Cambridge

Work of Schoen--Yau in the 70's/80's shows that area-minimizing (actually stable) two-sided surfaces in three-manifolds of non-negative scalar curvature are of a special topological type: a sphere, torus, plane or cylinder. The torus and cylinder cases are "borderline" for this estimate. It was shown by Cai--Galloway in the late 80's that the torus can only occur in a very special ambient three manifold. We complete the story by showing that a similar result holds for the cylinder. The talk should be accessible to those with a basic knowledge of curvature in Riemannian geometry.

Would you like to come see some toys?

'Toys' here have a special sense: objects of daily life which you can find or make in minutes, yet which, if played with imaginatively reveal surprises that keep scientists puzzling for a while. We will see table-top demos of many such toys and visit some of the science that they open up. The common theme is singularity.

Tadashi Tokieda is the Director of Studies in Mathematics at Trinity Hall, Cambridge and the Poincaré Professor in the Department of Mathematics, Stanford.

To book please email @email

When assigned with the task of extracting information from given image data the first challenge one faces is the derivation of a truthful model for both the information and the data. Such a model can be determined by the a-priori knowledge about the image (information), the data and their relation to each other. The source of this knowledge is either our understanding of the type of images we want to reconstruct and of the physics behind the acquisition of the data or we can thrive to learn parametric models from the data itself. The common question arises: how can we customise our model choice to a particular application? Or better how can we make our model adaptive to the given data?

Starting from the first modelling strategy this talk will lead us from nonlinear diffusion equations and subdifferential inclusions of total variation type functionals as the most successful image modeltoday to non-smooth second- and third-order variational models, with data models for Gaussian and Poisson distributed data as well as impulse noise. These models exhibit solution-dependent adaptivities in form of nonlinearities or non-smooth terms in the PDE or the variational problem, respectively. Applications for image denoising, inpainting and surface reconstruction are given. After a critical discussion of these different image and data models we will turn towards the second modelling strategy and propose to combine it with the first one using a PDE constrained optimisation method that customises a parametrised form of the model by learning from examples. In particular, we will consider optimal parameter derivation for total variation denoising with multiple noise distributions and optimising total generalised variation regularisation for its application in photography.

Abstract: This is a joint work with E. Breuillard.

A conjecture of Breuillard asserts that for every positive integer d, there is a positive constant c such that the following holds. Let S be a finite subset of GL(d,C) that generates a group, which is not virtually nilpotent. Then |S^n|>exp(cn) for all n.
Considering an algebraic number a that is not a root of unity and the semigroup generated by the affine transformations x-> ax+1, x-> ax+1, the above conjecture implies that the Mahler measure of a is at least 1+c' for some c'>0 depending on c. This property is known as Lehmer's conjecture.

I will talk about the converse of this implication, namely that Lehmer's conjecture implies the uniform growth conjecture of
Breuillard.

From a complex analytic perspective, both Teichmüller spaces and
symmetric spaces can be realised as contractible bounded domains, that
have several features in common but also exhibit many differences. In
this talk we will study isometric maps between these two important
classes of bounded domains equipped with their intrinsic Kobayashi metric.

Equations of quantum mechanics in the semiclassical regime present an enduring challenge for numerical analysts, because their solution is highly oscillatory and evolves on two scales. Standard computational approaches to the semiclassical Schrödinger equation do not allow for long time integration as required, for example, in quantum control of atoms by short laser bursts. This has motivated our approach of asymptotic splittings. Combining techniques from Lie-algebra theory and numerical algebra, we present a new computational paradigm of symmetric Zassenhaus splittings, which lends itself to a very precise discretisation in long time intervals, at very little cost. We will illustrate our talk by examples of quantum phenomena – quantum tunnelling and quantum scattering – and their computation and, time allowing, discuss an extension of this methodology to time-dependent semiclassical systems using Magnus expansions