Past

In this talk we present an overview of some recent developments concerning the a posteriori error analysis and adaptive mesh design of $h$- and $hp$-version discontinuous Galerkin finite element methods for the numerical approximation of second-order quasilinear elliptic boundary value problems. In particular, we consider the derivation of computable bounds on the error measured in terms of an appropriate (mesh-dependent) energy norm in the case when a two-grid approximation is employed. In this setting, the fully nonlinear problem is first computed on a coarse finite element space $V_{H,P}$. The resulting 'coarse' numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretization on the finer space $V_{h,p}$; thereby, only a linear system of equations is solved on the richer space $V_{h,p}$. Here, an adaptive $hp$-refinement algorithm is proposed which automatically selects the local mesh size and local polynomial degrees on both the coarse and fine spaces $V_{H,P}$ and $V_{h,p}$, respectively. Numerical experiments confirming the reliability and efficiency of the proposed mesh refinement algorithm are presented.

I will claim (and maybe show) that a lot of problems in differential geometry can be reformulated in terms of non-linear elliptic differential operators. After reviewing the theory of linear elliptic operators, I will show what can be said about the non-linear setting.

In this lecture I will report on joint work with J. Casado-Díaz, T. Chacáon Rebollo, V. Girault and M.~Gómez Marmol which was published in Numerische Mathematik, vol. 105, (2007), pp. 337-510.

As Herb Sutter predicted in 2005, "The Free Lunch is Over", software programmers can no longer rely on exponential performance improvements from Moore's Law.  Computationally intensive software now rely on concurrency for improved performance, as at the high end supercomputers are being built with millions of processing cores, and at the low end GPU-accelerated workstations feature hundreds of simultaneous execution cores.  It is clear that the numerical software of the future will be highly parallel, but what language will it be written in?

Over the past few decades, high-level scientific programming languages have become an important platform for numerical codes. Languages such as MATLAB, IDL, and R, offer powerful advantages: they allow code to be written in a language more familiar to scientists and they permit development to occur in an evolutionary fashion, bypassing the relatively slow edit/compile/run/plot cycle of Fortran or C. Because a scientist’s programming time is typically much more valuable than the computing cycles their code will use, these are substantial benefits. However, programs written in such languages are not portable to high performance computing platforms and may be too slow to be useful for realistic problems on desktop machines. Additionally, the development of such interpreted language codes is partially wasteful in the sense that it typically involves reimplementation (with associated debugging) of some algorithms that already exist in well-tested Fortran and C codes.  Python stands out as the only high-level language with both the capability to run on parallel supercomputers and the flexibility to interface with existing libraries in C and Fortran.

Our code, PyClaw, began as a Python interface, written by University of Washington graduate student Kyle Mandli, to the Fortran library Clawpack, written by University of Washington Professor Randy LeVeque.  PyClaw was designed to build on the strengths of Clawpack by providing greater accessibility.  In this talk I will describe the design and implementation of PyClaw, which incorporates the advantages of a high-level language, yet achieves serial performance similar to a hand-coded Fortran implementation and runs on the world's fastest supercomputers. It brings new numerical functionality to Clawpack, while making maximal reuse of code from that package.  The goal of this talk is to introduce the design principles we considered in implementing PyClaw, demonstrate our testing infrastructure for developing within PyClaw, and illustrate how we elegantly and efficiently distributed problems over tens of thousands of cores using the PETSc library for portable parallel performance.  I will also briefly highlight a new mathematical result recently obtained from PyClaw, an investigation of solitary wave formation in periodic media in 2 dimensions.

 Graph products of groups naturally generalize direct and free products and have a rich subgroup structure. Basic examples of graph products are right angled Coxeter and Artin groups. I will discuss various forms of Tits Alternative for subgroups and
their stability under graph products. The talk will be based on a joint work with Yago Antolin Pichel.

