Fall 2018

Organizer : Dr. Yongki Lee and Dr. Yuanzhen Shao

• • Date: Thursday, November 29

  • Time: 4:00 – 5:00 PM 

  • Location: Math/Physics Bldg 3028

  • Speaker: Dr. Xianghong Chen

  • Title: Spectral expansion on spheres: A borderline case

  • Abstract: 

    • • I will report on a recent joint work with Dashan Fan and Juan Zhang on spherical harmonic expansion. It is known that the corresponding partial sum expansion diverges for general integrable functions on spheres, due to superposition of eigenfunctions with close frequencies. However, convergence can be restored using summation methods of sufficiently high order. It is known that a critical order of such a summation method exists. At the critical order, Christ and Sogge (1988) showed that convergence holds in measure. We examine this borderline case and show that, on the other hand, almost everywhere divergence can happen. This can be regarded as a generalization of Kolmogorov's result for Fourier series on the unit circle.

• Joint talk with Department Colloquium 

  • Date: Friday, November 16

  • Time: 3:30 – 4:30 PM 

  • Location: Math/Physics Bldg 3314 

  • Speaker: Dr. Wen Feng(University of Kansas)

  • Title: Stability of Vortex Solitons for n-dimensional focusing NLS

  • Abstract: 

    • • • • We consider the nonlinear Schr ̈odinger equation in n space dimension

          • iut +△u+|u|p−1u=0, x∈Rn, t>0

          • and study the existence and stability of standing wave solutions of the form

          • eiwtei kj=1 mjθjφw(r1,r2,··· ,rk), n = 2k eiwtei kj=1mjθjφw(r1,r2,···,rk,z), n=2k+1

          • for n = 2k, (rj,θj) are polar coordinates in R2, j = 1,2··· ,k; for n = 2k + 1, (rj,θj) are polar coordinates in R2, (rk, θk, z) are cylindrical coordinates in R3, j = 1, 2 · · · , k − 1. We show the existence of such solutions, which are constructed variationally as minimizers of appropriate constrained functionals. These waves are shown to be spectrally stable (with respect with perturbations of the same type), if 1 < p < 1 + 4/n. 

• Joint talk with Department Colloquium 

  • Date: Wednesday, November 7 

  • Time: 3:30 – 4:30 PM 

  • Location: Math/Physics Bldg 3314 

  • Speaker: Dr. Shijun Zheng 

  • Title: Long Time Behavior for Hamiltonian Evolutionary Quantum Waves

  • Abstract: I will present an introduction to some basic models arising in conservation laws from quantum mechanics. Then, in the second half, I will show how to use Virial identity, a spatial-multiplier method, to obtain asymptotic behaviors and their stability for a class of dispersive equations including nonlinear rotational Schroedinger equation, fractional Hartree equation as well as Dirac equation.