Georgia Institute of Technology College of Sciences

One of the simplest and most natural ways of representing geometry and information in three and higher dimensions is using point clouds, such as scanned 3D points for shape modeling and feature vectors viewed as points embedded in high dimensions for general data analysis. Geometric understanding and analysis of point cloud data poses many challenges since they are unstructured, for which a global mesh or parametrization is difficult if not impossible to obtain in practice. Moreover, the embedding is highly non-unique due to rigid and non-rigid transformations. In this talk, I will present some of our recent work on geometric understanding and analysis of point cloud data. I will first discuss a multi-scale method for non-rigid point cloud registration based on the Laplace-Beltrami eigenmap and optimal transport. The registration is defined in distribution sense which provides both generality and flexibility. If time permits I will also discuss solving geometric partial differential equations directly on point clouds and show how it can be used to “connect the dots” to extract intrinsic geometric information for the underlying manifold.

The Caffarelli-Kohn-Nirenberg inequalities form a two parameter family of inequalities that interpolate between Sobolev's inequality and Hardy's inequality. The functional whose minimization yields the sharp constant is invariant under rotations. It has been known for some time that there is a region in parameter space where the optimizers for the sharp constant are {\it not} radial. In this talk I explain this and related problems andindicate a proof that, in the remaining parameter region, the optimizers are in fact radial. The novelty is the use of a flow that decreases the functional unless the function is a radial optimizer. This is joint work with Jean Dolbeault and Maria Esteban.

In the world of Hamiltonian partial differential equations, complete integrability is often associated to rare and peaceful dynamics, while wave turbulence rather refers to more chaotic dynamics. In this talk I will first try to give an idea of these different notions. Then I will discuss the example of the cubic Szegö equation, a nonlinear wave toy model which surprisingly displays both properties. The key is a Lax pair structure involving Hankel operators from classical analysis, leading to the inversion of large ill-conditioned matrices. .

The recent interest in network modeling has been largely driven by the prospect that network optimization will help us understand the workings of evolution in natural systems and the principles of efficient design in engineered systems. In this presentation, I will reflect on unanticipated properties observed in three classes of network optimization problems. First, I will discuss implications of optimization for the metabolic activity of living cells and its role in giving rise to the recently discovered phenomenon of synthetic rescues. I will then comment on the problem of controlling network dynamics and show that theoretical results on optimizing the number of driver nodes often only offer a conservative lower bound to the number actually needed in practice. Finally, I will discuss the sensitive dependence of network dynamics on network structure that emerges in the optimization of network topology for dynamical processes governed by eigenvalue spectra, such as synchronization and consensus processes. It follows that optimization is a double-edged sword for which desired and adverse effects can be exacerbated in network systems due to the high dimensionality of their phase spaces.