MenuMathematical Physics and Harmonic Analysis Seminar
Fall 2018
| Date: | September 7, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Irene Gamba, UT Austin |
| Title: | The Cauchy problem and BEC stability for the quantum Boltzmann-Condensation System at very low temperature |
| Abstract: | We discuss a quantum Boltzmann-Condensation system that describes the evolution of the interaction between a well formed Bose-Einstein Condensate (BEC) and the quasi-particles cloud. The kinetic model, derived as weak turbulence kinetic model from a quantum Hamiltonian, is valid for a dilute regime at which the temperature of a bosonic gas is very low compared to the Bose-Einstein condensation critical temperature. In particular, the system couples the density of the condensate from a Gross-Pitaevskii type equation to the kinetic equation through the dispersion relation in the kinetic model and the corresponding transition probability rate from pre to post collision momentum states. We show the well-posedness of the Cauchy problem for the system, find qualitative properties of the solution such as instantaneous creation of exponential tails, and prove the uniform condensate stability related to the initial mass ratio between condensed particles and quasi-particles. This stability result leads to global in time existence of the initial value problem for the quantum Boltzmann-Condensation system. |
| Date: | September 14, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Joonhyun La, Princeton University |
| Title: | Global well posedness of 2D diffusive Fokker-Planck-Navier-Stokes systems |
| Abstract: | In this talk, we prove that there is a unique global strong solution to the 2D Navier-Stokes system coupled with diffusive Fokker-Planck equation of a Hookean type potential. This system regards a polymeric fluid as a dilute suspension of polymers in an incompressible solvent, which is governed by the Navier-Stokes equation, and distribution of polymer configuration is governed by the Fokker-Planck equation, where spatial diffusion effects of polymers are also considered. Well-known Oldroyd-B models can be rewritten in the form of this system. Main conceptual difficulties include multi-scale nature of the system. We discuss an appropriate notion for the solution for this multi-scale system, and approximation scheme. |
| Date: | September 21, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Ziad Musslimani, Florida State University |
| Title: | PT symmetry, nonlocal integrable models and physical applications |
| Abstract: | In this talk, we shall review basic concepts related to the mathematics and physics of PT symmetry and non-self-adjoint eigenvalue problems. We shall also discuss recent activities in the newly emerging field of PT symmetric and reverse space-time integrable nonlocal models. |
| Date: | September 28, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Maciej Zworski, UC Berkeley |
| Title: | Magnetic oscillations in a model of graphene |
| Abstract: | We consider the simplest model for graphene in a magnetic field given by a hexagonal quantum graphs. Using semiclassical methods (with the strength of the magnetic field as the small parameter) we obtain a geometric description of the density of states showing asymmetry seen in physical experiments but not in commonly used perfect cone approximations. That density of states can then be used to see magnetic oscillations such as the de Haas--van Alphen effect. Joint work with S Becker. |
| Date: | October 5, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Lior Alon, Technion --- Israel Institute of Technology |
| Title: | Nodal and Neumann count statistics for quantum graphs |
| Abstract: | In this talk I will briefly go over the definitions and results from the work on the nodal count statistics on quantum graphs. Then I will introduce the concept of Neumann count, and properties of Neumann domains such as the spectral position of the restricted eigenfunction and analogous property to the area to length ratio (isoperimetric parameter). I will then state the results regarding the existence and symmetry of the probability distributions of the latter properties. If time allows I will present a simple but powerful result regarding the edge lengths dependence of the nodal and Neumann distributions for edge transitive combinatorial graphs, and I will finish with our latest results, showing that the nodal distributions for two specific graphs families converge to Gaussian distributions as the the number of edges grows to infinity. This talk is base on a joint work with R. Band (Technion) and G. Berkolaiko (Texas A&M). |
| Date: | October 19, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Mathew Johnson, University of Kansas |
| Title: | On the Stability of Roll Waves |
| Abstract: | Roll-waves are a well observed hydrodynamic instability occurring in inclined thin film flow, often mathematically described as periodic traveling wave solutions of either the viscous or inviscid St. Venant system. In this talk, I will discuss recent progress concerning the stability of both viscous and, if time allows, inviscid roll-waves in a variety of asymptotic regimes, including near the onset of hydrodynamic instability and large-Froude number analysis. This is joint work with Blake Barker (BYU), Pascal Noble (University of Toulouse), L. Miguel Rodrigues (University of Rennes), Zhao Yang (IU) and Kevin Zumbrun (IU). |
| Date: | October 26, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Dylan Allegretti, University of Sheffield |
| Title: | Categorified canonical bases and framed BPS states |
| Abstract: | In a famous paper from 2006, Fock and Goncharov introduced a moduli space of framed PGL(2,C)-local systems on a surface with boundary. This moduli space has the structure of a cluster variety, and the algebra of regular functions on this cluster variety has a canonical vector space basis. In this talk, I will describe a family of graded vector spaces which categorify Fock and Goncharov's canonical basis. In certain cases, these vector spaces arise as the cohomology of moduli spaces of stable quiver representations as predicted by the physics of BPS states in N=2 field theories. |
| Date: | November 2, 2018 |
| Time: | 11:00am |
| Location: | BLOC 628 |
| Speaker: | Anton Dzhamay, University of Northern Colorado |
| Title: | Geometry of Discrete Integrable Systems |
| Abstract: | Many interesting examples of discrete integrable systems can be studied from the geometric point of view. In this talk we will consider two classes of examples of such system: autonomous (QRT maps) and non-autonomous (discrete Painlevé equations). We introduce some geometric tools to study such systems, such as the blowup procedure to construct algebraic surfaces on which the mappings are regularized, linearization of the mapping on the Picard lattice of the surface and, for discrete Painlevé equations, the decomposition of the Picard lattice into complementary pairs of the surface and symmetry sublattices and construction of a binational representation of affine Weyl symmetry groups that gives a complete algebraic description of our non-linear dynamic. If time permits, we also explain the relationship between this picture and classical differential Painlevé equations. |
| Date: | November 16, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Jari Taskinen, University of Helsinki |
| Title: | Structure and existence of gaps of essential spectra of elliptic boundary problems in periodic waveguides |
| Abstract: | We review some recent joint works with Sergei Nazarov and others concerning spectral elliptic boundary value problems in periodic waveguides. We consider the structure of the essential spectrum e.g. for the Neumann Laplacian in the case of a doubly periodic perforated plane subject to periodic or non-periodic, non-compact perturbations. We also deal with the existence, number and position of spectral gaps in the case of the elasticity and piezoelectricity systems in waveguides with thin structures. |
