Past

According to Witten a gauge theory with a mass gap contains a possibly trivial Topological Field Theory  (TFT) in the infrared.  We show that in SU(N) YM it there exists a trivial TFT defined by   twistor Wilson loops whose v.e.v. is 1 in the large-N limit for any shape of the loops supported on certain Lagrangian submanifolds of space-time that lift to Lagrangian submanifolds of twistor space.

We derive a new version of the Makeenko-Migdal loop equation for the topological twistor Wilson loops, the holomorphic loop equation, that involves the change of variables in the YM functional integral from the connection to the anti-selfdual part of the curvature and the choice of a holomorphic gauge.

Employing the holomorphic loop equation and viewing Floer homology the other way around,
we associate to arcs asymptotic in both directions to the cusps of the Lagrangian submanifolds the critical points of an effective action implied by the holomorphic loop equation. The critical points of the effective action, being associated to the homology of the punctured Lagrangian submanifolds, consist of surface operators of the YM theory, supported on the punctures.  The correlators of surface operators in the TFT satisfy for large momentum the constraint that follows by the renormalization group and by the asymptotic freedom and they are saturated by an infinite sum of pure poles of scalar and pseudoscalar glueballs, whose joint spectrum is exactly linear in the mass squared.

For several physical purposes we outline  a related construction of a twistorial Topological String Theory dual to the TFT, that involves the Chern-Simons action on Lagrangian submanifolds of  
twistor space.

Relative entropy weighted optimization is convex optimization problem over the space of probability measures. Many convex optimization problems can be rephrased as such a problem. This is particularly useful since this problem type admits a quasi-explicit solution (i.e. as the expectation over a random variable), which immediately provides a Monte-Carlo method for numerically computing the solution of the optimization problem.

In this talk we discuss the background and application of this approach to stochastic optimal control problems, which may be considered as relative entropy weighted problems with Wiener space as probability space, and its connection with the theory of large deviations for Brownian functionals. As a particular application we discuss the minimization of the local time in a given point of Brownian motion with drift.

Recent years have seen rapid expansion of the capabilities 
to recreate in vivo conditions using in vitro microfluidic assays.  
A wide range of single cell and cell population behaviors can now 
be replicated, controlled and imaged for detailed studies to gain 
new insights.  Such experiments also provide useful fodder for 
computational models, both in terms of estimating model parameters 
and for testing model-generated hypotheses.  Our experiments have 
focused in several different areas.  
1) Single cell migration experiments in 3D collagen gels have 
revealed that interstitial flow can lead to biased cell migration 
in the upstream direction, with important implications to cancer 
invasion.  We show this phenomenon to be a consequence of 
integrin-mediated mechanotransduction.  
2) Endothelial cells seeded in fibrin gels form perfusable 
vascular networks within 2-3 days through a process termed 
“vasculogenesis”.  The process by which cells sense their 
neighbours, extend projections and form anastomoses, and 
generate interconnected lumens can be observed through time-lapse 
microscopy.  
3) These vascular networks, once formed, can be perfused with 
medium containing cancer cells that become lodged in the 
smaller vessels and proceed to transmigrate across the endothelial 
barrier and invade into the surrounding matrix.  High resolution 
imaging of this process reveals a fascinating sequence of events 
involving interactions between a tumour cell, endothelial cells, 
and underlying matrix.  These three examples will be presented 
with a view toward gaining new insights through computational 
modelling of the associated phenomena.

We will present in this lecture the global existence of weak solutions and the local existence and uniqueness of strong-in-time solutions for the fully-coupled Navier-Stokes/Q-tensor system on a bounded domain $\O\subset\mathbb{R}^d$ ($d=2,3$) with inhomogenerous Dirichlet and Neumann or mixed boundary conditions. Our result is valid for any physical parameter $\xi$ and we consider the Navier-Stokes equations with a general (but smooth) viscosity coefficient.

