Georgia Institute of Technology College of Sciences

A cobordism between 3-manifolds is ribbon if it is built from handles of index no greater than 2. Such cobordisms arise naturally from several different topological and geometric contexts. In this talk, we discuss these objects and present a few obstructions to their existence, from Thurston geometries, character varieties, and instanton and Heegaard Floer homologies. This is joint work with Aliakbar Daemi, Tye Lidman, and Mike Wong.

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.

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.
 

The Yamabe problem asks whether, given a closed Riemannian manifold, one can find a conformal metric of constant scalar curvature (CSC). An affirmative answer was given by Schoen in 1984, following contributions from Yamabe, Trudinger, and Aubin, by establishing the existence of a function that minimizes the so-called Yamabe energy functional; the minimizing function corresponds to the conformal factor of the CSC metric.

We address the quantitative stability of minimizing Yamabe metrics. On any closed Riemannian manifold we show—in a quantitative sense—that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close to a CSC metric. Generically, this closeness is controlled quadratically by the Yamabe energy deficit. However, we construct an example demonstrating that this quadratic estimate is false in the general. This is joint work with Max Engelstein and Luca Spolaor.

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).

Dense graph limit theory is essentially a first-order limit theory analogous to the classical Law of Large Numbers. Is there a corresponding central limit theorem? We believe so. Using the language of Gaussian Hilbert Spaces and the comprehensive theory of generalised U-statistics developed by Svante Janson in the 90s, we identify a collection of Gaussian measures (aka white noise processes) that describes the fluctuations of all orders of magnitude for a broad family of random graphs. We complement the theory with error bounds using a new variant of Stein’s method for multivariate normal approximation, which allows us to also generalise Janson’s theory in some important aspects. This is joint work with Gursharn Kaur.

Please note the unusual time/day.

Dense graph limit theory is essentially a first-order limit theory analogous to the classical Law of Large Numbers. Is there a corresponding central limit theorem? We believe so. Using the language of Gaussian Hilbert Spaces and the comprehensive theory of generalised U-statistics developed by Svante Janson in the 90s, we identify a collection of Gaussian measures (aka white noise processes) that describes the fluctuations of all orders of magnitude for a broad family of random graphs. We complement the theory with error bounds using a new variant of Stein’s method for multivariate normal approximation, which allows us to also generalise Janson’s theory in some important aspects. This is joint work with Gursharn Kaur.

Please note the unusual time: 5pm

This skit recounts one of the foundation stories of mathematics, the puzzle of the Seven Bridges of Königsberg, solved by Euler in 1726.  Except that it all takes place in a mad courtroom, and you are the jury!

A sunflower with p petals consists of p sets whose pairwise intersections are all the same set. The goal of the sunflower problem is to find the smallest r = r(p,k) such that every family of at least r^k k-element sets must contain a sunflower with p petals. Major breakthroughs within the last year by Alweiss-Lovett-Wu-Zhang and others show that r = O(p log(pk)) suffices. In this talk, after reviewing the history and significance of the Sunflower Problem, I will present our improvement to r = O(p log k), which we obtained during the 2020 REU at Georgia Tech. As time permits, I will elaborate on key lemmas and techniques used in recent improvements.

Based on joint work with Suchakree Chueluecha (Lehigh University) and Lutz Warnke (Georgia Tech), see https://arxiv.org/abs/2009.09327

We introduce a construction of oriented matroids from any triangulation of a product of two simplices, extending the regular case which follows from signed tropicalization. For this, we use the structure of such a triangulation in terms of polyhedral matching fields. The oriented matroid is composed of compatible chirotopes on the cells in a matroid subdivision of the hypersimplex, which might be of independent interest. We will also describe the extension to matroids over hyperfields and sketch some connections with optimization. This is joint work with Marcel Celaya and Georg Loho; Marcel Celaya will be giving a talk on the topological aspect of the work at the algebra seminar next week.

Please note the unusual time: 4pm