Past

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.

Given a polynomial function f defined on a variety X,

we consider two questions, which are non-commutative analogues

of the Prime Number Theorem and the Linnik Theorem:

- how often the values of f(x) at integral points in X are almost prime?

- can one effectively solve the congruence equation f(x)=b (mod q)

with f(x) being almost prime?

We discuss a solution to these questions when X is a homogeneous

variety (e.g, a quadratic surface).

Whenever we say the words "fluid flows" or "shape changes" we enter the realm of infinite-dimensional geometric mechanics. Water, for example, flows. In fact, Euler's equations tell us that water flows a particular way. Namely, it flows to get out of its own way as adroitly as possible. The shape of water changes by smooth invertible maps called diffeos (short for diffeomorphisms). The flow responsible for this optimal change of shape follows the path of shortest length, the geodesic, defined by the metric of kinetic energy. Not just the flow of water, but the optimal morphing of any shape into another follows one of these optimal paths.

The lecture will be about the commonalities between fluid dynamics and shape changes and will be discussed in the language most suited to fundamental understanding -- the language of geometric mechanics. A common theme will be the use of momentum maps and geometric control for steering along the optimal paths using emergent singular solutions of the initial value problem for a nonlinear partial differential equation called EPDiff, that governs metamorphosis along the geodesic flow of the diffeos. The main application will be in the registration and comparison of Magnetic Resonance Images for clinical diagnosis and medical procedures.

Starting from a Navier-Stokes-Cahn-Hilliard equation for a two-phase flow problem we discuss efficient numerical approaches based on adaptive finite element methods. Various extensions of the model are discussed: a) we consider the model on implicitly described geometries, which is used to simulate the sliding of droplets over nano-patterned surfaces, b) we consider the effect of soluble surfactants and show its influence on tip splitting of droplets under shear flow, and c) we consider bijels as a new class of soft matter materials, in which colloidal particles are jammed on the fluid-fluid interface and effect the motion of the interface due to an elastic force.

The work is based on joint work with Sebastian Aland (TU Dresden), John Lowengrub (UC Irvine) and Knut Erik Teigen (U Trondheim).

A K3 surface of degree two can be seen as a double cover of the complex projective plane, ramified over a nonsingular sextic curve. In this talk we explore two different methods for constructing explicit projective models of threefolds admitting a fibration by such surfaces, and discuss their relative merits.

Singular monopoles are solutions to the Bogomolny equation with prescribed singularities of Dirac monopole type. Previously such monopoles could be constructed only by the Nahm transform, with some difficulty. We therefore formulate a new construction of all singular monopoles. This construction relies on two ideas: Kronheimer's correspondence between singular monopoles on R^3 and self-dual connections on the multi-Taub-NUT space, and Cherkis' recent construction of self-dual connections on curved spaces using bow diagrams. As an example of our method we use it to obtain the explicit solution for a charge one SU(2) singular monopole with an arbitrary number of singularities.

We shall discuss a simple low order numerical integration scheme for ODEs and DAEs. The scheme has a parameter that allows for regularization of Jacobian of stiff problems and for numerically elucidating multi-scale response, if any, in some problems.

We discuss homological finiteness Bredon types FPm with respect to the class of finite subgroups and seperately with respect to the class of virtually cyclic subgroups. We will concentrate to the case of solubles groups and if the time allows to the case of generalized R. Thompson groups of type F. The results announced are joint work with Brita Nucinkis

(Southampton) and Conchita Martinez Perez (Zaragoza) and will appear in papers in Bulletin of LMS and Israel Journal of Mathematics.