Past

In this seminar we discuss the gas dynamics of chimneys, solar updraft towers and energy towers. The main issue is to discuss simple fluid dynamic models which still describe the main features of the mentioned applications. We focus first on one dimensional compressible models. Then we apply a small Mach number asymptotics to reduce to complexity and to avoid the known problems

of fully compressible models in the small Mach number regime. In case of the energy tower in addition we have to model the evaporation process.

Finally we obtain a much simpler fluid dynamic model which allows robust and very fast numerical simulations. We discuss the qualitative behaviour and the good agreement with expermental data (in cases such data are available).

Mirkovic-Vilonen polytopes are a combinatorial tool for studying
perfect bases for representations of semisimple Lie algebras.  They
were originally introduced using MV cycles in the affine Grassmannian,
but they are also related to the canonical basis.  I will explain how
MV polytopes can also be used to describe components of Lusztig quiver
varieties and how this allows us to generalize the theory of MV
polytopes to the affine case.

This talk will present a computationally efficient method of simulating cardiac electrical propagation using an

adaptive high-order finite element method. The refinement strategy automatically concentrates computational

effort where it is most needed in space on each time-step. We drive the adaptivity using a residual-based error

indicator, and demonstrate using norms of the error that the indicator allows to control it successfully. Our

results using two-dimensional domains of varying complexity demonstrate in that significant improvements in

efficiency are possible over the state-of-the-art, indicating that these methods should be investigated for

implementation in whole-heart scale software.

We present a variational model for the quasi-static evolution of brutal brittle damage for geometrically-linear elastic materials. We

allow for multiple damaged states. Moreover, unlike current formulations, the materials are allowed to be anisotropic and the

deformations are not restricted to anti-plane shear. The model can be formulated either energetically or through a strain threshold. We

explore the relationship between these formulations. This is joint work with Christopher Larsen, Worcester Polytechnic Institute.

Given a manifold $M$ and a basepointed labelling space $X$ the space of unordered finite configurations in $M$ with labels in $X$, $C(M;X)$ is the space of finite unordered tuples of points in $M$, each point with an associated point in $X$. The space is topologised so that particles cannot collide. Given a compact submanifold $M_0\subset M$ we define $C(M,M_0;X)$ to be the space of unordered finite configuration in which points `vanish' in $M_0$. The scanning map is a homotopy equivalence between the configuration space and a section space of a certain $\Sigma^nX$-bundle over $M$. Throughout the 70s and 80s this map has been given several unsatisfactory and convoluted definitions. A natural question to ask is whether the map is equivariant under the diffeomorphism group of the underlying manifold. However, any description of the map relies heavily on `little round $\varepsilon$-balls' and so only actions by isometry have any chance at equivariance. The goal of this talk is to give a more natural definition of the scanning map and show that diffeomorphism equivariance is an easy consequence.

I show how to construct some Fano 3-folds that have the same Hilbert series but different Betti numbers, and so lie on different components of the Hilbert scheme. I would like to show where these fit in to a speculative (indeed fantastical) geography of Fano 3-folds, and how the projection methods I use may apply to other questions in the geography.

 Higher-order transformations are ubiquitous within data management. In relational databases, higher-order queries appear in numerous aspects including query rewriting and query specification. In XML databases, higher-order functions are natural due to the close connection of XML query languages with functional programming. We investigate higher-order query languages that combine higher- order transformations with ordinary database query languages. We define higher-order query languages based on Relational Algebra and XQuery. We also study basic problems for these query languages including evaluation, containment, and type inference. We show that even though evaluating these higher-order query languages is non-elementary, there are subclasses that are polynomially reducible to evaluation for ordinary query languages.

Solitons are localised non-singular lumps of energy which describe particles non perturbatively. Finding the solitons usually involves solving nonlinear differential equations, but I shall show that in some cases the solitons emerge directly from the underlying space-time geometry: certain abelian vortices arise from surfaces of constant mean curvature in Minkowski space, and skyrmions can be constructed from the holonomy of gravitational instantons.

