Past

The inverse acoustic obstacle scattering problem, in its most general

form, seeks to determine the nature of an unknown scatterer from knowl-

edge of its far eld or radiation pattern. The problem which is the main

concern here is:

If the scattering cross section, i.e the absolute value of the radiation

pattern, of an unknown scatterer is known determine its shape.

In this talk we explore the problem from a number of points of view.

These include questions of uniqueness, methods of solution including it-

erative methods, the Minkowski problem and level set methods. We con-

clude by looking at the problem of acoustically invisible gateways and its

connections with cloaking

Standard numerical schemes for acoustic scattering problems suffer from the restriction that the number of degrees of freedom required to achieve a prescribed level of accuracy must grow at least linearly with respect to frequency in order to maintain accuracy as frequency increases. In this talk, we review recent progress on the development and analysis of hybrid numerical-asymptotic boundary integral equation methods for these problems. The key idea of this approach is to form an ansatz for the solution based on knowledge of the high frequency asymptotics, allowing one to achieve any required accuracy via the approximation of only (in many cases provably) non-oscillatory functions. In particular, we discuss very recent work extending these ideas for the first time to non-convex scatterers.

After giving a brief physical motivation I will define the notion of generalized pseudo-holomorphic curves, as well as tamed and compatible generalized complex structures. The latter can be used to give a generalization of an energy identity. Moreover, I will explain some aspects of the local and global theory of generalized pseudo-holomorphic curves.

I will explain how moduli spaces of quadratic differentials on Riemann surfaces can be interpreted as spaces of stability conditions for certain 3-Calabi-Yau triangulated categories. These categories are defined via quivers with potentials, but can also be interpreted as Fukaya categories. This work (joint with Ivan Smith) was inspired by the papers of  Gaiotto, Moore and Neitzke, but connections with hyperkahler metrics, Fock-Goncharov coordinates etc. will not be covered in this talk.

Rado introduced the following `lion and man' game in the 1930's: two players (the lion and the man) are in the closed unit disc and they can run at the same speed. The lion would like to catch the man and the man would like to avoid being captured.

This game has a chequered history with several false `winning strategies' before Besicovitch finally gave a genuine winning strategy.

We ask the surprising question: can both players win?

 Determining the price at which to conduct a trade is an age-old problem. The first (albeit primitive) pricing mechanism dates back to the Neolithic era, when people met in physical proximity in order to agree upon mutually beneficial exchanges of goods and services, and over time increasingly complex mechanisms have played a role in determining prices. In the highly competitive and relentlessly fast-paced markets of today’s financial world, it is the limit order book that matches buyers and sellers to trade at an agreed price in more than half of the world’s markets.  In this talk I will describe the limit order book trade-matching mechanism, and explain how the extra flexibility it provides has vastly impacted the problem of how a market participant should optimally behave in a given set of circumstances.

Abstract:

Motivated by the correlation functions-Wilson loop correspondence in maximally supersymmetric Yang-Mills theory, we will investigate a conjecture of Alday, Buchbinder, and Tseytlin regarding correlators of null polygonal Wilson loops with local operators in general position.  By translating the problem to twistor space, we can show that such correlators arise by taking null limits of correlation functions in the gauge theory, thereby providing a proof for the conjecture.  Additionally, twistor methods allow us to derive a recursive formula for computing these correlators, akin to the BCFW recursion for scattering amplitudes.

For a fixed n we will investigate homomorphisms Out(F_n) to

Out(F_m) (i.e. free representations) and Out(F_n) to

GL_m(K) (i.e. K-linear representations). We will

completely classify both kinds of representations (at least for suitable

fields K) for a range of values $m$.

 Abstract:  In this talk we will introduce a new particle approximation scheme to solve the stochastic filtering problem. This new scheme makes use of the Kusuoka-Lyons-Victoir (KLV) method to approximate the dynamics of the signal. In order to control the computational cost, a partial sampling procedure based on the tree based branching algorithm (TBBA) is performed. The novelty of the method lies in the fact that the weights used in the TBBA are computed combining the cubature weights and the filtering weights. In this way, we can avoid the sample degeneracy problem inherent to particle filters. We will also present some simulations showing the performance of the method.

It is a general belief that the heat kernels on fractals should exhibit highly oscillatory behaviors as opposed to the classical case of Riemannian manifolds.

For example, on a class of finitely ramified fractals, called (affine) nested fractals, a canonical ``Brownian motion" has been constructed and its transition density (heat kernel) $p_{t}(x,y)$ satisfies $c_{1} \leq t^{d_{s}/2} p_{t}(x,x) \leq c_{2}$ for $t \leq 1$ for any point $x$ of the fractal; here $d_{s}$ is the so-called spectral dimension. Then it is natural to ask whether the limit of this quantity as $t$ goes to 0 exists or not, and it has been conjectured NOT to exist by many people.

In this talk, I will present partial affirmative answers to this conjecture. First, for a general (affine) nested fractal, the non-existence of the limit is shown to be true for a ``generic" (in particular, almost every) point. Secondly, the same is shown to be valid for ANY point of the fractal in the particular cases of the $d$-dimensional standard Sierpinski gasket with $d\geq 2$ and of the $N$-polygasket with $N\geq 3$ odd, e.g. the pentagasket ($N=5$) and the heptagasket ($N=7$).

In this talk we will review M-theory dualities and recent attempts to make these dualities manifest in eleven-dimensional supergravity. We will review the work of Berman and Perry and then outline a prescription, called non-linear realisation, for making larger duality symmetries manifest. Finally, we will explain how the local symmetries are described by generalised geometry, which leads to a duality-covariant constraint that allows one to reduce from generalised space to physical space.

Image segmentation and a number of other problems from image processing and computer vision can be regarded

as interface problems. Recently, diffusive and sharp interface techniques have been used for these problems.

In this talk, we will first briefly explain these models and compare the advantages and disadvantages of these models. Numerically, these models can be solved through some PDEs. In the end, we will show some recent results on how to use graph cut to solve these interface problems. Moreover, the global minimizer can be guaranteed even the problem is nonconex and nonlinear. The use of max-flow in a network setting and also in an infinite dimensional setting will be explained.

We will first motivate and review some implicit schemes that arises from the discretization of non linear PDEs in finance or in optimal control problems - when using finite differences methods or finite element methods.

For the american option problem, we are led to compute the solution of a discrete obstacle problem, and will give some results for the convergence of nonsmooth Newton's method for solving such problems.

Implicit schemes are interesting for their stability properties, however they can be too costly in practice.

We will then present some novel schemes and ideas, based on the semi-lagrangian approach and on discontinuous galerkin methods, trying to be as much explicit as possible in order to gain practical efficiency.