Abstract:  In this talk, I will discuss a couple general mathematical frameworks that have arisen in the study of the cryo-electron microscopy problem.  I will start by recalling that cryo-EM is about determining the 3D shape of a protein from many noisy tomographic images of randomly oriented particles.  Motivated by this application, I will mostly focus on the use of non-Euclidean norms in manifold learning.  Synthetic results show, that when applying diffusion maps to volumetric data, it can be advantageous to base affinities on the Earth Mover’s Distance rather the Euclidean norm.  I will characterize the limiting differential operator that we obtain when we use non-Euclidean norms, and contrast this to the classical Laplace-Beltrami operator.  Time permitting, I may also mention the problem of parameter estimation in statistics in the presence of group actions, and some unexpected connections to invariant theory.  Based on joint works with several coauthors who will be named precisely during the talk.

Abstract:

Thresholds for increasing properties are a central concern in probabilistic combinatorics and elsewhere. (An increasing property, say F, is a superset-closed family of subsets of some [here finite] set X, and the “threshold question” for F asks, roughly, about how many random elements of X should one choose to make it likely that the resulting set lies in F?  For example:  about how many random edges from the complete graph on n vertices are typically required to produce a Hamiltonian cycle?)

We will try to give some sense of this area and then focus on a few recent highlights.

See Algebra and Geometry lecture series (yale.edu)

Abstract: 

If X is a del Pezzo surface and D is a smooth anti-canonical divisor, we can regard the complement X\D as a non-compact Calabi-Yau surface.  I will discuss a proof of a strong form of the Strominger-Yau-Zaslow mirror symmetry conjecture for these non-compact surfaces.  It turns out the mirror Calabi-Yau is a rational elliptic surface (in particular, it has an elliptic fibration onto P^1) with a singular fiber which is a chain of nodal spheres.  I will discuss how we can construct special Lagrangian fibrations on these manifolds, as well as moduli of complex and symplectic structures and how hyper-Kahler rotation allows us to construct an identification of these moduli spaces.  This is joint work with A. Jacob and Y.-S. Lin.