Past

The Bi-Conjugate Gradient method (BiCG) is a well-known iterative solver (Krylov method) for linear systems of equations, proposed about 35 years ago, and the basis for some of the most successful iterative methods today, like BiCGSTAB. Nevertheless, the convergence behavior is poorly understood. The method satisfies a Petrov-Galerkin property, and hence its residual is constrained to a space of decreasing dimension (decreasing one per iteration). However, that does not explain why, for many problems, the method converges in, say, a hundred or a few hundred iterations for problems involving a hundred thousand or a million unknowns. For many problems, BiCG converges not much slower than an optimal method, like GMRES, even though the method does not satisfy any optimality properties. In fact, Anne Greenbaum showed that every three-term recurrence, for the first (n/2)+1 iterations (for a system of dimension n), is BiCG for some initial 'left' starting vector. So, why does the method work so well in most cases? We will introduce Krylov methods, discuss the convergence of optimal methods, describe the BiCG method, and provide an analysis of its convergence behavior.

In the first part of this talk we introduce hypersymplectic manifolds and compare various aspects of their geometry with related notions in hyperkähler geometry. In particular, we explain the hypersymplectic quotient construction. Since many examples of hyperkähler structures arise from Yang-Mills moduli spaces via the hyperkähler quotient construction, we discuss the gauge theoretic equations for a (twisted) harmonic map from a Riemann surface into a compact Lie group. They can be viewed as the zero condition for a hypersymplectic moment map in an infinite-dimensional setup.

An agent is presented with an N Bandit (Fruit) machines. It is assumed that each machine produces successes or failures according to some fixed, but unknown Bernoulli distribution. If the agent plays for ever, how can he/she choose a strategy that ensures the average successes observed tend to the parameter of the "best" arm?

Alternatively suppose that the agent recieves a reward of a^n at the nth button press for a success, and 0 for a failure; now how can the agent choose a strategy to optimise his/her total expected rewards over all time? These are two examples of classic Bandit Problems.

We analyse the behaviour of two strategies, the Narendra Algorithm and the Gittins Index Strategy. The Narendra Algorithm is a "learning"

strategy, in that it answers the first question in the above paragraph, and we demonstrate this remains true when the sequences of success and failures observed on the machines are no longer i.i.d., but merely satisfy an ergodic condition. The Gittins Index Strategy optimises the reward stream given above. We demonstrate that this strategy does not "learn" and give some new explicit bounds on the Gittins Indices themselves.

 In this talk, I will present joint work with Uri Onn, Mark Berman, and Pirita Paajanen.

Let G be a linear algebraic group defined over the integers. Let O be a compact discrete valuation ring with a finite residue field of cardinality q and characteristic p. The group

G(O) has a filtration by congruence subgroups

G_m(O) (which is by definition the kernel of reduction map modulo P^m where P is the maximal ideal of O).

Let c_m=c_m(G(O))  denote the number of conjugacy classes in the finite quotient group G(O)/G_m(O) (which is called the mth congruence quotient of G(O)).  The conjugacy class zeta function of

G(O) is defined to be the Dirichlet series Z_{G(O)}(s)=\sum_{m=0,1,...} c_m q^_{-ms}, where s is a complex number with Re(s)>0. This zeta function was defined by du Sautoy when G is a p-adic analytic group and O=Z_p, the ring of p-adic integers, and he proved that in this case it is a rational function in p^{-s}.  We consider the question of dependence of this zeta function on p and more generally on the ring O.

We prove that for certain algebraic groups, for all compact discrete valuation rings with finite residue field of cardinality q and sufficiently large residue characteristic p, the conjugacy class zeta function is a rational function in q^{-s} which depends only on q and not on the structure of the ring. Note that this applies also to positive characteristic local fields.

A key in the proof is a transfer principle. Let \psi(x) and f(x) be resp.

definable sets and functions in Denef-Pas language.

For a local field K, consider the local integral Z(K,s)=\int_\psi(K)

|f(x)|^s dx, where | | is norm on K and dx normalized absolute value

giving the integers O of K volume 1. Then there is some constant

c=c(f,\psi) such that  for all local fields K of residue characteristic larger than c and residue field of cardinality q, the integral Z(K,s) gives the same rational function in q^{-s} and takes the same value as a complex function of s.

This transfer principle is more general than the specialization to local fields of the special case when there is no additive characters of the motivic transfer principle of Cluckers and Loeser since their result is the case when the integral is zero.

