Past

This is a special talk outside the normal Computational Mathematics and Application seminar series. Please note it takes place on a Wednesday.

In some options markets (eg. commodities), options are listed with only a single maturity for each underlying.

In others, (eg. equities, currencies),

options are listed with multiple maturities.

In this paper, we assume that the risk-neutral process for the underlying futures price is a pure jump Markov martingale and that European option prices are given at a continuum of strikes and at one or more maturities. We show how to construct a time-homogeneous process which meets a single smile and a piecewise time-homogeneous process, which can meet multiple smiles.

We also show that our construction leads to partial differential difference equations (PDDE's), which permit both explicit calibration and fast numerical valuation

I will describe in detail the first construction of infinite, finitely generated torsion groups due to Golod in early 60s --

these groups are special cases of the so-called Golod-Shafarevich groups. If time allows, I will discuss some related constructions and open problems.

Informally speaking, a finitely generated group G is said to be {\it Golod-Shafarevich} (with respect to a prime p) if it has a presentation with a ``small'' set of relators, where relators are counted with different weights depending on how deep they lie in the Zassenhaus p-filtration. Golod-Shafarevich groups are known to behave like (non-abelian) free groups in many ways: for instance, every Golod-Shafarevich group G has an infinite torsion quotient, and the pro-p completion of G contains a non-abelian free pro-p group. In this talk I will extend the list of known ``largeness'' properties of Golod-Shafarevich groups by showing that they always have an infinite quotient with Kazhdan's property (T). An important consequence of this result is a positive answer to a well-known question on non-amenability of Golod-Shafarevich groups.

Let $F$ be a uniformly distributed random $k$-SAT formula with $n$ variables and $m$ clauses. We present a polynomial time algorithm that finds a satisfying assignment of $F$ with high probability for constraint densities $m/n

The orientational order of a nematic liquid crystal in a spatially inhomogeneous flow situation is best described by a Q-tensor field because of the defects that inevitably occur. The evolution is determined by two equations. The flow is governed by a generalised Stokes equation in which the divergence of the stress tensor also depends on Q and its time derivative. The evolution of Q is governed by a convection-diffusion type equation that contains terms nonlinear in Q that stem from a Landau-de Gennes potential.

In this talk, I will show how the most general evolution equations can be derived from a dissipation principle. Based on this, I will identify a specific model with three viscosity coefficients that allows the contribution of the orientation to the viscous stress to be cast in the form of a Q-dependent body force. This leads to a convenient time-discretised strategy for solving the flow-orientation problem using two alternating steps. First, the flow field of the Stokes flow is computed for a given orientation field. Second, with the given flow field, one time step of the orientation equation is carried out. The new orientation field is then used to compute a new body force which is again used in the Stokes equation and so forth.

For some simple test applications at low Reynolds numbers, it is found that the non-homogeneous orientation of the nematic liquid crystal leads to non-linear flow effects similar to those known from Newtonian flow at high Reynolds numbers.

One of the major open challenges in general relativity is the classification of black hole solutions in higher dimensional theories. I will explain a recent result in this direction in the context of Kaluza-Klein spacetimes admitting a sufficient number N of commuting U(1)-symmetries. It turns out that the black holes in such a theory are characterized by the usual asymptotic charges, together with certain combinatorical data and moduli. The combinatorial data characterize the nature of the U(1)^N-action, and its analysis is closely related to properties of integer lattices and questions in the area of geometric number theory. I will also explain recent results on extremal black holes which show that such objects display remarkable ``symmetry enhancement'' properties

We define a chain complex B_*(C, M) (the "blob complex") associated to an n-category C and an n-manifold M. This is in some sense the derived category version of a TQFT. Various special cases of the blob complex are

familiar: (a) if M = S^1, then the blob complex is homotopy equivalent to the Hochschild complex of the 1-category C; (b) for * = 0, H_0 of the blob complex is the Hilbert space of the TQFT based on C; (c) if C is a commutative polynomial ring (viewed as an n-category), then the blob complex is homotopy equivalent to singular chains on the configuration (Dold-Thom) space of M. The blob complex enjoys various nice formal properties, including a higher dimensional generalization of the Deligne conjecture for Hochschild cohomology.

If time allows I will discuss applications to contact structures on 3-manifolds and Khovanov homology for links in the boundaries of 4-manifolds. This is joint work with Scott Morrison.

