Past

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