Georgia Institute of Technology College of Sciences

We investigate the nonlinear stability of elementary wave patterns (such as shock, rarefaction wave and contact discontinuity, etc) for bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the elementary wave patterns to the system. Then, the time-asymptotic stability of the planar rarefaction wave, viscous shock waves and viscous contact wave (viscous version of contact discontinuity) are proved for the 1D bipolar Vlasov-Poisson-Boltzmann system. These results imply that these basic wave patterns are still stable in the transportation of charged particles under the binary collision, mutual interaction, and the effect of the electrostatic potential force. The talk is based on the joint works with Hailiang Li (CNU, China), Tong Yang (CityU, Hong Kong) and Mingying Zhong (GXU, China).

We study the existence and branching patterns of wave trains in a two-dimensional lattice with linear and nonlinear coupling between nearest particles and a nonlinear substrate potential. The wave train equation of the corresponding discrete nonlinear equation is formulated as an advanced-delay differential equation which is reduced by a Lyapunov-Schmidt reduction to a finite-dimensional bifurcation equation with certain symmetries and an inherited Hamiltonian structure. By means of invariant theory and singularity theory, we obtain the small amplitude solutions in the Hamiltonian system near equilibria in non-resonance and $p:q$ resonance, respectively. We show the impact of the direction $\theta$ of propagation and obtain the existence and branching patterns of wave trains in a one-dimensional lattice by investigating the existence of travelling waves of the original two-dimensional lattice in the direction $\theta$ of propagation satisfying $\tan\theta$ is rational

This talk is an oral comprehensive exam in partial fulfillment of the requirements for a doctoral degree. To any topological surface we can assign a certain communtative algebra called a cluster algebra. A surface cluster algebra naturally records the geometry of the surface. The algebra is generated by arcs of the surface. Arcs carry a simplicial structure where the maximal simplices are triangulations. If you squint you can view a surface cluster algebra as a coordinate ring of decorated Teichmuller space with Penner's coordinate. Recent work from many authors has shown that the automorphisms of the surface cluster algebra which preserve triangulations arise from the mapping class group of the surface. But there are additional automorphisms that preserve meaningful structure of the cluster algebra. In this talk we will define surface cluster algebras and discuss future research toward understanding structure preserving automorphisms.

Science explains through systematic inquiry; magic celebrates that which defies explanation. This will be tour of sorts along the boundary between science and magic. We will explore the magic of quantum mechanics, the predictions of knot theory, and randomness, as well as the mysteries of the number 58008.