We show the solvability of a proposed Generalized Buckley-LeverettSystem, which is related to multidimensional Muskat Problem. More-over, we discuss some important questions concerning singular limitsof the proposed model.

Symplectic field theory (SFT) can be viewed as TQFT approach to Gromov-Witten theory. As in Gromov-Witten theory, transversality for the Cauchy-Riemann operator is not satisfied in general, due to the presence of multiply-covered curves. When the underlying simple curve is sufficiently nice, I will outline that the transversality problem for their multiple covers can be elegantly solved using finite-dimensional obstruction bundles of constant rank. By fixing the underlying holomorphic curve, we furthermore define a local version of SFT by counting only multiple covers of this chosen curve. After introducing gravitational descendants, we use this new version of SFT to prove that a stable hypersurface intersecting an exceptional sphere (in a homologically nontrivial way) in a closed four-dimensional symplectic manifold must carry an elliptic orbit. Here we use that the local Gromov-Witten potential of the exceptional sphere factors through the local SFT invariants of the breaking orbits appearing after neck-stretching along the hypersurface.

We discuss some recent developments on the following long-standing problem known as Ryser's

conjecture. Let $H$ be an $r$-partite $r$-uniform hypergraph. A matching in $H$ is a set of disjoint

edges, and we denote by $\nu(H)$ the maximum size of a matching in $H$. A cover of $H$ is a set of

vertices that intersects every edge of $H$. It is clear that there exists a cover of $H$ of size at

most $r\nu(H)$, but it is conjectured that there is always a cover of size at most $(r-1)\nu(H)$.

Abstract: In this talk, I will discuss the peeling behaviour of the Weyl tensor near null infinity for asymptotically flat higher dimensional spacetimes. The result is qualitatively different from the peeling property in 4d. Also, I will discuss the rewriting of the Bondi energy flux in terms of "Newman-Penrose" Weyl components.

This talk will consist of a pure PDE part, and an applied part. The unifying topic is mean curvature flow (MCF), and particularly mean curvature flow starting at cones. This latter subject originates from the abstract consideration of uniqueness questions for flows in the presence of singularities. Recently, this theory has found applications in several quite different areas, and I will explain the connections with Harnack estimates (which I will explain from scratch) and also with the study of the dynamics of charged fluid droplets.

There are essentially no prerequisites. It would help to be familiar with basic submanifold geometry (e.g. second fundamental form) and intuition concerning the heat equation, but I will try to explain everything and give the talk at colloquium level.

Joint work with Sebastian Helmensdorfer.

I will discuss some aspects of the simplicial theory of

infinity-categories which originates with Boardman and Vogt, and has

recently been developed by Joyal, Lurie and others. The main purpose of

the talk will be to present an extension of this theory which covers

infinity-operads. It is based on a modification of the notion of

simplicial set, called 'dendroidal set'. One of the main results is that

the category of dendroidal sets carries a monoidal Quillen model

structure, in which the fibrant objects are precisely the infinity

operads,and which contains the Joyal model structure for

infinity-categories as a full subcategory.

(The lecture will be mainly based on joint work with Denis-Charles

Cisinski.)

This talk is devoted to Talagrand's transport-entropy inequality and its deep connections to the concentration of measure phenomenon, large deviation theory and logarithmic Sobolev inequalities. After an introductive part on the field, I will present recent results obtained with P-M Samson and C. Roberto establishing the equivalence of Talagrand's inequality to a restricted version of the Log-Sobolev inequality. If time enables, I will also present some works in progress about transport inequalities in a discrete setting.

Abstract: First we provide a survey on the long-time behaviour of stochastic delay equations with bounded memory, addressing existence and uniqueness of invariant measures, Lyapunov spectra, and exponential growth rates.

Then, we study the very simple one-dimensional equation $dX(t)=X(t-1)dW(t)$ in more detail and establish the existence of a deterministic exponential growth rate of a suitable norm of the solution via a Furstenberg-Hasminskii-type formula.