The good use of condiments is one of the secrets of a tasty quiche. If you want to delight your guests, add a pinch of ground pepper or cinnamon to the yellow liquid formed by the mix of the eggs and the crème fraiche. Here, is a surprise : even if the liquid is at rest, the pinch of milled pepper spreads by itself at the surface of the mixture. It expands in a circular way, and within a few seconds, it covers an area equal to several times its initial one. Why does it spread like that ? What factors influence this dispersion ? I will present some experiments and mathematical models of this process.

*****     PLEASE NOTE THIS SEMINAR TAKES PLACE ON TUESDAY     *****

Reduced ODE models describing coarsening dynamics of droplets in nanometric polymer film interacting on solid substrate in the presence of large slippage at the liquid/solid interface are derived from one-dimensional lubrication equations. In the limiting case of the infinite slip length corresponding to the free suspended films a collision/absorption model then arises and is solved explicitly. The exact collision law is derived. Existence of a threshold at which the collision rates switch from algebraic to exponential ones is shown.

*****     PLEASE NOTE THIS SEMINAR TAKES PLACE ON TUESDAY     *****

*****     PLEASE NOTE THIS SEMINAR TAKES PLACE ON MONDAY     *****

Ultralow interfacial tension mixtures have interfacial tensions that are 1,000 times, or more, lower than typical oil-water systems. Despite the recent utility of ultralow interfacial tension mixtures in industry and research, quantifying the interfacial tension remains challenging. Here I describe a technique that measures ultralow interfacial tensions by magnetically deflecting paramagnetic spheres in a co-flow microfluidic device. This method involves the tuning of the distance between the co-flowing interface and the magnetic field source, and observing the behavior of the magnetic particles as they approach the liquid-liquid interface--the particles either pass through or are trapped. I demonstrate the effectiveness of this technique for measuring very low interfacial tensions by testing solutions of different surfactant concentrations, and show that the results are comparable with measurements made using a spinning drop tensiometer.

*****     PLEASE NOTE THIS SEMINAR TAKES PLACE ON MONDAY     *****

*****     PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON MONDAY 24TH JUNE 2013     *****

Energy equations describing magnetic and inertial confinement functions (ICF) are strongly coupled, time dependent non-linear PDEs. The huge disparity of the coefficients in the coupled non-linear equations brings tremendous numerical difficulties to get high resolution solutions. It results in highly ill-conditioned linear systems in each non-linear iteration. Solving the resulted non-linear systems is time-consuming which takes up to 90% in the total simulation time. Many customized numerical techniques have to be employed to get a robust and accurate solution.This talk will present an inexact Newton-Krylov-Schwarz framework to solve the problem, demonstrating how to integrate preconditioning, partial Jacobian matrix forming techniques, parallel computing techniques with the Newton-Krylov solvers to solve the challenging problem. The numerical results will be shown and other numerical problems will be mentioned.

*****     If anyone is planning to take the 11.36 train after the seminar to the NA conference in Glasgow a taxi from the Gibson building is being arranged. Please contact Jude, @email, to book a place in the taxi.     *****

The Bayesian approach to inverse problems is of paramount importance in quantifying uncertainty about the input to and the state of a system of interest given noisy observations. Herein we consider the forward problem of the forced 2D Navier Stokes equation. The inverse problem is inference of the forcing, and possibly the initial condition, given noisy observations of the velocity field. We place a prior on the forcing which is in the form of a spatially correlated temporally white Gaussian process, and formulate the inverse problem for the posterior distribution. Given appropriate spatial regularity conditions, we show that the solution is a continuous function of the forcing. Hence, for appropriately chosen spatial regularity in the prior, the posterior distribution on the forcing is absolutely continuous with respect to the prior and is hence well-defined. Furthermore, the posterior distribution is a continuous function of the data.

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This is a joint work with Andrew Stuart and Kody Law (Warwick)

*****     PLEASE NOTE THIS SEMINAR WILL COMMENCE AT 12.00     *****

I will present experimental work on collective dynamics in two different systems: (i) a collection of self propelled droplets and (ii) coupled mechanical oscillators.  

