Tue, 21 Nov 2017

16:00 - 17:00
L5

Distal Shelah Expansions

Lotte Kestner
(Imperial College)
Abstract

 

(Joint with Gareth Boxall) In this talk I will introduce some properties of distal theories. I will remark that distality is preserved neither under reducts nor expansions of the language. I will then go on to discuss a recent result that the Shelah expansion of a theory is distal if and only if the theory itself is distal. 

Thu, 15 Jun 2017
12:00
L3

Two-phase model of crowd propagation

Ewelina Zatorska
(Imperial College)
Abstract

I will talk about the fluid equations used to model pedestrian motion and traffic. I will present the compressible-incompressible Navier-Stokes two phase system describing the flow in the free and in the congested regimes, respectively. I will also show how to approximate such system by the compressible Navier-Stokes equations with singular pressure for the fixed barrier densities and also some recent developments for the barrier densities varying in the space and time.
This is a talk based on several papers in collaboration with: D. Bresch, C. Perrin, P. Degond, P. Minakowski, and L. Navoret.
 

Fri, 09 Jun 2017

11:00 - 12:00
C3

A homotopy exact sequence for overconvergent isocrystals

Ambrus Pal
(Imperial College)
Abstract

I will explain how to prove the exactness of the homotopy sequence of overconvergent p-adic fundamental groups for a smooth and projective morphism in characteristic p. We do so by first proving a corresponding result for rigid analytic varieties in characteristic 0, following dos Santos in the algebraic case. In characteristic p we proceed by a series of reductions to the case of a liftable family of curves, where we can apply the rigid analytic result. Joint work with Chris Lazda.

Mon, 20 Feb 2017

14:15 - 15:15
L3

A Statistical Model of Urban Retail Structure

LOUIS ELLAM
(Imperial College)
Abstract

One of the challenges of 21st-century science is to model the evolution of complex systems.  One example of practical importance is urban structure, for which the dynamics may be described by a series of non-linear first-order ordinary differential equations.  Whilst this approach provides a reasonable model of urban retail structure, it is somewhat restrictive owing to uncertainties arising in the modelling process.

We address these shortcomings by developing a statistical model of urban retail structure, based on a system of stochastic differential equations.   Our model is ergodic and the invariant distribution encodes our prior knowledge of spatio-temporal interactions.  We proceed by performing inference and prediction in a Bayesian setting, and explore the resulting probability distributions with a position-specific metrolpolis-adjusted Langevin algorithm.

Tue, 29 Nov 2016
14:00
L3

Stochastic discrete Hamiltonian variational integrators

Tom Tyranowski
(Imperial College)
Abstract

Stochastic Hamiltonian systems with multiplicative noise are a mathematical model for many physical systems with uncertainty. For example, they can be used to describe synchrotron oscillations of a particle in a storage ring. Just like their deterministic counterparts, stochastic Hamiltonian systems possess several important geometric features; for instance, their phase flows preserve the canonical symplectic form. When simulating these systems numerically, it is therefore advisable that the numerical scheme also preserves such geometric structures. In this talk we propose a variational principle for stochastic Hamiltonian systems and use it to construct stochastic Galerkin variational integrators. We show that such integrators are indeed symplectic, preserve integrals of motion related to Lie group symmetries, demonstrate superior long-time energy behavior compared to nonsymplectic methods, and they include stochastic symplectic Runge-Kutta methods as a special case. We also analyze their convergence properties and present the results of several numerical experiments. 