Parts of the talk are based on joint work with Martin Hairer and Jonathan Mattingly. 

The AdS/CFT correspondence is a powerful tool to analyse strongly coupled quantum field

theories. Over the past few years there has been a surge of activity aimed at finding

possible applications to condensed matter systems. One focus has been to holographically

realise various kinds of phases via the construction of fascinating new classes of black

hole solutions. In this framework, I will discuss the possibility of describing finite

temperature phase transitions leading to spontaneous breaking of translational invariance of

the dual field theory at strong coupling. Along with the general setup I will also discuss

specific string/M theory embeddings of the corresponding symmetry breaking modes leading to

the description of such phases.

The part of the West Antarctic Ice Sheet that drains into the Amundsen Sea is currently thinning at such a rate that it contributes nearly 10 percent of the observed rise in global mean sea level. Acceleration of the outlet glaciers means that the sea level contribution has grown over the past decades, while the likely future contribution remains a key unknown. The synchronous response of several independent glaciers, coupled with the observation that thinning is most rapid at their downstream ends, where the ice goes afloat, hints at an oceanic driver. The general assumption is that the changes are a response to an increase in submarine melting of the floating ice shelves that has been driven in turn by an increase in the transport of ocean heat towards the ice sheet. Understanding the causes of these changes and their relationship with climate variability is imperative if we are to make quantitative estimates of sea level into the future.

Observations made since the mid‐1990s on the Amundsen Sea continental shelf have revealed that the seabed troughs carved by previous glacial advances guide seawater around 3‐4°C above the freezing point from the deep ocean to the ice sheet margin, fuelling rapid melting of the floating ice. This talk summarises the results of several pieces of work that investigate the chain of processes linking large‐scale atmospheric processes with ocean circulation over the continental shelf and beneath the floating ice shelves and the eventual transfer of heat to the ice. While our understanding of the processes is far from complete, the pieces of the jigsaw that have been put into place give us insight into the potential causes of variability in ice shelf melting, and allow us to at least formulate some key questions that still need to be answered in order to make reliable projections of future ice sheet evolution in West Antarctica.

Algorithmic trade execution has become a standard technique

for institutional market players in recent years,

particularly in the equity market where electronic

trading is most prevalent. A trade execution algorithm

typically seeks to execute a trade decision optimally

upon receiving inputs from a human trader.

A common form of optimality criterion seeks to

strike a balance between minimizing pricing impact and

minimizing timing risk. For example, in the case of

selling a large number of shares, a fast liquidation will

cause the share price to drop, whereas a slow liquidation

will expose the seller to timing risk due to the

stochastic nature of the share price.

We compare optimal liquidation policies in continuous time in

the presence of trading impact using numerical solutions of

Hamilton Jacobi Bellman (HJB)partial differential equations

(PDE). In particular, we compare the time-consistent

mean-quadratic-variation strategy (Almgren and Chriss) with the

time-inconsistent (pre-commitment) mean-variance strategy.

The Almgren and Chriss strategy should be viewed as the

industry standard.

We show that the two different risk measures lead to very different

strategies and liquidation profiles.

In terms of the mean variance efficient frontier, the

original Almgren/Chriss strategy is signficently sub-optimal

compared to the (pre-commitment) mean-variance strategy.

This is joint work with Stephen Tse, Heath Windcliff and

Shannon Kennedy.

In recent years, surprising connections between type theory and homotopy theory have been discovered. In this talk I will recall the notions of intensional type theories and identity types. I will describe "infinity groupoids", formal algebraic models of topological spaces, and explain how identity types carry the structure of an infinity groupoid. I will finish by discussing categorical semantics of intensional type theories.

The talk will take place in Lecture Theatre B, at the Department of Computer Science.

Problem #1: (marker-less scaling) Poikos ltd. has created algorithms for matching photographs of humans to three-dimensional body scans. Due to variability in camera lenses and body sizes, the resulting three-dimensional data is normalised to have unit height and has no absolute scale. The problem is to assign an absolute scale to normalised three-dimensional data.

