L4

For many applications in science and engineering, the ability to efficiently and accurately approximate solutions to elliptic PDEs dictates what physical phenomena can be simulated numerically.  In this seminar, we present a high-order accurate discretization technique for variable coefficient PDEs with smooth coefficients.  The technique comes with a nested dissection inspired direct solver that scales linearly or nearly linearly with respect to the number of unknowns.  Unlike the application of nested dissection methods to classic discretization techniques, the constant prefactors do not grow with the order of the discretization.  The discretization is robust even for problems with highly oscillatory solutions.  For example, a problem 100 wavelengths in size can be solved to 9 digits of accuracy with 3.7 million unknowns on a desktop computer.  The precomputation of the direct solver takes 6 minutes on a desktop computer.  Then applying the computed solver takes 3 seconds.  The recent application of the algorithm to inverse media scattering also will be presented.

 Locality is not expected to be a fundamental aspect of a full theory of quantum gravity; it should be emergent in an appropriate semiclassical limit.  In the context of general holography, I'll define a new construct - the causal state - which provides a necessary and sufficient condition for a boundary state to have a holographic semiclassical dual causal geometry (and thus be "local").  This definition illuminates some general features of holographic quantum gravity: for instance, I'll show that the emergence of locality is "all or nothing" in the sense that it exhibits features of quantum error correction and quantum secret sharing.  In the special case of AdS/CFT, I'll also argue that the causal state is the natural boundary dual to the so-called causal wedge of a region. 

Liquid crystals are classical examples of mesophases or materials that are intermediate in character between conventional solids and liquids. There are different classes of liquid crystals and we focus on the simplest and most widely used nematic liquid crystals. Nematic liquid crystals are simply put, anisotropic liquids with distinguished directions and are the working material of choice for the multi-billion dollar liquid crystal display industry. In this workshop, we briefly review the mathematical theories for nematic liquid crystals, the modelling framework and some recent work on modelling experiments on confined liquid crystalline systems conducted by the Aarts Group (Chemistry Oxford) and experiments on nematic microfluidics by Anupam Sengupta (ETH Zurich). This is joint work with Alexander Lewis, Peter Howell, Dirk Aarts, Ian Griffiths, Maria Crespo Moya and Angel Ramos.
We conclude with a brief overview of new experiments on smectic liquid crystals in the Aarts laboratory and questions related to the recycling of liquid crystal displays originating from informal discussions with Votechnik ( a company dealing with automated recycling technologies , http://votechnik.com/).
 

The goal of this talk will be to give an overview of recent work, joint with Kim Moore, on a short time existence problem in Lagrangian mean curvature flow. More specifically, we consider a compact initial Lagrangian submanifold with a finite number of singularities, each asymptotic to a pair of transversely intersecting planes. We show it is possible to construct a smooth Lagrangian mean curvature flow, existing for positive times, that attains the singular Lagrangian as its initial condition in a suitable weak sense.  The construction uses a family of smooth solutions whose initial conditions approximate the singular Lagrangian. In order to appeal to compactness theorems and produce the desired solution, it is necessary to first establish uniform curvature estimates on the approximating family. As time allows I hope to focus in particular on the proof of these estimates, and their role in the proof of the main theorem.