Date | Speaker | Topic |
---|---|---|

M Sep 12 | – | Organizational Meeting |

M Sep 19 | Chris Dock (Tufts) | Lipschitz stability of U(r) Phase Retrieval
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. |

M Sep 26 | Clare Wickman Lau (JHU APL) | Wasserstein Gradient Flows for Potentials in Frame Theory
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. |

M Oct 3 |
Noel Walkington (CMU) | Numerical Approximation of Multiphase Flows in Porous Media
Abstract: This talk will review models and structural properties of the |

T Oct 11 4pm on zoom |
Erin C. Munro Krull (Ripon College) |
Why strengthening electronic connections between cells may hinder propagation between them. Gap junctions are channels that connect cell membranes allowing electric ions to pass directly between cells. They connect cells throughout the body, including heart myocytes, neurons, and astrocytes. Voltage propagation mediated by gap junctions can be passive or active. In passive propagation, the voltage of one cell affects the voltage of neighboring cells without triggering action potentials (APs). In active propagation, an AP in one cell triggers APs in neighboring cells; this occurs in cardiac tissue and throughout the nervous system. It is known experimentally that there is an ideal gap junction conductance for AP propagation — weaker or stronger conductance can block propagation. We present a theory explaining this phenomenon by analyzing an idealized model that focuses exclusively on gap junctional and spike-generating currents. We also find a novel type of behavior that we call semi-active propagation, where cells in the network are so strongly connected that they are not excitable at rest, but still propagate action potentials. |

M Oct 17 | Yekaterina Epshteyn (University of Utah) | Grain Structure, Grain Growth and Evolution of the Grain Boundary Network
Abstract: Cellular networks are ubiquitous in nature. Most technologically useful The evolution of grain boundaries and associated grain growth is a very |

M Oct 24 | Karamatou Yacoubou Djima (Wellesley) | Extracting Autism's Biomarkers in Placenta Using Multiscale Methods
Abstract:
The placenta is the essential organ of maternal-fetal interactions, where nutrient, oxygen, and waste exchange occur. In recent studies, differences in the morphology of the placental chorionic surface vascular network (PCSVN) have been associated with developmental disorders such as autism. This suggests that the PCSVN could potentially serve as a biomarker for the early diagnosis and treatment of autism. Studying PCSVN features in large cohorts requires a reliable and automated mechanism to extract the vascular networks. In this talk, we present a method for PCSVN extraction. Our algorithm builds upon a directional multiscale mathematical framework based on a combination of shearlets and Laplacian eigenmaps and can isolate vessels with high success in high-contrast images such as those produced in CT scans. |

M Oct 31 11am-12pm JCC 610 | Hung M. Phan (UMass Lowell) | Splitting algorithms and applications in spatial design problems.
Abstract: In computer graphic applications or design problems, a three-dimensional object is often represented by a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy spatial constraints that are imposed either by observation from the real world, or by concrete design specifications of the object. In this talk, we model various geometric constraints as convex sets in Euclidean spaces. We then apply splitting algorithms in these constructions to find optimal solutions. |

M Nov 7 | Patrick Farrell (Oxford) | Optimal-complexity multigrid Solvers for high-order discretisations of the de Rham Complex
Abstract: The Riesz maps of the spaces H^1, H(curl) and H(div) frequently arise as |

M Nov 14 | Arkadz Kirshtein (Tufts) | Biomedical and Physical Applied Modeling
Abstract: |

M Nov 21 | Radu Balan (UMD) |
Optimal l1 factorizations of positive semidefinite matrices Abstract: Among infinitely many factorizations A=VV* of a psd matrix A, we seek the factor V that has the smallest (1,2) norm. In this talk we review the origin of this problem as well as existing results regarding the optimal value. We prove two new results describing two properties of the optimal factorization: full continuity over the set of psd matrices (previously, the continuity was shown over strictly positive matrices), and equivalence to a linear program over the space of Borel measures. We discuss also the conjecture that the squared (1,2) norm of V is equivalent to the (1,1) norm of A (a.k.a. the projective norm). |

M Nov 28 | Tal Shnitzer (MIT) | Diffusion Operators for Data Fusion with Symmetric Positive Definite Geometry
Due to recent technological advances, multi-modal data have become abundant as many data-acquisition systems record information through multiple sensors simultaneously. Coupled with an unknown system model and high data dimensions, such data poses a significant challenge for common analysis techniques. We approach this problem from a geometric perspective, using manifold learning to recover the underlying data structures, yet, comparing and combining such representations is non-trivial. I will discuss two recent approaches to data fusion, with known and unknown dataset alignment, in which we exploit the Riemannian geometry of symmetric positive definite matrices to compare diffusion operators, a manifold learning tool. |

M Dec 5 | Canceled |