Max Xu : Random multiplicative functions and applications
- Probability ( 191 Views )Random multiplicative functions are probabilistic models for multiplicative arithmetic functions, such as Dirichlet characters or the Liouville function. In this talk, I will first quickly give an overview of the area, and then focus on some of the recent works on proving central limit theorems, connections to additive combinatorics, as well as some other deterministic applications. Part of the talk is based on joint work with Soundararajan, with Harper and Soundararajan (in progress) and with Angelo and Soundararajan (in progress).
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.
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.