Tuesday, January 23, 2018
01/23/2018 - 4:00pm to 5:00pm
Cryo-electron microscopy (cryo-EM) is an imagingtechnology that is revolutionizing structural biology, enabling reconstruction of molecules at near-atomic resolution. Cryo-EM produces a large number of very noisy two-dimensional tomographic projection images of a molecule, taken at unknown viewing directions. The extreme levels of noise make classical tasks in statistics and signal processing, such as alignment, detection and classification, impossible. I will start the talk by studying the multi-reference alignment problem, which can be interpreted as a simplified model for cryo-EM. In multi-reference alignment, we aim to estimate multiple signals from their circularly-translated, unlabeled, copies. In high noise regimes, the measurements cannot be alignedor clustered. Nonetheless, accurate estimation can be achieved using non-convex optimization by considering the invarariants of the problem. Then, I will introduce the analog invariants of the cryo-EM problem and focus on two applications, called 2D classification and ab initio modeling.
01/23/2018 - 4:15pm to 5:15pm
This talk will illustrate some patterns in the homology of the configuration space Fk(M), the space of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expectthe topology of these configuration spaces to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which the homology groups of these spaces stabilize. In this talk I will explain these stability patterns, and describe higher-order stability phenomena -- relationships between unstable homology classes in different degrees -- establishedin recent work joint with Jeremy Miller. This project was inspired by work-in-progress of Galatius--Kupers--Randal-Williams.