We consider the analysis for a class of random differential equations driven by rough noise and with a trajectory that is influenced by its own law. Having described the mathematical setup with great precision, we will illustrate how such equations arise naturally as the limits of a cloud of interacting particles. Finally, we will provide examples to show the ubiquity of such systems across a range of physical and economic phenomena and hint at possible extensions.

Diffusion limits of MCMC methods in high dimensions provide a useful theoretical tool for studying efficiency.

In particular they facilitate precise estimates of the number of steps required to explore the target measure, in stationarity, as a function of the dimension of the state space. However, to date such results have only been proved for target measures with a product structure, severely limiting their applicability to real applications. The purpose of this talk is to desribe a research program aimed at identifying diffusion limits for a class of naturally occuring problems, found by finite dimensional approximation of measures on a Hilbert space which are absolutely continuous with respect to a Gaussian reference measure.

The diffusion limit to a Hilbert space valued SDE (or SPDE) is proved.

Joint work with Natesh Pillai (Warwick) and Jonathan Mattingly (Duke)

String theory suggests the simultaneous presence of many ultralight axions possibly populating each decade of mass down to the Hubble scale 10^-33eV. Conversely the presence of such a plenitude of axions (an "axiverse") would be evidence for string theory, since it arises due to the topological complexity of the extra-dimensional manifold and is ad hoc in a theory with just the four familiar dimensions. We investigate how upcoming astrophysical experiments will explore the existence of such axions over a vast mass range from 10^-33eV to 10^-10eV. Axions with masses between 10^-33eV to 10^-28eV cause a rotation of the CMB polarization that is constant throughout the sky. The predicted rotation angle is of order \alpha~1/137. Axions in the mass range 10^-28eV to 10^-18eV give rise to multiple steps in the matter power spectrum, that will be probed by upcoming galaxy surveys and 21 cm line tomography. Axions in the mass range 10^-22eV to 10^-10eV affect the dynamics and gravitational wave emission of rapidly rotating astrophysical black holes through the Penrose superradiance process. When the axion Compton wavelength is of order of the black hole size, the axions develop "superradiant" atomic bound states around the black hole "nucleus". Their occupation number grows exponentially by extracting rotational energy from the ergosphere, culminating in a rotating Bose-Einstein axion condensate emitting gravitational waves. This mechanism creates mass gaps in the spectrum of rapidly rotating black holes that diagnose the presence of axions. The rapidly rotating black hole in the X-ray binary LMC X-1 implies an upper limit on the decay constant of the QCD axion f_a

We consider the classical problem of forming portfolios of vanilla options in order to hedge more exotic derivatives. In particular, we focus on a model in which the agent can trade a stock and a family of variance swaps written on that stock. The market is only approximately complete in the sense that any submarket consisting of the stock and the variance swaps of a finite set of maturities is incomplete, yet every bounded claim is in the closure of the set of attainable claims. Taking a Hilbert space approach, we give a characterization of hedging portfolios for a certain class of contingent claims. (Joint work with Francois Berrier)

Transient or unstable behavior is often ignored in considering long time dynamics in the deterministic world. However, stochastic effects can change the picture dramatically, so that the transients can dominate the long range behavior.

Coherence resonance is one relatively simple example of this transformation, and we consider others such as noise-driven synchronization in networks, recurrence of diseases, and stochastic stabilization in systems with delay.

The challenge is to identify common features in these phenomena, leading to new approaches for other systems of this type. Some recurring themes include the influence of multiple time scales, cooperation of both discrete and continuous aspects in the dynamics, and the remnants of underlying bifurcation structure visible through the noise.

An iterative substructuring method with balancing Neumann-Neumann preconditioner is known as an efficient parallel algorithm for the elasticity equations. This method was extended to the Stokes equations by Pavarino and Widlund [2002]. In their extension, Q2/P0-discontinuous elements are used for velocity/pressure and a Schur complement system within "benign space", where incompressibility satisfied, is solved by CG method.