In the first part, I will talk about microswimmers made from water-in-oil emulsion droplets. Following a brief description of the swimming mechanism, I will discuss some of the collective effects that emerge in quasi 1 and 2 dimensional confinements of swimming droplets. Specifically, I dwell on hydrodynamic and volume exclusion interactions, only through which these droplets can couple their motions. 

In the second part, I will present recent results about an intriguing dynamic known as a chimera state. In the world of coupled oscillators, a chimera state is the co-existence of synchrony and asynchrony in a population of identical oscillators, which are coupled nonlocally. Following nearly 10 years of intense theoretical research, it has been an imminent question whether these chimera states exist in real systems. Recently, we built an experiment with of springs, swings and metronomes and realised, for the first time, these symmetry breaking states in a purely physical system.

*****     PLEASE NOTE THIS SEMINAR WILL COMMENCE AT 12.00     *****

*****     PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON MONDAY 17TH JUNE 2013     *****

Computing is an exercise of discretization of the real world into space, time, and value. While discretization in time and space is well understood in the sciences, discretization of value is a scientific domain full of opportunity. Maxeler's Multiscale Dataflow Computing allows the programmer to finely trade off discretization of value with real performance measured in wallclock time.

In this talk I will show the connection between discretization of value and Kolmogorov Complexity on one hand and approximation theory on the other. Utilizing the above concepts together with building general purpose computing systems based on dataflow concepts, has enabled us to deliver production systems for Oil & Gas imaging (modelling, multiple elimination, RTM, Geomechanics), Finance Risk (derivatives modelling and scenario analysis), as well as many scientific application such as computing weather models, Astrochemistry, and brain simulations. Algorithms range from 3D Finite Difference, Finite Elements (sparse matrix solvers), pattern matching, conjugate gradient optimization, to communication protocols and bitcoin calculations. Published results of users of our machines show a 20-50x total advantage in computations per unit space (1U) and computations per Watt.

*****     PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON MONDAY 17TH JUNE 2013     *****

Advanced models such as Lévy models require advanced numerical methods for developing efficient pricing algorithms. Here we focus on PIDE based methods. There is a large arsenal of numerical methods for solving parabolic equations that arise in this context. Especially Galerkin and Galerkin inspired methods have an impressive potential. In order to apply these methods, what is required is a formulation of the equation in the weak sense.

We therefore classify Lévy processes according to the solution spaces of the associated parabolic PIDEs. We define the Sobolev index of a Lévy process by a certain growth condition on the symbol. It follows that for Lévy processes with a certain Sobolev index b the corresponding evolution problem has a unique weak solution in the Sobolev-Slobodeckii space with index b/2. We show that this classification applies to a wide range of processes. Examples are the Brownian motion with or without drift, generalised hyperbolic (GH), CGMY and (semi) stable Lévy processes.

A comparison of the Sobolev index with the Blumenthal-Getoor index sheds light on the structural implication of the classification. More precisely, we discuss the Sobolev index as an indicator of the smoothness of the distribution and of the variation of the paths of the process.

An application to financial models requires in particular to admit pure jump processes as well as unbounded domains of the equation. In order to deal at the same time with the typical payoffs which can arise, the weak formulation of the equation has to be based on exponentially weighted Sobolev-Slobodeckii spaces. We provide a number of examples of models that are covered by this general framework. Examples of options for which such an analysis is required are calls, puts, digital and power options as well as basket options.

The talk is based on joint work with Ernst Eberlein.

Colloidal suspensions do not freeze uniformly; rather, the frozen phase (e.g. ice) becomes segregated, trapping bulk regions of the colloid within, which leads to a fascinating variety of patterns that impact both nature and technology. Yet, despite the central importance of ice segregation in several applications, the physics are poorly understood in concentrated systems and continuum models are available only in restricted cases. I will discuss a particular set of steady-state ice segregation patterns that were obtained during a series of directional solidification experiments on concentrated suspensions. As a case study, I will focus of one of these patterns, which is very reminiscent of ice lenses observed in freezing soils and rocks; a form of ice segregation which underlies frost heave and frost weathering. I will compare these observations against an extended version of a 'rigid-ice' model used in previous frost heave studies. The comparison between theory and experiment is qualitatively correct, but fails to quantitatively predict the ice-lensing pattern. This leaves open questions about the validity of the assumptions in 'rigid-ice' models. Moreover, 'rigid-ice' models are inapplicable to the study of other ice segregation patterns. I conclude this talk with some possibilities for a more general model of freezing colloidal suspensions.

