Abstract:

The classical phase retrieval problem arises in contexts ranging from speech recognition to x-ray crystallography and quantum state tomography. The generalization to matrix frames that are U(r) phase retrievable is natural in the sense that it corresponds to quantum tomography of impure states. The U(r) phase retrieval problem is also essentially equivalent to low rank matrix recovery, and in particular to the Euclidean Distance Geometry problem. We provide computable global stability bounds for the quasi-linear analysis map β and a path forward for understanding related problems in terms of the differential geometry of the semi-algebraic variety of positive semi-definite matrices. In particular, we manifest a Whitney stratification of the PSD matrices of low rank, which allows us to “stratify” the computation of the global stability bound. We show that in contrast to the rank 1 case, for the impure state case no such global stability bounds can be obtained for the non-linear analysis map α with respect to certain natural distance metrics. Finally, our computation of the global lower Lipschitz constant for the β analysis map provides novel conditions for a matrix frame to be U(r) phase retrievable.  

Abstract: In this talk, I will discuss some key ideas and methods of two disparate fields of mathematical research, frame theory and optimal transport, using the methods of the second to answer questions posed in the first. In particular, I will discuss construction of gradient flows in the Wasserstein space for a new potential, the tightness potential, which is a modification of the probabilistic frame potential. The potential is suited for the application of a gradient descent scheme from optimal transport that can be used as the basis of an algorithm to evolve a given frame toward a tight probabilistic frame.

Abstract:

This talk will review models and structural properties of the
equations used to model geophysical flows which involve multiple
components undergoing phase transitions. Simulations of these problems
only model the gross properties of these flows since a precise
description of the physical system is neither available nor
computationally tractable. In this context mathematics is essential
if phenomenology, physical intuition, mechanics, thermodynamics, and
thought experiments, are to be integrated into well posed models.
Numerical schemes which inherit the essential structural and physical
properties of the underlying models can then be developed.

Abstract:

Cellular networks are ubiquitous in nature. Most technologically useful
materials arise as polycrystalline microstructures, composed of a myriad
of small monocrystalline cells or grains, separated by interfaces, or
grain boundaries of crystallites with different lattice orientations. A
central problem in materials science is to develop technologies capable of
producing an arrangement of grains that provides for a desired set of
material properties. One method by which the grain structure can be
engineered is through grain growth (also termed coarsening) of a starting
structure.

The evolution of grain boundaries and associated grain growth is a very
complex multiscale process. It involves, for example, dynamics of grain
boundaries, triple junctions, and the dynamics of lattice misorientations.
Grain growth can be viewed as the evolution of a large metastable network,
and can be mathematically modeled by a set of deterministic local
evolution laws for the growth of an individual grain combined with
stochastic models to describe the interaction between them. In this talk,
we will discuss recent progress in mathematical modeling, simulation and
analysis of the evolution of the grain boundary network in polycrystalline
materials.