The conjugacy class zeta function is related to the representation zeta function which counts number of irreducible complex representations in each degree (provided there are finitely many or finitely many natural classes) as was shown in the work of Lubotzky and Larsen, and gives information on analytic properties of latter zeta function.

This talk will be an introduction to property (T). It was originally introduced by Kazhdan as a method of showing that certain discrete subgroups of Lie groups are finitely generated, but has expanded to become a widely used tool in group theory. We will take a short tour of some of its uses.

We derive solutions of the Kirchhoff equations for a knot tied on an infinitely long elastic rod subjected to combined tension and twist. We consider the case of simple (trefoil) and double (cinquefoil) knots; other knot topologies can be investigated similarly. The rod model is based on Hookean elasticity but is geometrically non-linear. The problem is formulated as a non-linear self-contact problem with unknown contact regions. It is solved by means of matched asymptotic expansions in the limit of a loose knot. Without any a priori assumption, we derive the topology of the contact set, which consists of an interval of contact flanked by two isolated points of contacts. We study the influence of the applied twist on the equilibrium and find an instability for a threshold value of the twist.

Geoghegan's stack construction is a tool for analysing groups

that act on simply connected CW complexes, by providing a topological

description in terms of cell stabilisers and the quotient complex,

similar to what Bass-Serre theory does for group actions on trees. We

will introduce this construction and see how it can be used to give

results on finiteness properties of groups.

I will describe recent work on motivic DT invariants for 3-manifolds, which are expected to be a refinement of Chern-Simons theory. The conclusion will be that these should be possible to define and work with, but there will be some interesting problems along the way. There will be a discussion of the problem of upgrading the description of the moduli space of flat connections as a critical locus to the problem of describing the fundamental group algebra of a 3-fold as a "noncommutative critical locus," including a recent topological result on obstructions for this problem. I will also address the question of how a motivic DT invariant may be expected to pick up a finer invariant of 3-manifolds than just the fundamental group.

This work demonstrates how we can extract clinically useful patterns

extracted from time series data (speech signals) using nonlinear signal
processing and how to exploit those patterns using robust statistical
machine learning tools, in order to estimate remotely and accurately
average Parkinson's disease symptom severity. 

String field theory is a candidate for a full non-perturbative definition

of string theory. We aim to define string field theory on a space-time

lattice to investigate its behaviour at the quantum level. Specifically, we

look at string field theory in a one dimensional linear dilaton background,

using level truncation to restrict the theory to a finite number of fields.

I will report on our preliminary results at level-0 and level-1.

The maps $u$ which are continuous in ${\mathbb R}^n$ and circle-valued are precisely the maps of the form $u=\exp (i\varphi)$, where the phase $\varphi$ is continuous and real-valued.

In the context of Sobolev spaces, this is not true anymore: a map $u$ in some Sobolev space $W^{s,p}$ need not have a phase in the same space. However, it is still possible to describe all the circle-valued Sobolev maps. The characterization relies on a factorization formula for Sobolev maps, involving three objects: good phases, bad phases, and topological singularities. This formula is the analog, in the circle-valued context, of Weierstrass' factorization theorem for holomorphic maps.

The purpose of the talk is to describe the factorization and to present a puzzling byproduct concerning sums of Dirac masses.

The Siegel-Walfisz theorem gives an asymptotic estimate for the number of primes in an arithmetic progression, provided the modulus of the progression is small in comparison with the length of the progression. Counting primes is harder when the modulus is not so small compared to the length, but estimates such as Linnik's constant and the Brun-Titchmarsh theorem give us some information. We aim to look in particular at upper bounds for the number of primes in such a progression, and improving the Brun-Titchmarsh bound.

There is a close but underexploited analogy between the Euler characteristic

of a topological space and the cardinality of a set. I will give a quite

general definition of the "magnitude" of a mathematical structure, framed

categorically. From this single definition can be derived many

cardinality-like invariants (some old, some new): the Euler characteristic

of a manifold or orbifold, the Euler characteristic of a category, the

magnitude of a metric space, the Euler characteristic of a Koszul algebra,

and others. A conjecture states that this purely categorical definition

also produces the classical invariants of integral geometry: volume, surface

area, perimeter, .... No specialist knowledge will be assumed.

