Past

The Faraday cage effect is the phenomenon whereby electrostatic and electromagnetic fields are shielded by a wire mesh "cage". Nick Trefethen, Jon Chapman and I recently carried out a mathematical analysis of the two-dimensional electrostatic problem with thin circular wires, demonstrating that the shielding effect is not as strong as one might infer from the physics literature. In this talk I will present new results generalising the previous analysis to the electromagnetic case, and to wires of arbitrary shape. The main analytical tool is the asymptotic method of multiple scales, which is used to derive continuum models for the shielding, involving homogenized boundary conditions on an effective cage boundary. In the electromagnetic case one observes interesting resonance effects, whereby at frequencies close to the natural frequencies of the equivalent solid shell, the presence of the cage actually amplifies the incident field, rather than shielding it. We discuss applications to radiation containment in microwave ovens and acoustic scattering by perforated shells. This is joint work with Ian Hewitt.

A subgroup Gamma of a semisimple algebraic group G is called strongly dense if every subgroup of Gamma is either cyclic or Zariski-dense. I will describe a method for building strongly dense free subgroups inside a given Zariski-dense subgroup  Gamma of G, thus providing a refinement of the Tits alternative. The method works for a large class of G's and Gamma's. I will also discuss connections with word maps and expander graphs. This is joint work with Bob Guralnick and Michael Larsen.

 In the last few years, it has become clear that there are striking connections between supersymmetry and geometric representation theory.  In this talk, I will discuss boundary conditions in three dimensional gauge theories with N = 4 supersymmetry.  I will then outline a physical understanding of a remarkable conjecture in representation theory known as `symplectic duality.

In acoustic materials (non null sound velocity), there is a clear separation of scale between the relaxation to mechanical equilibrium, governed by Euler equations, and the slower relaxation to thermal equilibrium, governed by heat equation if thermal conductivity is finite. In one dimension in acoustic systems, thermal conductivity is diverging and the thermal equilibrium is reached by a superdiffusion governed by a fractional heat equation. In non-acoustic materials it seems that there is not such separation of scales, and thermal and mechanical equilibriums are reached at the same time scale, governed by a Euler-Bernoulli beam equation. We prove such macroscopic behaviors in chains of oscillators with dynamics perturbed by a random local exchange of momentum, such that energy and momentum are conserved. (Works in collaborations with T. Komorowski).

Using Vovk's game-theoretic approach to mathematical finance and probability, it is possible to obtain new results in both areas.We first prove that one can make an arbitrarily large profit by investing in those one-dimensional paths which do not possess a local time of finite p-variation.  Additionally, we provide pathwise Tanaka formulas suitable for our local times and for absolutely continuous functions with sufficient regular derivatives. In the second part we derive a model-independent super-replication theorem in continuous time. Our result covers a broad range of exotic derivatives, including look-back options, discretely monitored Asian options, and options on realized variance.
 This talk is based on joint works with M. Beiglböck, A.M.G. Cox, M. Huesmann and N. Perkowski.

 

We show how the theories of paracontrolled distributions and regularity structures can be implemented on manifolds, to solve singular SPDEs like the parabolic Anderson model.

This is ongoing work with Bruce Driver (UCSD) and Antoine Dahlqvist (Cambridge)

I will present several new  3d N=2 dualities with super-potentials involving monopole operators. Some of the theories that I will discuss describe systems of D3 branes ending on pq-webs. In these cases  3d mirror symmetry is a consequence of S-duality.

 

In his talk, Kerry will explore the pressing practical problem of how hurricane activity will respond to global warming, and how hurricanes could in turn be influencing the atmosphere and ocean

In his talk, Kerry will explore the pressing practical problem of how hurricane activity will respond to global warming, and how hurricanes could in turn be influencing the atmosphere and ocean.

This talk will discuss work-in-progress on the numerical approximation
of reflected diffusions arising from applications in engineering, finance
and network queueing models.  Standard numerical treatments with
uniform timesteps lead to 1/2 order strong convergence, and hence
sub-optimal behaviour when using multilevel Monte Carlo (MLMC).

In simple applications, the MLMC variance can be improved by through
a reflection "trick".  In more general multi-dimensional applications with
oblique reflections an alternative method uses adaptive timesteps, with
smaller timesteps when near the boundary.  In both cases, numerical
results indicate that we obtain the optimal MLMC complexity.

This is based on joint research with Eike Muller, Rob Scheichl and Tony
Shardlow (Bath) and Kavita Ramanan (Brown).

I will discuss some reaction-diffusion equations of bistable type motivated by biology and medicine. The aim is to understand the effect of the shape of the domain on propagation or on blocking of advancing waves. I will first describe the motivations of these questions and present a result about the existence of generalized “transition waves”. I will then discuss various geometric conditions that lead to either blocking, or partial propagation, or complete propagation. These questions involve new qualitative results for some non-linear elliptic and parabolic partial differential equations. I report here on joint work with Juliette Bouhours and Guillemette Chapuisat.

On the railway network, for example, there is a large base of installed equipment with a useful life of many years.  This equipment has condition monitoring that can flag a fault when a measured parameter goes outside the permitted range.  If we can use existing measurements to predict when this would occur, preventative maintenance could be targeted more effectively and faults reduced.  As an example, we will consider the current supplied to a points motor as a function of time in each operational cycle.

We say that a real closed field is an IPA-real closed field if it admits an integer part (IP) which is a model of Peano Arithmetic (PA). In [2] we prove that the value group of an IPA-real closed field must satisfy very restrictive conditions (i.e. must be an exponential group in the residue field, in the sense of [4]). Combined with the main result of [1] on recursively saturated real closed fields, we obtain a valuation theoretic characterization of countable IPA-real closed fields. Expanding on [3], we conclude the talk by considering recursively saturated o-minimal expansions of real closed fields and their IPs.

References:
[1] D'Aquino, P. - Kuhlmann, S. - Lange, K. : A valuation theoretic characterization ofrecursively saturated real closed fields ,
Journal of Symbolic Logic, Volume 80, Issue 01, 194-206 (2015)
[2] Carl, M. - D'Aquino, P. - Kuhlmann, S. : Value groups of real closed fields and
fragments of Peano Arithmetic, arXiv: 1205.2254, submitted
[3] D'Aquino, P. - Kuhlmann, S : Saturated o-minimal expansions of real closed fields, to appear in Algebra and Logic (2016)
[4] Kuhlmann, S. :Ordered Exponential Fields, The Fields Institute Monograph Series, vol 12. Amer. Math. Soc. (2000)
 

I will describe how the moduli of various congruences between Hecke eigenvalues of automorphic forms ought to show up in ratios of critical values of $\text{GSP}_2 \times \text{GL}_2$ L-functions. To test this experimentally requires the full force of Farmer and Ryan's technique for approximating L-values given few coefficients in the Dirichlet series.