Georgia Institute of Technology College of Sciences

I will discuss joint work with Hutchings which gives a rigorousconstruction of cylindrical contact homology via geometric methods. Thistalk will highlight our use of non-equivariant constructions, automatictransversality, and obstruction bundle gluing. Together these yield anonequivariant homological contact invariant which is expected to beisomorphic to SH^+ under suitable assumptions. By making use of familyFloer theory we obtain an S^1-equivariant theory defined with coefficientsin Z, which when tensored with Q recovers the classical cylindrical contacthomology, now with the guarantee of well-definedness and invariance. Thisintegral lift of contact homology also contains interesting torsioninformation.

This talk addresses an important problem in arctic engineering due to interesting dynamic phenomena in a forced linear system. The nonautonomous system considered is representative of a whole class of engineering problems that are not approachable by standard techniques from dynamical system theory.The background are ice-induced vibrations of structures (e.g. wind turbines or measurement masts) in regions with active sea ice. Ice is a complex material and the mechanism for ice-induced vibrations is not fully clear at present. In particular, the conditions under which the observed, qualitatively different vibration regimes are active cannot be predicted accurately so far. A recent mathematical model developed by Delft University of Technology assumes that a number of parallel ice strips are pushing with a constant velocity against a flexible structure. The structure is modelled as a single degree of freedom harmonic oscillator. The contact force acts on the structure, but at the same time slows down the advancement of the ice, thereby introducing a dynamic nonlinearity in the otherwise linear system. When the local contact force becomes large enough, the ice crushes and the corresponding strip is reset to a random offset in front of the structure.This is the first mathematical model that exhibits all three different dynamic regimes that are observed in reality: for slow ice velocities the structure undergoes quasi-static sawtooth responses where all ice strips fail at the same time (a kind of synchronization phenomenon), for large ice velocities the structure response appears random, and for intermediate ice velocities the system exhibits vibrations at the structure eigenfrequency, commonly called frequency lock-in behavior. The latter type of vibrations causes a lot of damage to the structure and poses a safety and economic risk, so its occurrence needs to be predicted accurately.As I will show in this talk, the descriptive terms for the three vibration regimes are slightly misleading, as the mechanisms behind the observed behaviors are somewhat different than intuition suggests. I will present first results in analyzing the system and offer some explanations of the observed behaviors, as well as some simple criteria for the switch between the different vibration regimes.

A conversation with Dan Margalit, GT math professor and inaugural CoS Leddy Family Faculty Fellow, who was a tenure-track assistant professor at Tufts University for two years prior to coming to Tech.

Networks, or graphs, can represent a great variety of systems in the real world including neural networks, power grid, the Internet, and our social networks. Mathematical models for such systems naturally reflect the graph theoretical information of the underlying network. This talk explores some common themes in such models from the point of view of systems of nonlinear equations.