Georgia Institute of Technology College of Sciences

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.

Reaction diffusion equations are a common tool in mathematical biology, and are used in diverse fields such as ecology, epidemiology and developmental biologyI will show some examples of reaction diffusion equations and what their solutions look like. I will focus on the problem of pattern formation during development and the mathematics that underly it, a problem first studied by Alan Turing more than 60 years ago. I will present a basic example that we can solve together using techniques from differential equations and linear algebra.