Recently, two new Calabi-Yau threefolds have been discovered which have small Hodge numbers, and give rise to three chiral generations of fermions via the so-called 'standard embedding' compactification of the heterotic string.
In this talk I will describe how to deform the standard embedding on these manifolds in order to achieve the correct gauge group.  I will also describe how to calculate the resulting spectrum and interactions, which is still work in progress.

A stock loan is a contract between two parties: the lender, usually a bank or other financial institution providing a loan, and the borrower, represented by a client who owns one share of a stock used as collateral for the loan. Several reasons might motivate the client to get into such a deal. For example he might not want to sell his stock or even face selling restrictions, while at the same time being in need of available funds to attend to another financial operation. In Xia and Zhou (2007), a stock loan is modeled as a perpetual American option with a time varying strike and analyzed in detail within the Black-Scholes framework. In this paper, we extend the valuation of such loans to an incomplete market setting, which takes into account the natural trading restrictions faced by the client. When the maturity of the loan is infinite we obtain an exact formula for the value of the loan fee to be charged by the bank based on a result in Henderson (2007). For loans of finite maturity, we characterize its value using a variational inequality first presented in Oberman and Zariphopoulou (2003). In both cases we show analytically how the fee varies with the model parameters and illustrate the results numerically. This is joint work with Cesar G. Velez (Universidad Nacional de Colombia).

John Allen: The Bennett Pinch revisited

Abstract: The original derivation of the well-known Bennett relation is presented. Willard H. Bennett developed a theory, considering both electric and magnetic fields within a pinched column, which is completely different from that found in the textbooks. The latter theory is based on simple magnetohydrodynamics which ignores the electric field.

The discussion leads to the interesting question as to whether the possibility of purely electrostatic confinement should be seriously considered.

Angela Mihai: A mathematical model of coupled chemical and electrochemical processes arising in stress corrosion cracking

Abstract: A general mathematical model for the electrochemistry of corrosion in a long and narrow metal crack is constructed by extending classical kinetic models to also incorporate physically realistic kinematic conditions of metal erosion and surface film growth. In this model, the electrochemical processes are described by a system of transport equations coupled through an electric field, and the movement of the metal surface is caused, on the one hand, by the corrosion process, and on the other hand, by the undermining action of a hydroxide film, which forms by consuming the metal substrate. For the model problem, approximate solutions obtained via a combination of analytical and numerical methods indicate that, if the diffusivity of the metal ions across the film increases, a thick unprotective film forms, while if the rate at which the hydroxide produces is increased, a thin passivating film develops.

We will try to cover the following problems in the workshop:

(1) Modelling of aortic aneurisms showing the changes in blood flow / wall loads before and after placements of aortic stents;

(2) Modelling of blood flows / wall loads in interracial aneurisms when flow diverters are used;

(3) Metal artefact reduction in computer tomography (CT).

If we run out of time the third topic may be postponed.

Motivic exponential integrals are an abstract version of p-adic exponential integrals for big p. The latter in itself is a flexible tool to describe (families of) finite expontial sums. In this talk we will only need the more concrete view of "uniform in p p-adic integrals"

instead of the abstract view on motivic integrals. With F. Loeser, we obtained a first transfer principle for these integrals, which allows one to change the characteristic of the local field when one studies equalities of integrals, which appeared in Ann. of Math (2010). This transfer principle in particular applies to the Fundamental Lemma of the Langlands program (see arxiv). In work in progress with Halupczok and Gordon, we obtain a second transfer principle which allows one to change the characteristic of the local field when one studies integrability conditions of motivic exponential functions. This in particular solves an open problem about the local integrability of Harish-Chandra characters in (large enough) positive characteristic.

While studying growth of lattices in semisimple Lie groups we

encounter many interesting number theoretic problems. In some cases we

can show an equivalence between the two classes of problems, while in

the other the true relation between them is unclear. On the talk I

will give a brief overview of the subject and will then try to focus

on some particularly interesting examples.

Many growing plant cells undergo rapid axial elongation with negligible radial expansion. Growth is driven by high internal turgor pressure causing viscous stretching of the cell wall, with embedded cellulose microfibrils providing the wall with strongly anisotropic properties. We present a theoretical model of a growing cell, representing the primary cell wall as a thin axisymmetric fibre-reinforced viscous sheet supported between rigid end plates. Asymptotic reduction of the governing equations, under simple sets of assumptions about the fibre and wall properties, yields variants of the traditional Lockhart equation, which relates the axial cell growth rate to the internal pressure. The model provides insights into the geometric and biomechanical parameters underlying bulk quantities such as wall extensibility and shows how either dynamical changes in wall material properties or passive fibre reorientation may suppress cell elongation. We then investigate how the action of enzymes on the cell wall microstructure can lead to the required dynamic changes in macroscale wall material properties, and thus demonstrate a mechanism by which hormones may regulate plant growth.

