Georgia Institute of Technology College of Sciences

We consider the problem of computing unstable manifolds for equilibrium solutions of parabolic PDEs posed on irregular spatial domains. This new approach is based on the parameterization method, a general functional analytic framework for studying invariant manifolds of dynamical systems. The method leads to an infinitesimal invariance equation describing the unstable manifold. A recursive scheme leads to linear homological equations for the jets of the manifold which are solved using the finite element method. One feature of the method is that we recover the dynamics on the manifold in addition to its embedding.  We implement the method for some example problems with polynomial and non-polynomial nonlinearities posed on various non-convex two dimensional domains. We provide numerical support for the accuracy of the computed manifolds using the natural notion of a-posteriori error admitted by the parameterization method. This is joint work with J.D. Mireles-James and Necibe Tuncer. 

A question going back to Serre asks which groups arise as fundamental groups of smooth, complex projective varieties, or more generally, compact Kaehler manifolds.  The most basic examples of these are surface groups, arising as fundamental groups of 1-dimensional projective varieties.  We will survey known examples and restrictions on such groups and explain the special role surface groups play in their classification. Finally, we connect this circle of ideas to more general questions about surface bundles and mapping class groups. 

It is a fundamental problem in computer vision to describe the geometric relations between two or more cameras that view the same scene -- state of the art methods for 3D reconstruction incorporate these geometric relations in a nontrivial way. At the center of the action is the essential variety: an irreducible subvariety of P^8 of dimension 5 and degree 10 whose homogeneous ideal is minimal generated by 10 cubic equations. Taking a linear slice of complementary dimension corresponds to solving the "minimal problem" of 5 point relative pose estimation. Viewed algebraically, this problem has 20 solutions for generic data: these solutions are elements of the special Euclidean group SE(3) which double cover a generic slice of the essential variety. The structure of these 20 solutions is governed by a somewhat mysterious Galois group (ongoing work with Regan et. al.)

We may ask: what other minimal problems are out there? I'll give an overview of work with Kohn, Pajdla, and Leykin on this question. We have computed the degrees of many minimal problems via computer algebra and numerical methods. I am inviting algebraic geometers at large to attack these problems with "pen and paper" methods: there is still a wide class of problems to be considered, and the more tools we have, the better.

Kodaira, and independently Atiyah, gave the first examples of surface bundles over surfaces whose signature does not vanish, demonstrating that signature need not be multiplicative.  These examples, called Kodaira fibrations, are in fact complex projective surfaces admitting a holomorphic submersion onto a complex curve, whose fibers have nonconstant moduli. After reviewing the Atiyah-Kodaira construction, we consider Kodaira fibrations with nontrivial holomorphic invariants in degree one. When the dimension of the invariants is at most two, we show that the total space admits a branched covering over a product of curves.

I will describe a few classical problems in capillarity and the associated classical variational framework.  These problems include the well-known shape and rise height problems for the meniscus in a tube as well as the problems associated with sessile and pendent drops. I will briefly discuss elements of recent modifications of the variational theory allowing floating objects.  Finally, I will describe a few open problems. 

We establish an upper bound on the spectral gap for compact quantum graphs which depends only on the diameter and total number of vertices. This bound is asymptotically sharp for pumpkin chains with number of edges tending to infinity. This is a joint work with D. Borthwick and L. Corsi.

In this talk, we discuss about methods for proving existence and uniqueness of a root of a square analytic system in a given region. For a regular root, Krawczyk method and Smale's $\alpha$-theory are used. On the other hand, when a system has a multiple root, there is a separation bound isolating the multiple root from other roots. We define a simple multiple root, a multiple root whose deflation process is terminated by one iteration, and establish its separation bound. We give a general framework to certify a root of a system using these concepts.

Using what we have studied in the Brändén-Huh paper, we will go over the proof of the ultra-log-concavity version of Mason's conjecture.

We study the geometry of minimizers of the interaction energy functional. When the interaction potential is mildly repulsive, it is known to be hard to characterize those minimizers due to the fact that they break the rotational symmetry, suggesting that the problem is unlikely to be resolved by the usual convexity or symmetrization techniques from the calculus of variations. We prove that, if the repulsion is mild and the attraction is sufficiently strong, the minimizer is unique up to rotation and exhibits a remarkable simplex-shape rigid structure. As the first crucial step we consider the maximum variance problem of probability measures under the constraint of bounded diameter, whose answer in one dimension was given by Popoviciu in 1935.

We give a formula relating various notions of heights of abelian varieties. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener, and it extends the Faltings-Silverman formula for elliptic curves. We also discuss the case of Jacobians in some detail, where graphs and electrical networks will play a key role.   Based on joint works with Robin de Jong (Leiden).

