MC 20 Titles and Abstracts
Numerical Methods for Stochastic Differential Equations, withApplications in Neuroscience
Abstract: In this talk I will provide some background on stochastic differentialequations and on how to develop and analyse numerical methods forsimulating solutions. To illustrate the ideas, I will discuss some applications in neuroscience.
Algebraic Geometry, Numerical Linear Algebra and Structure from X
The structure from motion problem in computer vision is the problem of determining camera position and orientation as well as the 3D positions of scene features using the motion of image features only. The analogous problem for audio and radio is the problem of determining sender and receiver positions using the received audio or radio signal only. For both video, audio and radio there are a number of challenges, e.g. feature detection, robust feature matching and robust parameter estimation. The problem is challenging because of the non-linear nature of the problem. A key component of such systems are so called minimal problems. Methods that combine algebraic geometry, numerical linear algebra are key in solving such problems. In the talk a summary of results of both theoretical and applied nature is presented.
Close interactions of drops and particles in viscous flow
Integral equation based numerical methods are attractive for thesimulations of fluid mechanics at the micro scale such as in droplet-based microfluidics, with tiny water drops dispersed in oil,
stabilized by surfactants. We have developed highly accuracte numerical methods for drops with insoluble surfactants, both in two and three dimensions with the latter recently extended to include also electric fields.
This involves addressing several fundamental challenges that are highly relevant also to other applications: accurate quadrature methods for singular and nearly singular integrals, adaptive
time-stepping, and reparameterization of time-dependent surfaces for high quality discretization of the drops throughout the simulations. In this talk, particular emphasis will be on quadrature methods applied to the evaluation of nearly singular layer potentials inlcuding error estimates and their use for adaptive parameter selection.
The additive structure of the spectra of quantum graphs and discrete measures
It is proven that the spectrum of the Laplacian on a metric graph Γ contains arithmetic sequences if and only if the graph has a loop – an edge connected to one vertex by both end points. Moreover the length of the longest possible arithmetic subsequence is estimated using the corresponding discrete graph G. Our main tool is diophantine analysis, specifically ”Lang’s Gm Conjecture” concerning the intersection of the division group of a finitely generated subgroup of (C ∗ ) N with a subvariety of (C ∗ ) N . On our way we prove recent Colin de Verdi´er’s Conjecture concerning structure of polynomials associated with metric graphs. The trace formula connecting spectra of standard Laplacians on metric graphs to the sets of periodic orbits allows us to construct a large family of exotic crystalline measures, studied recently by Y. Meyer. Crystalline measures are discrete measures with Fourier transform being a discrete measure as well. Our analysis in the first part imply that constructed measures are not just combinations of Poisson summation formulae. This is a joint work with Peter Sarnak.
Mathematics in Water Waves: An overview of questions asked and problems answered in the analytic field of dispersive equations
We give a brief overview of the fundamentals of mathematics for water waves. Starting from a linear case, what governs the evolution of a wave? What is the relationship between a travelling and a time-dependent solution? We see how linear theory alone is insufficient in this setting, as it neither explains or captures how waves break. That naturally leads us to John Scott Russell, the concept of a solitary wave, and equations for water waves that better balance the effects of nonlinearity and dispersion. After a detour to the Euler equations and their relationship to model equations, and an oversight of problems typically asked for these equations, we finally turn to a specific instance – the Whitham equation – showing how a seemingly simple-looking equation capture several of the features of the full water-wave problem.
Chaos measures, random matrices, and the Riemann zeta function
In the talk we will introduce the notion of Gaussian random multiplicative chaos and try to explain what is has to do with random matrices and the Riemann zeta function.