We describe an invariant manifold of the equations of molecular dynamics associated to a given discrete group of isometries. It is a time-dependent manifold, but its dependence on time is explicit. In the case of the translation group, it has dimension 6N, where N is an assignable positive integer. The manifold is independent of the description of the atomic forces within a general framework. Most of continuum mechanics inherits some version of this manifold, as do theories in-between molecular dynamics and continuum mechanics, even though they do not inherit the time reversibility of molecular dynamics on this manifold. The manifold implies a natural statistics of molecular motion, which suggests a simplifying ansatz for the Boltzmann equation which, in turn, leads to new explicit far-from-equilibrium solutions of this equation. In some way the manifold underlies experimental science, i.e., the viscometric flows of fluids and the bending and twisting of beams in solids and the procedures commonly used to measure constitutive relations, this being related to the fact that the form of the manifold can be prescribed independent of the atomic forces.

Which positive integers are the area of a right angled triangle with rational sides? In this talk I will discuss this classical problem, its reformulation in terms of rational points on elliptic curves and Tunnell's theorem which gives a complete solution to this problem assuming the Birch and Swinnerton-Dyer conjecture.

 The unknotting number of a knot is an incredibly difficult invariant to compute.
In fact, there are many knots which are conjectured to have unknotting number 2 but for
which no proof of this is currently available. It therefore remains an unsolved problem to find an
algorithm that determines whether a knot has unknotting number one. In my talk, I will
show that an analogous problem for links is soluble. We say that a link has splitting number
one if some crossing change turns it into a split link. I will give an algorithm that
determines whether a link has splitting number one. (In the case where the link has
two components, we must make a hypothesis on their linking number.) The proof
that the algorithm works uses sutured manifolds and normal surfaces.

abstract:In this talk, we describe how to compute the critical point for various lattice models of planar statistical physics. We will first introduce the percolation, Ising, Potts and random-cluster models on the square lattice. Then, we will discuss how critical points of these different models are related. In a final part, we will compute the critical point of these models. This last part harnesses two main ingredients that we will describe in details: duality and sharp threshold theorems. No background is necessary.

Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process which takes values in the vertex set of a graph G, and is more likely to cross edges it has visited before. We show that it can be represented in terms of a Vertex-reinforced jump process (VRJP) with independent gamma

conductances: the VRJP was conceived by Werner and first studied by Davis and Volkov (2002,2004), and is a continuous-time process favouring sites with more local time. We show that the VRJP is a mixture of time-changed Markov jump processes and calculate the mixing measure. The mixing measure is interpreted as a marginal of the supersymmetric hyperbolic sigma model introduced by Disertori, Spencer and Zirnbauer.

This enables us to deduce that VRJP and ERRW are strongly recurrent in any dimension for large reinforcement (in fact, on graphs of bounded degree), using a localisation result of Disertori and Spencer (2010).

(Joint work with Pierre Tarrès.)

Instantons and W-bosons in 5d N=2 Yang-Mills theory arise from a circle

compactification of the 6d (2,0) theory as Kaluza-Klein modes and winding

self-dual strings, respectively. We study an index which counts BPS

instantons with electric charges in Coulomb and symmetric phases. We first

prove the existence of unique threshold bound state of U(1) instantons for

any instanton number. By studying SU(N) self-dual strings in the Coulomb

phase, we find novel momentum-carrying degrees on the worldsheet. The total

number of these degrees equals the anomaly coefficient of SU(N) (2,0) theory.

We finally propose that our index can be used to study the symmetric phase of

this theory, and provide an interpretation as the superconformal index of the

sigma model on instanton moduli space. 

The geometric Lévy model (GLM) is a natural generalisation of the geometric Brownian motion (GBM) model. The theory of such models simplifies considerably if one takes a pricing kernel approach. In one dimension, once the underlying Lévy process has been specified, the GLM has four parameters: the initial price, the interest rate, the volatility and the risk aversion. The pricing kernel is the product of a discount factor and a risk aversion martingale. For GBM, the risk aversion parameter is the market price of risk. In this talk I show that for a GLM, this interpretation is not valid: the excess rate of return above the interest rate is a nonlinear function of the volatility and the risk aversion such that it is positive, and is increasing with respect to these variables. In the case of foreign exchange, Siegel’s paradox implies that one can construct foreign exchange models for which the excess rate of return is positive for both the exchange rate and the inverse exchange rate. Examples are worked out for a range of Lévy processes. (The talk is based on a recent paper: Brody, Hughston & Mackie, Proceedings of the Royal Society London, to appear in May 2012).  