Algebraic multigrid methods are nowadays popular to solve linear systems arising from the discretization of elliptic PDEs. They try to combine the efficiency of well tuned specific schemes like classical (geometric-based) multigrid methods, with the ease of use of general purpose preconditioning techniques. This requires to define automatic coarsening procedures, which set up an hierarchy of coarse representations of the problem at hand using only information from the system matrix.

In this talk, we focus on aggregation-based algebraic multigrid methods. With these, the coarse unknowns are simply formed by grouping variables in disjoint subset called aggregates.

In the first part of the talk, we consider symmetric M-matrices with nonnegative row-sum. We show how aggregates can then be formed in such a way that the resulting method satisfies a prescribed bound on its convergence rate. That is, instead of the classical paradigm that applies an algorithm and then performs its analysis, the analysis is integrated in the set up phase so as to enforce minimal quality requirements. As a result, we obtain one of the first algebraic multigrid method with full convergence proof. The efficiency of the method is further illustrated by numerical results performed on finite difference or linear finite element discretizations of second order elliptic PDEs; the set of problems includes problems with jumps, anisotropy, reentering corners and/or unstructured meshes, sometimes with local refinement.

In the second part of the talk, we discuss nonsymmetric problems. We show how the previous approach can be extended to M-matrices with row- and column-sum both nonnegative, which includes some stable discretizations of convection-diffusion equations with divergence free convective flow. Some (preliminary) numerical results are also presented.

This is joint work with Artem Napov.

We will recall basic definitions and facts about homogeneous Riemannian manifolds and we will discuss the Einstein condition on this kind of spaces. In particular, we will talk about non existence results of invariant Einstein metrics. Finally, we will talk briefly about the Ricci flow equation in the homogeneous setting.

We solve the problem of static hedging of (upper) barrier options (we concentrate on up-and-out put, but show how the other cases follow from this one) in models where the underlying is given by a time-homogeneous diffusion process with, possibly, independent stochastic time-change.

The main result of the paper includes analytic expression for the payoff of a (single) European-type contingent claim (which pays a certain function of the underlying value at maturity, without any pathdependence),

such that it has the same price as the barrier option up until hitting the barrier. We then consider some examples, including the Black-Scholes, CEV and zero-correlation SABR models, and investigate an approximation of the exact static hedge with two vanilla (call and put) options.

It has long been known that many materials are crystalline when in their energy-minimizing states. Two of the most common crystalline structures are the face-centred cubic (fcc) and hexagonal close-packed (hcp) crystal lattices. Here we introduce the problem of crystallization from a mathematical viewpoint and present an outline of a proof that the ground state of a large system of identical particles, interacting under a suitable potential, behaves asymptotically like fcc or hcp, as the number of particles tends to infinity. An interesting feature of this result is that it holds under no initial assumption on the particle positions. The talk is based upon a joint work in progress with Florian Theil.

From a categorical point of view, the standard Zermelo-Frankel set theoretic approach to the foundations of mathematics is fundamentally deficient: it is based on the notion of equality of objects in a set. Equalities between objects are not preserved by equivalences of categories, and thus the notion of equality is 'incorrect' in category theory. It should be replaced by the notion of 'isomorphism'.

Moving higher up the categorical ladder, the notion of isomorphism between objects is 'incorrect' from the point of view of 2-category, and should be replaced by the notion of 'equivalence'...

Recently, people have started to take seriously the idea that one should be less dogmatic about working with set-theoretic axiomatisiations of mathematics, and adopt the more fluid point of view that different foundations of mathematics might be better suited to different areas of mathematics. In particular, there are currently serious attempts to develop foundations for mathematics built on homotopy types, or, in another language, ∞-groupoids.

An (∞,1)-topos should admit an internal 'homotopical logic', just as an ordinary (1-)topos admits an internal logic modelling set theory.

