The talk will begin with a brief account of the construction of string topology operations. I will point out some mysteries with the formulation of these operations, such as the role of (moduli) of surfaces, and pose some questions. The remainder of the talk will address these issues. In particular, I will sketch some ideas for a higher-dimensional version of string topology. For instance, (1) I will describe an E_{d+1} algebra structure on the (shifted) homology of the free mapping space H_*(Map(S^d,M^n)) and (2) I will outline how to obtain operations H_*(Map(P,M)) -> H_*(Map(Q,M)) indexes by a moduli space of zero-surgery data on a smooth d-manifold P with resulting surgered manifold Q.
Take a group G and split it as the fundamental group of a graph of groups, then take the vertex groups and split them as fundamental groups of graphs of groups etc. If at some point you end up with a collection of unsplittable groups, then you have a hierarchy. Haken showed that for any 3-manifold M with an incompressible surface S, one can cut M along S and and then find other incompressible surfaces in M\S and cut again, and repeating this process one eventually obtains a collection of balls. Analogously, Delzant and Potyagailo showed that for any finitely presented group without 2-torsion and a certain sensible class E of subgroups of G, G admits a hierarchy where the edge groups of the splittings lie in E. I really like their proof and I will present it.
In this talk we consider the Boltzmann equation arising in gas dynamics with long-range interactions. Mathematically, it involves bilinear singular integral operators known as collision operators with non-cutoff collision kernels. As for the associated Cauchy problem, we develop a theory of weak solutions and present some of its a priori estimates related with physical quantities including the energy and moments.
Shifted Laplace preconditioners have attracted considerable attention as
a technique to speed up convergence of iterative solution methods for the
Helmholtz equation. In the talk we present a comprehensive spectral
analysis of the discrete Helmholtz operator preconditioned with a shifted
Laplacian. Our analysis is valid under general conditions. The propagating
medium can be heterogeneous, and the analysis also holds for different types
of damping, including a radiation condition for the boundary of the computational
domain. By combining the results of the spectral analysis of the
preconditioned Helmholtz operator with an upper bound on the GMRES-residual
norm we are able to derive an optimal value for the shift, and to
explain the mesh-depency of the convergence of GMRES preconditioned
with a shifted Laplacian. We will illustrate our results with a seismic test
problem.
Joint work with: Yogi Erlangga (University of British Columbia) and Kees Vuik (TU Delft)
(IN: LADY MARGARET HALL)
As part of the Conference on Geometric Model Theory in honour of Professor Boris Zilber
( IN: LADY MARGARET HALL)
As part of the Conference on Geometric Model Theory in honour of Professor Boris Zilber
Abstract. Even in simple models in which thevolatility is only known to stay in two bounds, it is quite hard to price andhedge derivatives which are not Markovian. The main reason for thisdifficulty emanates from the fact that the probability measures are singular toeach other. In this talk we will prove a martingale representation theoremfor this market. This result provides a complete answer to the questionsof hedging and pricing. The main tools are the theory of nonlinearG-expectations as developed by Peng, the quasi-sure sto chastic artini and thesecond order backward stochastic differential equations.
This is jointwork with Nizar Touzi from Ecole Polytechnique and Jianfeng Zhang fromUniversity of Southern California.
• Amy Smith presents: “Multiscale modelling of coronary blood flow derived from the microstructure”
• Laura Gallimore presents: “Modelling Cell Motility”
• Jean-Charles Seguis presents: “Coupling the membrane with the cytosol: a first encounter”
This will not be a normal workshop with a single scientist presenting an unsolved problem where mathematics may help. Instead it is more of a discussion meeting with a few speakers all interested in a single theme. So far we have:
Lenny Smith (LSE) on Using Empirically Inadequate Models to inform Your Subjective Probabilities: How might Solvency II inform climate change decisions?
Dan Rowlands (AOPP, Oxford) on "objective" climate forecasting;
Tim Palmer (ECMWF and AOPP, Oxford) on Constraining predictions of climate change using methods of data assimilation;
Chris Farmer (Oxford) about the problem of how to ascertain the error in the equations of a model when in the midst of probabilistic forecasting and prediction.
We study dynamics from models for the human tear film in one and two dimensional domains.
The tear film is roughly a few microns thick over a domain on a centimeter scale; this separation of scales makes lubrication models desirable. Results on one-dimensional blinking domains are presented for multiple blink cycles. Results on two-dimensional stationary domains are presented for different boundary conditions. In all cases, the results are sensitive to the boundary conditions; this is intuitively satisfying since the tear film seems to be controlled primarily from the boundary and its motion. Quantitative comparison with in vivo measurement will be given in some cases. Some discussion of tear film properties will also be given, and results for non-Newtonian models will be given as available, as well as some future directions.