Georgia Institute of Technology College of Sciences

It was shown by Keith Ball that the maximal section of an n-dimensional cube is \sqrt{2}. We show the analogous sharp bound for a maximal marginal of a product measure with bounded density. We also show an optimal bound for all k-codimensional marginals in this setting, conjectured by Rudelson and Vershynin. This bound yields a sharp small ball inequality for the length of a projection of a random vector. This talk is based on the joint work with G. Paouris and P. Pivovarov.

Numerical optimal transport is an important area of research, but most problems are too large and complex for easy computation. Because continuous transport problems are generally solved by conversion to either discrete or semi-discrete forms, I focused on methods for those two. I developed a discrete algorithm specifically for fast approximation with controlled error bounds: the general auction method. It works directly on real-valued transport problems, with guaranteed termination and a priori error bounds. I also developed the boundary method for semi-discrete transport. It works on unaltered ground cost functions, rapidly identifying locations in the continuous space where transport destinations change. Because the method computes over region boundaries, rather than the entire continuous space, it reduces the effective dimension of the discretization. The general auction is the first relaxation method designed for compatibility with real-valued costs and weights. The boundary method is the first transport technique designed explicitly around the semi-discrete problem and the first to use the shift characterization to reduce dimensionality. No truly comparable methods exist. The general auction and boundary method are able to solve many transport problems that are intractible using other approaches. Even where other solution methods exist, in testing it appears that the general auction and boundary method outperform them.

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.