Georgia Institute of Technology College of Sciences

Alexandru Oancea: Title: Symplectic homology for cobordisms Abstract: Symplectic homology for a Liouville cobordism - possibly filled at the negative end - generalizes simultaneously the symplectic homology of Liouville domains and the Rabinowitz-Floer homology of their boundaries. I will explain its definition, some of its properties, and give a sample application which shows how it can be used in order to obstruct cobordisms between contact manifolds. Based on joint work with Kai Cieliebak and Peter Albers. Basak Gürel: Title: From Lusternik-Schnirelmann theory to Conley conjecture Abstract: In this talk I will discuss a recent result showing that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree two with positive symplectic area and positive integral of the first Chern class. This theorem encompasses all known cases of the Conley conjecture (symplectic CY and negative monotone manifolds) and also some new ones (e.g., weakly exact symplectic manifolds with non-vanishing first Chern class). The proof hinges on a general Lusternik–Schnirelmann type result that, under some natural additional conditions, the sequence of mean spectral invariants for the iterations of a Hamiltonian diffeomorphism never stabilizes. Based on joint work with Viktor Ginzburg.

One dimensional discrete Schrödinger operators arise naturally in modeling the motion of quantum particles in a disordered medium. The medium is described by potentials which may naturally be generated by certain ergodic dynamics. We will begin with two classic models where the potentials are periodic sequences and i.i.d. random variables (Anderson Model). Then we will move on to quasi-periodic potentials, of which the randomness is between periodic and i.i.d models and the phenomena may become more subtle, e.g. a metal-insulator type of transition may occur. We will show how the dynamical object, the Lyapunov exponent, plays a key role in the spectral analysis of these types of operators.

A classical theorem of Arnold, Moser shows that in analytic families of maps close to a rotation we can find maps which are smoothly conjugate to rotations. This is one of the first examples of the KAM theory. We aim to present an efficient numerical algorithm, and its implementation, which approximate the conjugations given by the Theorem

A conversation with Adam Fox, former GT postdoc who secured his "dream job" as a tenure-track assistant professor at Western New England University, but who recently moved into industry as a Data Scientist.