Georgia Institute of Technology College of Sciences

In the first part of the talk, I will show that for two bivariate polynomials $P(x,y)$ and $Q(x,y)$ with coefficients in fields with char 0 to simultaneously exhibit small expansion, they must exploit the underlying additive or multiplicative structure of the field in nearly identical fashion. This in particular generalizes the main result of Shen and yields an Elekes-Ronyai type structural result for symmetric nonexpanders, resolving a question mentioned by de Zeeuw (Joint with S. Roy and C-M. Tran). In the second part of the talk, I will show how this sum-product phenomena helps us avoid color-isomorphic even cycles in proper edge colorings of complete graphs (Joint with G. Ge, Z. Xu, and T. Zhang).

Random and irregular growth is all around us: tumor growth, fluid flow through porous media, and the spread of bacterial colonies. Simple models for these processes originated in the '50s with percolation theory and have since given rise to many new models and interesting mathematics. I will introduce a few models (percolation, invasion percolation, first-passage percolation, diffusion-limited aggregation, ...), along with some of their basic properties.
 

Cryo-Electron Microscopy (cryo-EM) is an imaging technology that is revolutionizing structural biology. Cryo-electron microscopes produce a large number of very noisy two-dimensional projection images of individual frozen molecules; unlike related methods, such as computed tomography (CT), the viewing direction of each particle image is unknown. The unknown directions, together with extreme levels of noise and additional technical factors, make the determination of the structure of molecules challenging. While other methods for structure determination, such as x-ray crystallography and nuclear magnetic resonance (NMR), measure ensembles of molecules, cryo-electron microscopes produce images of individual molecules. Therefore, cryo-EM could potentially be used to study mixtures of different conformations of molecules. Indeed, current algorithms have been very successful at analyzing homogeneous samples, and can recover some distinct conformations mixed in solutions, but, the determination of multiple conformations, and in particular, continuums of similar conformations (continuous heterogeneity), remains one of the open problems in cryo-EM. In practice, some of the key components in “molecular machines” are flexible and therefore appear as very blurry regions in 3-D reconstructions of macro-molecular structures that are otherwise stunning in resolution and detail.

We will discuss “hyper-molecules,” the mathematical formulation of heterogenous 3-D objects as higher dimensional objects, and the machinery that goes into recovering these “hyper-objects” from data. We will discuss some of the statistical and computational challenges, and how they are addressed by merging data-driven exploration, models and computational tools originally built for deep-learning.

This is joint work with Joakim Andén and Amit Singer.

 Recall that an excedance of a permutation $\pi$ is any position $i$
 such that $\pi_i > i$. Inspired by the work of Hopkins, McConville and
 Propp (arXiv:1612.06816) on sorting using toppling, we say that
 a permutation is toppleable if it gets sorted by a certain sequence of
 toppling moves. For the most part of the talk, we will explain the
 main ideas in showing that the number of toppleable permutations on n
 letters is the same as those for which excedances happen exactly at
 $\{1,\dots, \lfloor (n-1)/2 \rfloor\}$. Time permitting, we will give
 some ideas showing that this is also the number of acyclic
 orientations with unique sink (also known as the Ursell function) of the
 complete bipartite graph $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor + 1}$.

 This is joint work with D. Hathcock (CMU) and P. Tetali (Georgia Tech).