Please note that this is a joint seminar with the William Dunn School of Pathology and will be held in the EPA Seminar Room

Please note the unusual start-time.

In order to run accurate electrochemical models of batteries (and other devices) it is necessary to know a priori the values of many geometric, electrical and electrochemical parameters (10-100 parameters) e.g. diffusion coefficients, electrode thicknesses etc. However a basic difficulty is that the only external measurements that can be made on cells without deconstructing and destroying them are surface temperature plus electrical measurements (voltage, current, impedance) at the terminals. An interesting research challenge therefore is the accurate, robust estimation of physically realistic model parameters based only on external measurements of complete cells. System identification techniques (from control engineering) including ‘electrochemical impedance spectroscopy’ (EIS) may be applied here – i.e. small signal frequency response measurement. However It is not clear exactly why and how impedance correlates to SOC/ SOH and temperature for each battery chemistry due to the complex interaction between impedance, degradation and temperature.

I will give a brief overview of some of the recent work in this area and try to explain some of the challenges in the hope that this will lead to a fruitful discussion about whether this problem can be solved or not and how best to tackle it.

I will explain how to define a notion of stable-independence in NIP

theories, which is an attempt to capture the "stable part" of types.

The displacement of a liquid by an air finger is a generic two-phase flow that

underpins applications as diverse as microfluidics, thin-film coating, enhanced

oil recovery, and biomechanics of the lungs. I will present two intriguing

examples of such flows where, firstly, oscillations in the shape of propagating

bubbles are induced by a simple change in tube geometry, and secondly, flexible

vessel boundaries suppress viscous fingering instability.

1) A simple change in pore geometry can radically alter the behaviour of a

fluid displacing air finger, indicating that models based on idealized pore

geometries fail to capture key features of complex practical flows. In

particular, partial occlusion of a rectangular cross-section can force a

transition from a steadily-propagating centred finger to a state that exhibits

spatial oscillations via periodic sideways motion of the interface at a fixed

location behind the finger tip. We characterize the dynamics of the

oscillations and show that they arise from a global homoclinic connection

between the stable and unstable manifolds of a steady, symmetry-broken

solution.

2) Growth of complex dendritic fingers at the interface of air and a viscous

fluid in the narrow gap between two parallel plates is an archetypical problem

of pattern formation. We find a surprisingly effective means of suppressing

this instability by replacing one of the plates with an elastic membrane. The

resulting fluid-structure interaction fundamentally alters the interfacial

patterns that develop and considerably delays the onset of fingering. We

analyse the dependence of the instability on the parameters of the system and

present scaling arguments to explain the experimentally observed behaviour.

The Arnoldi method for standard eigenvalue problems possesses several

attractive properties making it robust, reliable and efficient for

many problems. We will present here a new algorithm equivalent to the

Arnoldi method, but designed for nonlinear eigenvalue problems

corresponding to the problem associated with a matrix depending on a

parameter in a nonlinear but analytic way. As a first result we show

that the reciprocal eigenvalues of an infinite dimensional operator.

We consider the Arnoldi method for this and show that with a

particular choice of starting function and a particular choice of

scalar product, the structure of the operator can be exploited in a

very effective way. The structure of the operator is such that when

the Arnoldi method is started with a constant function, the iterates

will be polynomials. For a large class of NEPs, we show that we can

carry out the infinite dimensional Arnoldi algorithm for the operator

in arithmetic based on standard linear algebra operations on vectors

and matrices of finite size. This is achieved by representing the

polynomials by vector coefficients. The resulting algorithm is by

construction such that it is completely equivalent to the standard

Arnoldi method and also inherits many of its attractive properties,

which are illustrated with examples.