For the construction of the coarse space for the balancing preconditioner, some supplementary solvability conditions are considered. In our algorithm for 3-D computation, P1/P1 elements for velocity/pressure with pressure stabilization are used to save computational cost in the stiffness matrix. We introduce a simple coarse space similar to the one of elasticity equations. Owing to the stability term, solvabilities of local Dirichlet problem for a Schur complement system, of Neumann problem for the preconditioner, and of the coarse space problem are ensured. In our implementation, local Dirichlet and Neumann problems are solved by a 4x4-block modified Cholesky factorization procedure with an envelope method, which leads to fast computation with small memory requirement. Numerical result on parallel efficiency with a shared memory computer will be shown. Direct use of the Stokes solver in an application of Earth's mantle convection problem will be also shown.

We take a leisurely look at some mathematical models from population genetics and the ways that they can be analysed. Some of the models have a very familiar form - for example diffusion models of population size look a lot like interest rate models. But hopefully there will also be something new.

I will address the celebrated and long standing “No-Hair” conjecture that aims for

the classification of stationary, regular, electro-vacuum black hole space-times.

Besides reviewing some of the necessary concepts from general relativity I will

focus on the analysis of the singular harmonic map to which the source free Einstein-Maxwell

equations reduce in the stationary and axisymmetric case.

In this talk I will outline the two constructions of the Brauer group Br($X$) of a scheme $X$, namely via etale cohomology and Azumaya algebras and briefly describe how one may compute this group using the Hochschild-Serre spectral sequence. In the early '70s Manin observed that one can use the Brauer group of a projective variety $X/k$ to define an obstruction to the existence of rational points on $X$. I will discuss this arithmetic application and time permitting, outline an example for $X$ a K3 surface.

Using the theory of formal groups, Landweber´s exactness theorem provides means to construct interesting invariants of topological spaces out of geometric objects. I will illustrate the resulting connection between algebraic geometry and stable homotopy theory in the special case of elliptic curves.

I shall describe the notion of finite decomposition complexity (FDC), introduced in joint work with Romain Tessera and Guoliang Yu on the Novikov and related conjectures. The talk will focus on the definition of FDC and examples of groups having FDC.

In a quantum network model, unitary matrices are assigned to each edge and node of a graph.  The quantum amplitude for a particle to propagate from node A to node B is the sum over all random walks (Feynman paths) from A to B, each walk being weighted by the ordered product of matrices along the path.  In most cases these models are too difficult to solve analytically, but I shall argue that when the matrices are random elements of SU("), independently drawn from the invariant measure on that group, then averages of these quantum amplitudes are equal to the probability that a certain kind of self-avoiding *classical* random walk reaches B when started at A.  This leads to various conjectures about the generic behaviour of such network models on regular lattices in two and three dimensions.

It a classical result from Kleinian groups that a discrete group, $G$, of isometries of hyperbolic k-space $\Bbb H^k$ will act on the

boundary sphere, $S^{k-1}$, of $\Bbb H^k$ as a convergence group.

That is:

For every sequence of distinct isometries $(g_i)\subset G$ there is a subsequence ${g_i{_j})$ and points $n,p \in \S^{k-1}$ such that for $ x \in S^{k-1} -\{n\}$, $g_i_{j}(x) \to p$ uniformly on compact subsets

The accuracy of the Nos-Hoover thermostat to sample the Gibbs measure depends on the

dynamics being ergodic. It has been observed for a long time that this dynamics is

actually not ergodic for some simple systems, such as the harmonic oscillator.

In this talk, we rigourously prove the non-ergodicity of the Nos-Hoover thermostat, for

the one-dimensional harmonic oscillator.

We will also show that, for some multidimensional systems, the averaged dynamics for the limit

of infinite thermostat "mass" has many invariants, thus giving

theoretical support for either non-ergodicity or slow ergodization.

Our numerical experiments for a two-dimensional central force problem

and the one-dimensional pendulum problem give evidence for

non-ergodicity.

We also present numerical experiments for the Nose-Hoover chain with

two thermostats applied to the one-dimensional harmonic

oscillator. These experiments seem to support the non-ergodicity of the

dynamics if the masses of the reservoirs are large enough and are

consistent with ergodicity for more moderate masses.

Joint work with Frederic Legoll and Richard Moeckel

In this talk we discuss the main results of my PhD thesis. We begin by giving some background on Moufang polygons. This is followed by a short introduction of the basic model theoretic notions related to the thesis, such as asymptotic classes of finite structures, measurable structures, (superstable) supersimple theories and (finite Morley rank) S_1 rank. We also mention the relation between Moufang polygons and the associated little projective groups.

Moufang polygons have been classified by Tits and Weiss, and a complete list is given in their book `Moufang polygons'.

