Thu, 09 Jun 2016
12:00
L6

Ancient solutions of Geometric Flows

Panagiota Daskalopoulos
(Columbia University)
Abstract
Some of the most important problems in geometric flows are related to the understanding of singularities. This usually happens through a blow up procedure near the potential singularity which uses the scaling properties of the partial differential equation involved. In the case of a parabolic equation the blow up analysis often leads to special solutions which are defined for all time $-\infty < t \leq T$ for some $T \leq +\infty$. The classification of such solutions often sheds new insight to the singularity analysis. 
In this talk we will discuss Uniqueness Theorems for ancient solutions to geometric partial differential equations such as the Mean curvature flow, the Ricci flow and the Yamabe flow. We will also discuss the construction of new ancient solutions from the parabolic gluing of one or more solitons.
Thu, 16 Jun 2016
12:00
L6

Minimal hypersurfaces with bounded index

Ben Sharp
(University of Pisa)
Abstract
An embedded hypersurface in a Riemannian manifold is said to be minimal if it is a critical point with respect to the induced area. The index of a minimal hypersurface (roughly speaking) tells us how many ways one can locally deform the surface to decrease area (so that strict local area-minimisers have index zero). We will give an overview of recent works linking the index, topology and geometry of closed and embedded minimal hypersurfaces. The talk will involve separate joint works with Reto Buzano, Lucas Ambrozio and Alessandro Carlotto. 
Thu, 02 Jun 2016
12:00
L6

Regularity Theory for Symmetric-Convex Functionals of Linear Growth

Franz Gmeineder
(Oxford)
Abstract
In this talk I will report on regularity results for convex autonomous functionals of linear growth which depend on the symmetric gradients. Here, generalised minimisers will be attained in the space BD of functions of bounded of deformation which consists of those summable functions for which the distributional symmetric gradient is a Radon measure of finite total variation. Due to Ornstein's Non--Inequality, BD contains BV as a proper subspace and thus the full weak gradients of BD--functions might not exist even as Radon measures. In this talk, I will discuss conditions on the variational integrand under which partial regularity or higher Sobolev regularity for minima and hence the existence and higher integrability of the full gradients of minima can be established. This is joint work with Jan Kristensen.
Thu, 19 May 2016
12:00
L6

Stochastic Conservation Laws

Kenneth Karlsen
(University of Oslo)
Abstract
Stochastic partial differential equations arise in many fields, such as biology, physics, engineering, and economics, in which random phenomena play a crucial role. Recently many researchers have been interested in studying the effect of stochastic perturbations on hyperbolic conservation laws and other related nonlinear PDEs possessing shock wave solutions, with particular emphasis on existence and uniqueness questions (well-posedness). In this talk I will attempt to review parts of this activity.
Thu, 12 May 2016
12:00
L6

Quantization of time-like energy for wave maps into spheres

Roland Grinis
(Oxford)
Abstract
In this talk, we shall discuss how building upon the threshold theorem for wave maps, techniques inspired by the blow-up analysis of supercritical harmonic maps, can lead to a decomposition of the map into a decoupled sum of rescaled solitons, along a suitably chosen sequence of time slices converging to the maximal time of existence, with a term having asymptotically vanishing energy in the interior of the light cone, and when the target manifold is an Euclidean sphere. This work is motivated by the soliton resolution conjecture, on which spectacular progress has been achieved recently for equivariant wave maps, radial Yang-Mills fields and semi-linear critical wave equations.
Thu, 05 May 2016
12:00
L6

Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions

Marshall Slemrod
(University of Wisconsin)
Abstract
We will discuss some underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the interconnected nonlinear partial differential equations.
Risk in a large claims insurance market with bipartite graph structure
 Reinert, G Kley, O Klueppelberg, C Operations Research volume 64 issue 5 1159-1176 (22 Jul 2016)
Tue, 26 Apr 2016
14:00
L3

Best L1 polynomial approximation

Yuji Nakatsukasa
(University of Oxford)
Abstract

An important observation in compressed sensing is the exact recovery of an l0 minimiser to an underdetermined linear system via the l1 minimiser, given the knowledge that a sparse solution vector exists. Here, we develop a continuous analogue of this observation and show that the best L1 and L0 polynomial approximants of a corrupted function (continuous analogue of sparse vectors) are equivalent. We use this to construct best L1 polynomial approximants of corrupted functions via linear programming. We also present a numerical algorithm for computing best L1 polynomial approximants to general continuous functions, and observe that compared with best L-infinity and L2 polynomial approximants, the best L1 approximants tend to have error functions that are more localized.

Joint work with Alex Townsend (MIT).

Thu, 16 Jun 2016
16:00
L6

Gaps Between Smooth Numbers

Roger Heath-Brown
(Oxford University)
Abstract

Let $a_1, \cdots, a_N$ be the sequence of y-smooth numbers up to x (i.e. composed only of primes up to y). When y is a small power of x, what can one say about the size of the gaps $a_{j+1}-a_j$? In particular, what about

$$\sum_1^N (a_{j+1}-a_j)^2?$$

Subscribe to