Please see http://www.cs.ox.ac.uk/seminars/CompBioPublicSeminars/

The optimal function spaces for the local-in-time well-posedness theory of the Navier-Stokes equations are closely related to the scaling symmetry of the equations. This might appear to be tied to particular methods used in the proofs, but in this talk we will raise the possibility that the equations are actually ill-posed for finite-energy initial data just at the borderline of some of the most benign scale-invariant spaces. This is related to debates about the adequacy of the Leray-Hopf weak solutions for predicting the time evolution of the system. (Joint work with Hao Jia.)

• Fabian Spill - Stochastic and continuum modelling of angiogenesis
• Matt Saxton - Modelling the contact-line dynamics of an evaporating drop
• Almut Eisentraeger - Water purification by (high gradient) magnetic separation

Note early start to avoid a clash with the OCCAM group meeting.

Sela showed that the theory of the non abelian free groups is stable. In a joint work with Sklinos, we give some characterization of the forking independence relation between elements of the free group F over a set of parameters A in terms of the Grushko and cyclic JSJ decomposition of F relative to A. The cyclic JSJ decomposition of F relative to A is a geometric group theory tool that encodes all the splittings of F as an amalgamated product (or HNN extension) over cyclic subgroups in which A lies in one of the factors.

So far, the circle method has been a very useful tool to prove
many cases of Manin's conjecture. Work of B. Birch back in 1961 establishes
this for smooth complete intersections in projective space as soon as the
number of variables is large enough depending on the degree and number of
equations. In this talk we are interested in subvarieties of biprojective
space. There is not much known so far, unless the underlying polynomials are
of bidegree (1,1). In this talk we present recent work which combines the
circle method with the generalised hyperbola method developed by V. Blomer
and J. Bruedern. This allows us to verify Manin's conjecture for certain
smooth hypersurfaces in biprojective space of general bidegree.

Lattice rules are equal-weight quadrature/cubature rules for the approximation of multivariate integrals which use lattice points as the cubature nodes. The quality of such cubature rules is directly related to the discrepancy between the uniform distribution and the discrete distribution of these points in the unit cube, and so, they are a kind of low-discrepancy sampling points. As low-discrepancy based cubature rules look like Monte Carlo rules, except that they use cleverly chosen deterministic points, they are sometimes called quasi-Monte Carlo rules.

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The talk starts by motivating the usage of Monte Carlo and then quasi-Monte Carlo methods after which some more recent developments are discussed. Topics include: worst-case errors in reproducing kernel Hilbert spaces, weighted spaces and the construction of lattice rules and sequences.

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In the minds of many, quasi-Monte Carlo methods seem to share the bad stanza of the Monte Carlo method: a brute force method of last resort with slow order of convergence, i.e., $O(N^{-1/2})$. This is not so.

While the standard rate of convergence for quasi-Monte Carlo is rather slow, being $O(N^{-1})$, the theory shows that these methods achieve the optimal rate of convergence in many interesting function spaces.

E.g., in function spaces with higher smoothness one can have $O(N^{-\alpha})$, $\alpha > 1$. This will be illustrated by numerical examples.

More than half of the world's financial markets use a limit order book

mechanism to facilitate trade. For markets where trade is conducted

through a central counterparty, trading platforms disseminate the same

information about the limit order book to all market participants in

real time, and all market participants are able to trade with all

others. By contrast, in markets that operate under bilateral trade

agreements, market participants are only able to view the limit order

book activity from their bilateral trading partners, and are unable to

trade with the market participants with whom they do not possess a

bilateral trade agreement. In this talk, I discuss the implications

of such a market structure for price formation. I then introduce a

simple model of such a market, which is able to reproduce several

important empirical properties of traded price series. By identifying and

matching several robust moment conditions to the empirical data, I make

model-based inference about the network of bilateral trade partnerships

in the market. I discuss the implications of these findings for market

stability and suggest how the regulator might improve market conditions

by implementing simple restrictions on how market participants form their

bilateral trade agreements.

