Events for 04/12/2024 from all calendars

Mathematical Physics and Harmonic Analysis Seminar

Time: 1:50PM - 2:50PM

Location: BLOC 302

Speaker: Wei Li, DePaul University

Title: Edge States on Sharply Joined Photonic Crystals

Abstract: Edge states are important in transmitting information and transporting energy. We investigate edge states in continuous models of photonic crystals with piecewise constant coefficients, which are more realistic and controllable for manufacturing optical devices. First, we show the existence of Dirac points on honeycomb structures with suitable symmetries. Then we show that when perturbed in two appropriate ways, the perturbed honeycomb structures have a common band gap, and when joined along suitable interfaces, there exist edge states which propagate along the interfaces and exponentially decay away from the interfaces. The main tools used are layer potentials, asymptotic analysis, the Gohberg-Sigal theory and Lyapunov-Schmidt reductions.

This is joint work with Junshan Lin, Jiayu Qiu, Hai Zhang.

Geometry Seminar

Time: 4:00PM - 5:00PM

Location: BLOC 302

Speaker: Brandon Ashley, Southern Oregon University

Title: A Group-Theoretic Approach to Darboux Integrable $f$-Gordon Equations

Abstract: Classically, a partial differential equation is said to be Darboux integrable if its general solution can be found by only solving a system of ordinary differential equations. In this talk, we describe a transformation group-theoretic approach to the study of Darboux integrable equations and highlight how this approach can be used to solve the equivalence problem for so-called $f$-Gordon equations of the form $u_{xy} = f(x,y,u,u_x,u_y)$.