# Mathematics Colloquia and Seminars

Over the past several years, the study of Bose-Einstein condensates (BECs) has become one of the most important areas of atomic and molecular physics. Their study has begun to yield an increased understanding of superfluidity and superconductivity, and their eventual engineering applications also hold great promise. In this talk, I will discuss my recent research on the macroscopic dynamics of coherent structures in BECs loaded into lattice and superlattice potentials, for which I employ methods from dynamical systems and perturbation theory. The macroscopic behavior of BECs near zero temperature is modeled very well by the Gross-Pitaevskii (GP) equation, which is a time-dependent nonlinear Schr\"{o}dinger equation with an external potential. In the case of (periodic) lattice potentials, I will discuss the use of Hamiltonian perturbation theory to study spatial resonances in coherent structure solutions of the GP equation. I thereby derive analytical estimates of the size of resonance bands and the location in phase space of period-multiplied'' wavefunctions, whose periods are integer multiples of the underlying lattice period. I will also briefly present recent work on multiple-component BECs and BECs loaded into (periodic and quasiperiodic) superlattice potentials. I will summarize with a discussion of some of my current research efforts.