Thu, 03 Nov 2016
12:00
L5

A new approach to study strong advection problems

Harsha Hutridurga
(Imperial College)
Abstract
In this talk, I shall be attempting to give an overview of a new weak convergence type tool developed by myself, Thomas Holding (Warwick) and Jeffrey Rauch (Michigan) to handle multiple scales in advection-diffusion type models used in the turbulent diffusion theories. Loosely speaking, our strategy is to recast the advection-diffusion equation in moving coordinates dictated by the flow associated with a mean advective field. Crucial to our analysis is the introduction of a fast time variable. We introduce a notion of "convergence along mean flows" which is a weak multiple scales type convergence -- in the spirit of two-scale convergence theory. We have used ideas from the theory of "homogenization structures" developed by G. Nguetseng. We give a sufficient structural condition on the "Jacobain matrix" associated with the flow of the mean advective field which guarantees the homogenization of the original advection-diffusion problem as the microscopic lengthscale vanishes. We also show the robustness of this structural condition by giving an example where the failure of such a structural assumption leads to a degenerate limit behaviour. More details on this new tool in homogenzation theory can be found in the following paper: T. Holding, H. Hutridurga, J. Rauch. Convergence along mean flows, in press SIAM J Math. Anal., arXiv e-print: arXiv:1603.00424, (2016). In a sequel to the above mentioned work, we are preparing a work where we address the growth in the Jacobain matrix -- termed as Lagrangian stretching in Fluid dynamics literature -- and its consequences on the vanishing microscopic lengthscale limit. To this effect, we introduce a new kind of multiple scales convergence in weighted Lebesgue spaces. This helps us recover some results in Freidlin-Wentzell theory. This talk aims to present both these aspects of our work in an unified manner.
Thu, 01 Dec 2016

14:00 - 15:00
L5

A multilevel method for semidefinite programming relaxations of polynomial optimization problems with structured sparsity

Panos Parpas
(Imperial College)
Abstract

We propose a multilevel paradigm for the global optimisation of polynomials with sparse support. Such polynomials arise through the discretisation of PDEs, optimal control problems and in global optimization applications in general. We construct projection operators to relate the primal and dual variables of the SDP relaxation between lower and higher levels in the hierarchy, and theoretical results are proven to confirm their usefulness. Numerical results are presented for polynomial problems that show how these operators can be used in a hierarchical fashion to solve large scale problems with high accuracy.

Mon, 06 Jun 2016

12:00 - 13:00
L5

Black Holes and Higher Derivative Gravity

Kellogg Stelle
(Imperial College)
Abstract
Quantum corrections to the gravitational action generically include quadratic terms in the curvature. Moreover, these terms are distinguished with respect to other corrections in that their inclusion renders the theory renormalisable. The talk will discuss the changes their inclusion make to black hole solutions and the occurrence of other spherically symmetric solutions, such as wormholes and horizonless solutions.
Mon, 16 May 2016

15:45 - 16:45
L6

Volumes of minimal hypersurfaces and stationary geodesic nets

Yevgeni Liokumovich
(Imperial College)
Abstract

We will prove an upper bound for the volume of a minimal
hypersurface in a closed Riemannian manifold conformally equivalent to
a manifold with $Ric > -(n-1)$.  In the second part of the talk we will
construct a sweepout of a closed 3-manifold with positive Ricci
curvature by 1-cycles of controlled length and prove an upper bound
for the length of a stationary geodesic net. These are joint works
with Parker Glynn-Adey (Toronto) and Xin Zhou (MIT).

Mon, 25 Jan 2016
15:45
L6

Minimal surfaces in 3-manifold topology

Dan Ketover
(Imperial College)
Abstract

I will explain some recent work using minimal surfaces to address problems in 3-manifold topology.  Given a Heegaard splitting, one can sweep out a three-manifold by surfaces isotopic to the splitting, and run the min-max procedure of Almgren-Pitts and Simon-Smith to construct a smooth embedded minimal surface.   If the original splitting were strongly irreducible (as introduced by Casson-Gordon), H. Rubinstein sketched an argument in the 80s showing that the limiting minimal surface should be isotopic to the original splitting.  I will explain some results in this direction and how jointly with T. Colding and D. Gabai we can use such min-max minimal surfaces to complete the classification problem for Heegaard splittings of non-Haken hyperbolic 3-manifolds.

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