Symplectic reflection algebras are an important class of algebras related to an incredibly high number of different topics such as combinatorics, noncommutative geometry and resolutions of singularities and have themselves a rich representation theory. We will recall their definition and classification coming from symplectic reflection groups and outline some of the results that have characterised their representation theory over the last decade, focusing on the link with representations of quivers.

We review the theory of $E_n$-algebras (roughly, algebras with $n$ compatible multiplications) and discuss $E_n$-deformation theory in the sense of Lurie. We then describe, to the best of our ability, the use of $E_n$-deformation theory in the on-going work of Calaque, Pantev, Toen, Vezzosi, and Vaquie about deformation quantization of derived stacks with shifted Poisson structure.

We review the theory of $E_n$-algebras (roughly, algebras with $n$ compatible multiplications) and discuss $E_n$-deformation theory in

the sense of Lurie. We then describe, to the best of our ability, the use of $E_n$-deformation theory in the on-going work of Calaque, Pantev, Toen,

Vezzosi, and Vaquie about deformation quantization of derived stacks with shifted Poisson structure.

***** PLEASE NOTE THAT THIS WILL TAKE PLACE ON TUESDAY 11TH JUNE ****

Signal transduction pathways are sophisticated information processing machinery in the cell that is arguably taking advantage of highly non-idealistic natures of intracellular environments for its optimum operations. In this study, we focused on effects of intracellular macromolecular crowding on signal transduction pathways using single-particle simulations. We have previously shown that rebinding of kinases to substrates can remarkably increase processivity of dual-phosphorylation reactions and change both steady-state and transient responses of the reaction network. We found that molecular crowding drastically enhances the rebinding effect, and it shows nonlinear time dependency although kinetics at the macroscopic level still follows the conventional model in dilute media. We applied the rate law revised on the basis of these calculations to MEK-ERK system and compared it with experimental measurements.

***** PLEASE NOTE THAT THIS WILL TAKE PLACE ON TUESDAY 11TH JUNE ****

The class of sofic groups was introduced by Gromov in 1999. It
includes all residually finite and all amenable groups. In fact, no group has been proved
not to be sofic, so it remains possible that all groups are sofic. Their
defining property is that, roughly speaking, for any finite subset F of
the group G, there is a map from G to a finite symmetric group, which is
approximates to an injective homomorphism on F. The widespread interest in
these group results partly from their connections with other branches of
mathematics, including dynamical systems. In the talk, we will concentrate
on their definition and algebraic properties.

In this talk, we consider the setting: a random realization of an evolving dynamical system, and explain how, using notions common in the theory of rough paths, such as the signature, and shuffle product, one can provide a new united approach to the fundamental problem of predicting the conditional distribution of the near future given the past. We will explain how the problem can be reduced to a linear regression and least squaresanalysis. The approach is clean and systematic and provides a clear gradation of finite dimensional approximations. The approach is also non-parametric and very general but still presents itself in computationally tractable and flexible restricted forms for concrete problems. Popular techniques in time series analysis such as GARCH can be seen to be restricted special cases of our approach but it is not clear they are always the best or most informative choices. Some numerical examples will be shown in order to compare our approach and standard time series models.

This talk is based on a joint work with Céline Labart. We are interested in this paper in the numerical simulation of solutions to Backward Stochastic Differential Equations. There are several existing methods to handle this problem and one of the main difficulty is always to compute conditional expectations.

Even though our approach can also be applied in the case of the dynamic programmation equation, our starting point is the use of Picard's iterations that we write in a forward way

In order to compute the conditional expectations, we use Wiener Chaos expansions of the underlying random variables. From a practical point of view, we keep only a finite number of terms in the expansions and we get explicit formulas.

We will present numerical experiments and results on the error analysis.