Alex Blumenthal : Chaotic regimes for random dynamical systems
- Probability ( 154 Views )It is anticipated that chaotic regimes (e.g., strange attractors) arise in a wide variety of dynamical systems, including those arising from the study of ensembles of gas particles and fluid mechanics. However, in most cases the problem of rigorously verifying asymptotic chaotic regimes is notoriously difficult. For volume-preserving systems (e.g., incompressible fluid flow or Hamiltonian systems), these issues are exemplified by coexistence phenomena: even in quite simple models which should be chaotic, e.g. the Chirikov standard map, completely opposite dynamical regimes (elliptic islands vs. hyperbolic sets) can be tangled together in phase space in a convoluted way. Recent developments have indicated, however, that verifying chaos is tractable for systems subjected to a small amount of noise— from the perspective of modeling, this is not so unnatural, as the real world is inherently noisy. In this talk, I will discuss two recent results: (1) a large positive Lyapunov exponent for (extremely small) random perturbations of the Chirikov standard map, and (2) a positive Lyapunov exponent for the Lagrangian flow corresponding to various incompressible stochastic fluids models, including stochastic 2D Navier-Stokes and 3D hyperviscous Navier-Stokes on the periodic box. The work in this talk is joint with Jacob Bedrossian, Samuel Punshon-Smith, Jinxin Xue and Lai-Sang Young.
Lei Li : Some algorithms and analysis for first order interacting particle systems
- Applied Math and Analysis ( 112 Views )We focus on first order interacting particle systems, which can be viewed as overdamped Langevin equations. In the first part, we will look at the so-called random batch methods (RBM) for simulating the interacting particle systems. The algorithms are motivated by the mini-batch idea in machine learning. For some special cases, we show the convergence of RBMs for the first marginal under Wasserstein distance. In the second part, we look at the Coulomb interaction in 3D space. We show that as the number of particles go to infinity, almost surely, the empirical measure converges in law to weak solutions of the limiting nonlinear Fokker-Planck equation. This talk is based on joint works with Shi Jin (Shanghai Jiao Tong), Jian-Guo Liu (Duke University) and Pu Yu (Peking University).
Terry Loring : Multivariate pseudospectrum and topological physics
- Applied Math and Analysis ( 101 Views )The usual pseudospectrum acquires an additional feature when restricted to matrices with a certain symmetry. The new feature is a simple form of K-theory which can be used to compute the index of some one-dimensional topological insulators. The usual pseudospectrum applies to a single matrix, or equivalently to two Hermitian matrices. Generalized to apply to more Hermitian matrices, the nature of the pseudospectrum changes radically, often having interesting geometry. Examples come from D-branes and higher-dimensional topological insulators. The algorithm to compute the pseudospectrum also produces common approximate eigenvectors for a collection of almost commuting Hermitian matrices. Applied to a basic model of a finite volume topological insulator it produces vectors that are approximately stationary and somewhat localized in position.
Jimmy He : Shift invariance of half space integrable models
- Probability ( 76 Views )I'll discuss work on shift invariance in a half space setting. These are non-trivial symmetries allowing certain observables of integrable models with a boundary to be shifted while preserving their joint distribution. The starting point is the colored stochastic six vertex model in a half space, from which we obtain results on the asymmetric simple exclusion process, as well as for the beta polymer through a fusion procedure, both in a half space setting. An application to the asymptotics of a half space analogue of the oriented swap process is also given.