Prior Knowledge: A database of similar (but different) reference objects with known scales. An imperfect 1:1 mapping from the input coordinates to the coordinates of each object within the reference database. A projection matrix mapping the three-dimensional data to the two-dimensional space of the photograph (involves a non-linear and non-invertible transform; x=(M*v)_x/(M*v)_z, y=(M*v)_y/(M*v)_z).

Problem #2: (improved silhouette fitting) Poikos ltd. has created algorithms for converting RGB photographs of humans in (approximate) poses into silhouettes. Currently, a multivariate Gaussian mixture model is used as a first pass. This is imperfect, and would benefit from an improved statistical method. The problem is to determine the probability that a given three-component colour at a given two-component location should be considered as "foreground" or "background".

Prior Knowledge: A sparse set of colours which are very likely to be skin (foreground), and their locations. May include some outliers. A (larger) sparse set of colours which are very likely to be clothing (foreground), and their locations. May include several distributions in the case of multi-coloured clothing, and will probably include vast variations in luminosity. A (larger still) sparse set of colours which are very likely to be background. Will probably overlap with skin and/or clothing colours. A very approximate skeleton for the subject.

Limitations: Sample colours are chosen "safely". That is, they are chosen in areas known to be away from edges. This causes two problems; highlights and shadows are not accounted for, and colours from arms and legs are under-represented in the model. All colours may be "saturated"; that is, information is lost about colours which are "brighter than white". All colours are subject to noise; each colour can be considered as a true colour plus a random variable from a gaussian distribution. The weight of this gaussian model is constant across all luminosities, that is, darker colours contain more relative noise than brighter colours.

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.

Please note the unusual start-time.

Algebraic theories, locally presentable categories and their application to type theories. The seminar will take place in Lecture Theatre A of the Department of Computer Science.

A SMEC device is an array of aerofoil-shaped parallel hollow vanes forming linear venturis, perforated at the narrowest point where the vanes most nearly touch. When placed across a river or tidal flow, the water accelerates through the venturis between each pair of adjacent vanes and its pressure drops in accordance with Bernoulli’s Theorem. The low pressure zone draws a secondary flow out through the perforations in the adjacent hollow vanes which are all connected to a manifold at one end. The secondary flow enters the manifold through an axial flow turbine.

SMEC creates a small upstream head uplift of, say 1.5m – 2.5m, thereby converting some of the primary flow’s kinetic energy into potential energy. This head difference across the device drives around 80% of the flow between the vanes which can be seen to act as a no-moving-parts venturi pump, lowering the head on the back face of the turbine through which the other 20% of the flow is drawn. The head drop across this turbine, however, is amplified from, say, 2m up to, say, 8m. So SMEC is analogous to a step-up transformer, converting a high-volume low-pressure flow to a higher-pressure, lower-volume flow. It has all the same functional advantages of a step-up transformer and the inevitable transformer losses as well.

The key benefit is that a conventional turbine (or Archimedes Screw) designed to work efficiently at a 1.5m – 2.5m driving head has to be of very large diameter with a large step-up gearbox. In many real-World locations, this makes it too expensive or simply impractical, in shallow water for example.

The work we did in 2009-10 for DECC on a SMEC across the Severn Estuary concluded that compared to a conventional barrage, SMEC would output around 80% of the power at less than half the capital cost. Crucially, however, this greatly superior performance is achieved with minimal environmental impact as the tidal signal is preserved in the upstream lagoon, avoiding the severe damage to the feeding grounds of migratory birdlife that is an unwelcome characteristic of a conventional barrage.

To help successfully commercialise the technology, however, we will eventually want to build a reliable (CFD?) computer model of SMEC which even if partly parametric, would benefit hugely from an improved understanding of the small-scale turbulence and momentum transfer mechanisms in the mixing section.