This work is inspired by a paper of Kramer, Tent and van Maldeghem called "Simple groups of finite Morley rank and Tits buildings". The authors work in a superstable context. They show that Moufang polygons of finite Morley rank are exactly Pappian polygons, i.e., projective planes, symplectic quadrangles and split Cayley hexagons, provided that they arise over algebraically closed fields.

We work under the weaker assumption of supersimplicity. Therefore, we expect more examples. Indeed, apart from those already occuring in the finite Morley rank case, there are four further examples, up to duality, of supersimple Moufang polygons; namely, Hermitian quadrangles in projective dimension 3 and 4, the twisted triality hexagon and the (perfect) Ree-Tits octagon, provided that the underlying field (or `difference' field in the last case) is supersimple.

As a result, we obtain the nice characterization that supersimple Moufang polygons are exactly those Moufang polygons belonging to families which also arise over finite fields.

Examples of supersimple Moufang polygons are constructed via asymptoticity

arguments: every class C of finite Moufang polygons forms an asymptotic class, and every non-principal ultraproduct of C gives rise to a measurable structure, thus supersimple (of finite S_1 rank). For the remaining cases one can proceed as follows: let \Gamma be any Moufang polygon belonging to a family which does not arise over finite fields, and call K its underlying field; then K is

(first-order) definable in \Gamma, and by applying some model theoretic facts this definability is inconsistent with supersimplicity".

Layered (or laminated) structures are increasingly used in modern industry (e.g., in microelectronics and aerospace engineering). Integrity of such structures is mainly determined by the quality of their interfaces: poor adhesion or delamination can lead to a catastrophic failure of the whole structure. Can nonlinear waves help us to detect such defects? We study the dynamics of a nonlinear longitudinal bulk strain wave in a split, layered elastic bar, made of nonlinearly hyperelastic Murnaghan material. We consider a symmetric two-layered bar and assume that there is perfect interface for x 0, where the x-axis is directed along the bar. Using matched asymptotic multiple-scales expansions and the integrability theory of the KdV equation by the Inverse Scattering Transform, we examine scattering of solitary waves and show that the defect causes generation of more than one secondary solitary waves from a single incident soliton and, thus, can be used to detect the defect. The theory is supported by experimental results. Experiments have been performed in the Ioffe Institute in St. Petersburg (Russia), using holographic interferometry and laser induced generation of an incident compression solitary wave in two- and three-layered polymethylmethacrylate (PMMA) bars, bonded using ethyl cyanoacrylate-based (CA) adhesive.

Artin formalism gives an equality of certain L-functions of elliptic curves or of zeta-functions of number fields. When combined with the Birch and Swinnerton-Dyer conjecture, this can give interesting results about the Galois module structure of the Selmer group of an elliptic curve. When combined with the analytic class number formula, this can help determine the Galois module structure of the group of units of a number field. In this talk, I will introduce the main technique, which is completely representation theoretic, for extracting such information

Work with N.K. Nichols (Reading), C. Boess & A. Bunse-Gerstner (Bremen)

The Gauss-Newton (GN) method is a well known iterative technique for solving nonlinear least squares problems subject to dynamical system constraints. Such problems arise commonly from applications in optimal control and state estimation. Variational data assimilation systems for weather, ocean and climate prediction currently use approximate GN methods. The GN method solves a sequence of linear least squares problems subject to linearized system constraints. For very large systems, low resolution linear approximations to the model dynamics are used to improve the efficiency of the algorithm. We propose a new method for deriving low order system approximations based on model reduction techniques from control theory. We show how this technique can be combined with the GN method to give a state estimation technique that retains more of the dynamical information of the full system. Numerical experiments using a shallow-water model illustrate the superior performance of model reduction to standard truncation techniques.

Scalar balance laws with monostable reaction, possibly non-convex flux, and

viscosity $\varepsilon$ are known to admit so-called entropy travelling fronts for all velocities greater than or equal to an $\varepsilon$-dependent minimal value, both when $\varepsilon$ is positive, when all fronts are smooth, and for $\varepsilon =0$, when the possibly non-convex flux results in fronts of speed close to the minimal value typically having discontinuities where jump conditions hold.

I will discuss the vanishing-viscosity limit of these fronts.

I shall discuss joint work with Angus Macintyre on the model theory of the ring of adeles of a number field