An expectation-maximization (EM) algorithm is a powerful clustering method that was initially developed to fit Gaussian mixture distributions. In the absence of a particular probability density function, an EM algorithm aims to estimate the "best" function that maximizes the likelihood of data being generated by the model. We present an EM algorithm which addresses the problem of clustering "mutated" substrings of similar parent strings such that each substring is correctly assigned to its parent string. This problem is motivated by the process of simultaneously reading similar RNA sequences during which various substrings of the sequence are produced and could be mutated; that is, a substring may have some letters changed during the reading process. Because the original RNA sequences are similar, a substring is likely to be assigned to the wrong original sequence. We describe our EM algorithm and present a test on a simulated benchmark which shows that our method yields a better assignment of the substrings than what has been achieved by previous methods. We conclude by discussing how this assignment problem applies to RNA structure prediction.

The classical isoperimetric inequality states that in Euclidean space spheres form the least perimeter enclosures for any give volume. We will review the historic development of this result in mathematics, and various approaches to proving it. Then we will discuss how one of these approaches, which is a variational argument, may be extended to spaces of nonpositive curvature, known as Cartan-Hadamard manifolds, in order to generalize the isoperimetric inequality.

Stephen Smale’s h-cobordism Theorem was a landmark result in the classification of smooth manifolds. It paved the way towards solutions for the topological Poincaré and Schoenflies conjectures in dimensions greater than 5. Later, building on this, Freedman’s work applied these techniques to 4 manifolds. I shall discuss the ideas relating to h-cobordisms and the proof, which is a wonderful application of handlebody theory and the Whitney trick. Time permitting, we shall explore the world of smooth 4 manifolds further, and talk about cork twists.

We will show the sharp estimate on the behavior of the smallest singular value of random matrices under very general assumptions. One of the steps in the proof is a result about the efficient discretization of the unit sphere in an n-dimensional euclidean space. Another step involves the study of the regularity of the behavior of lattice sets. Some elements of the proof will be discussed. Based on the joint work with Tikhomirov and Vershynin.

The classical isoperimetric inequality states that in Euclidean space spheres provide unique enclosures of least perimeter for any given volume. In this talk we discuss how this inequality may be extended to spaces of nonpositive curvature, known as Cartan-Hadamard manifolds, as conjectured by Aubin, Gromov, Burago, and Zalgaller in 1970s and 80s. The proposed proof is based on a comparison formula for total curvature of level sets in Riemannian manifolds, and estimates for the smooth approximation of the signed distance function, via inf-convolution and Reilly type formulas among other techniques. Immediate applications include sharp extensions of Sobolev and Faber-Krahn inequalities to spaces of nonpositive curvature. This is joint work with Joel Spruck.

A graph is $k$-critical if its chromatic number is $k$ but any its proper subgraph has chromatic number less than $k$. Let $k\geq 4$. Gallai asked in 1984 if any $k$-critical graph on $n$ vertices contains at least $n$ distinct $(k-1)$-critical subgraphs. Improving a result of Stiebitz, Abbott and Zhou proved in 1995 that every such graph contains $\Omega(n^{1/(k-1)})$ distinct $(k-1)$-critical subgraphs. Since then no progress had been made until very recently, Hare resolved the case $k=4$ by showing that any $4$-critical graph on $n$ vertices contains at least $(8n-29)/3$ odd cycles. We mainly focus on 4-critical graphs and develop some novel tools for counting cycles of specified parity. Our main result shows that any $4$-critical graph on $n$ vertices contains $\Omega(n^2)$ odd cycles, which is tight up to a constant factor by infinite many graphs. As a crucial step, we prove the same bound for 3-connected non-bipartite graphs, which may be of independent interest. Using the tools, we also give a very short proof to the Gallai's problem for the case $k=4$. Moreover, we improve the longstanding lower bound of Abbott and Zhou to $\Omega(n^{1/(k-2)})$ for the general case $k\geq 5$. Joint work with Tianchi Yang.

Expander decomposition has been a central tool in designing graph algorithms in many fields (including fast centralized algorithms, approximation algorithms and property testing) for decades. Recently, we found that it also gives many impressive applications in dynamic graph algorithms and distributed graph algorithms. In this talk, I will survey these recent results based on expander decomposition, explain the key components for using this technique, and give some toy examples on how to apply these components.

In 1959 Mark Kac introduced a simple model for the evolution 
of a gas of hard spheres undergoing elastic collisions. The main 
simplification consisted in replacing deterministic collisions with 
random Poisson distributed collisions.

It is possible to obtain many interesting results for this simplified 
dynamics, like estimates on the rate of convergence to equilibrium and 
validity of the Boltzmann equation. The price paid is that this system 
has no space structure.

I will review some classical results on the Kac model and report on an 
attempt to reintroduce some form of space structure and non-equilibrium 
evolution in a way that preserve the mathematical tractability of the 
system.