked out by blue and red dots and surrounded by green rays. The picture, drawn by Yasushi Yamashita, illustrates Gaven Martin’s search for the smallest volume 3-dimensional hyperbolic orbifold. It represents a family of two generator groups of isometries of hyperbolic 3-space which was recently studied, for quite different reasons, by myself, Yamashita and Ser Peow Tan.

After explaining the coloured dots and their role in Martin’s search, we concentrate on the green rays. These are Keen-Series pleating rays which are used to locate spaces of discrete groups. The theory also suggests why groups represented by the red dots on the rays in the inner part of the eye display some interesting geometry.
 

Puzzling things happen in human perception when ambiguous or incomplete information is presented to the eyes. Rivalry occurs when two different images, presented one to each eye, lead to alternating percepts, possibly of neither image separately. Illusions, or multistable figures, occur when a single image can be perceived in several ways. The Necker cube is the most famous example. Impossible objects arise when a single image has locally consistent but globally inconsistent geometry. Famous examples are the Penrose triangle and etchings by Maurits Escher.

In this lecture Ian Stewart will demonstrate how these phenomena provide clues about the workings of the visual system, with reference to recent research in the field which has modelled simplified, systematic methods by which the brain can make decisions. In these models a neural network is designed to interpret incoming sensory data in terms of previously learned patterns. Rivalry occurs when different interpretations are confused, and illusions arise when the same data have several interpretations.

The lecture will be non-technical and highly illustrated, with plenty of examples.

Please email @email to register

We consider the non-linear equation $T^{-1} u+\partial_tu-\partial_x^2\pi(u)=\xi$

driven by space-time white noise $\xi$, which is uniformly parabolic because we assume that $\pi'$ is bounded away from zero and infinity. Under the further assumption of Lipschitz continuity of $\pi'$ we show that the stationary solution is - as for the linear case - almost surely Hölder continuous with exponent $\alpha$ for any $\alpha<\frac{1}{2}$ w. r. t. the parabolic metric. More precisely, we show that the corresponding local Hölder norm has stretched exponential moments.

On the stochastic side, we use a combination of martingale arguments to get second moment estimates with concentration of measure arguments to upgrade to Gaussian moments. On the deterministic side, we first perform a Campanato iteration based on the De Giorgi-Nash Theorem as well as finite and infinitesimal versions of the $H^{-1}$-contraction principle, which yields Gaussian moments for a weaker Hölder norm. In a second step this estimate is improved to the optimal

Hölder exponent at the expense of weakening the integrability to stretched exponential.

This is joint work with Felix Otto.