It turns out that formalising such a logic is rather closely related to the problem of finding good foundations for 'intensional dependent type theory' in theoretical computer science/logic. This is sometimes referred to as the attempt to construct a 'homotopy lambda calculus'.

It is expected that a homotopy theoretic formalisation of the foundations of mathematics would be of genuine practical significance to the average mathematician!

In this talk we will give an introduction to these ideas, and to the recent work of Vladimir Voevodsky and others in this area.

Symplectic cohomology is an invariant of symplectic manifolds with contact type boundary. For example, for disc cotangent bundles it recovers the

homology of the free loop space. The aim of this talk is to describe algebraic operations on symplectic cohomology and to deduce applications in

symplectic topology. Applications range from describing the topology of exact Lagrangian submanifolds, to proving existence theorems about closed

Hamiltonian orbits and Reeb chords.

Combinatorial pattern matching is a subject which has given us fast and elegant algorithms for a number of practical real world problems as well as being of great theoretical interest. However, when single character wildcards or so-called "don't know" symbols are introduced into the input, classic methods break down and it becomes much more challenging to find provably fast solutions. This talk will give a brief overview of recent results in the area of pattern matching with don't knows and show how techniques from fields as disperse FFTs, group testing and algebraic coding theory have been required to make any progress. We will, if time permits, also discuss the main open problems in the area.

I will make a short review of some continous approximations to the Navier-Stokes equations, especially with the aim of introducing alpha models for the Large Eddy Simulation of turbulent flows.

Next, I will present some recent results about approximate deconvolution models, derived with ideas similar to image processing. Finally, I will show the rigorous convergence of solutions towards those of the averaged fluid equations.

The sequence {nα}, where α is an irrational number and {.} denotes fractional part, plays

a fundamental role in probability theory, analysis and number theory. For suitable α, this sequence provides an example for "most uniform" infinite sequences, i.e. sequences whose discrepancy has the

smallest possible order of magnitude. Such 'low discrepancy' sequences have important applications in Monte Carlo integration and other problems of numerical mathematics. For rapidly increasing n_k the behaviour of {n_kα} is similar to that of independent random variables, but its asymptotic properties depend strongly also on the number theoretic properties of n_k, providing a simple example for pseudorandom behaviour. Finally, for periodic f the sequence f(nα) provides a generalization of the trig-onometric system with many interesting  properties. In this lecture, we give a survey of the field  (going back more than 100 years) and formulate new results.

We consider a non linear Schrödinger equation on a compact manifold of dimension d subject to some multiplicative random perturbation. Using some stochastic Strichartz inequality, we prove the existence and uniqueness of a maximal solution in H^1 under some general conditions on the diffusion coefficient. Under stronger conditions on the noise, the nonlinearity and the diffusion coefficient, we deduce the existence of a global solution when d=2. This is a joint work with Z. Brzezniak.

The is the second part of the Analysis and Geometry Seminar today.

The isoperimetric inequality is a fundamental tool in many geometric and analytical issues, beside being the starting point for a great variety of other important inequalities.

We shall present some recent results dealing with the quantitative version of this inequality, an old question raised by Bonnesen at the beginning of last century. Applications of the sharp quantitative isoperimetric inequality to other classic inequalities and to eigenvalue problems will be also discussed.

The theory and computation of convex measures of financial risk has been a very active area of Financial Mathematics, with a rich history in a short number of years. The axioms specify sensible properties that measures of risk should possess (and which the industry's favourite, value-at-risk, does not). The most common example is related to the expectation of an exponential utility function.

A basic application is hedging, that is taking off-setting positions, to optimally reduce the risk measure of a portfolio. In standard continuous-time models with dynamic hedging, this leads to nonlinear PDE problems of HJB type. We discuss so-called static-dynamic hedging of exotic options under convex risk measures, and specifically the existence and uniqueness of an optimal position. We illustrate the computational challenge when we move away from the risk measure associated with exponential utility.

Joint work with Aytac Ilhan (Goldman Sachs) and Mattias Jonsson (University of Michigan).

In 1974 Haim Gaifman conjectured that if a first-order theory T is relatively categorical over T(P) (the theory of the elements satisfying P), then every model of T(P) expands to one of T.

The conjecture has long been known to be true in some special cases, but nothing general is known. I prove it in the case of abelian groups with distinguished subgroups. This is some way outside the previously known cases, but the proof depends so heavily on the Kaplansky-Mackey proof of Ulm's theorem that the jury is out on its generality.