Georgia Institute of Technology College of Sciences

Perspectives from numerical optimization and computational algebra are  
brought to bear on a scientific application and a data science  
application.  In the first part of the talk, I will discuss  
cryo-electron microscopy (cryo-EM), an imaging technique to determine  
the 3-D shape of macromolecules from many noisy 2-D projections,  
recognized by the 2017 Chemistry Nobel Prize.  Mathematically, cryo-EM  
presents a particularly rich inverse problem, with unknown  
orientations, extreme noise, big data and conformational  
heterogeneity. In particular, this motivates a general framework for  
statistical estimation under compact group actions, connecting  
information theory and group invariant theory.  In the second part of  
the talk, I will discuss tensor rank decomposition, a higher-order  
variant of PCA broadly applicable in data science.  A fast algorithm  
is introduced and analyzed, combining ideas of Sylvester and the power  
method.

We have seen that Hankel index of varieties can be controlled by some invariants such as $$N_{2,p}$$ or p-base point free property. Also, we know that the Hankel index of (a linear join of) variety of minimal degree is infinity (and all invariant above are same as infinity). As next case, I will share some computations of invariants on a variety that projecting rational normal curve away from a point (which is a variety of almost minimal degree).

A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, the existence of a Stein structure is essentially a homotopical question, but for 4-manifolds the situation is more subtle. In this talk we will consider the existence of such structures in the ambient settings, that is, manifolds/domains with various degree of convexity as open/compact subsets of a complex manifold, e.g. complex 2-space C^2. In particular, I will discuss the following question: Which homology spheres embed in C^2 as the boundary of a Stein domain? This question was first considered and explored in detail by Gompf. At that time, he made a fascinating conjecture that no non-trivial Brieskorn homology sphere, with either orientation, embeds in C^2 as a Stein boundary. In this talk, I will survey what we know about this conjecture, and report on some closely related recent work in progress that ties to an interesting symplectic rigidity phenomena in low dimensions.

While deviation estimates above the mean is a very well studied subject in high-dimensional probability, for their lower analogues far less are known. However, it has been observed, in several key situations, that lower deviation inequalities exhibit very different and stronger behavior. In this talk I will discuss how convexity can serve as a key feature to (a) explain this distinction, (b) obtain improved lower tail bounds, and (c) characterize the tightness of Gaussian concentration.