Georgia Institute of Technology College of Sciences

Online link: https://gatech.zoom.us/j/93504092832?pwd=V29FVVFlcEtwNWhkTnUyMnFqbVYyUT09

Many proposed interplanetary space missions, including Europa Lander and Dragonfly, involve trajectory design in environments where multiple large bodies exert gravitational influence on the spacecraft, such as the Jovian and Saturnian systems as well as cislu- nar space. In these contexts, an analysis based on the mathematical theory of dynamical systems provides both better insight as well as new tools to use for the mission design compared to classic two-body Keplerian methods. Indeed, a rich variety of dynamical phenomena manifest themselves in such systems, including libration point dynamics, stable and unstable mean-motion resonances, and chaos. To understand the previously mentioned dynamical behaviors, invariant manifolds such as periodic orbits, quasi-periodic invariant tori, and stable/unstable manifolds are the major objects whose interactions govern the local and global dynamics of relevant celestial systems.

This work is focused on the development of numerical methodologies for computing such invariant manifolds and investigating their interactions. After a study of persistence of mean-motion resonances in the planar circular restricted 3-body problem (PCRTBP), techniques for computing the stable/unstable manifolds attached to resonant periodic orbits and heteroclinics corresponding to resonance transitions are presented. Next, I will focus on the development of accurate and efficient parameterization methods for numerical calculation of whiskered quasi-periodic tori and their attached stable/unstable manifolds, for periodically-forced PCRTBP models. As part of this, a method for Levi- Civita regularization of such periodically-forced systems is introduced. Finally, I present methods for combining the previously mentioned parameterizations with knowl- edge of the objects’ internal dynamics, collision detection algorithms, and GPU computing to very rapidly compute propellant-free heteroclinic connecting trajectories between them, even in higher dimensional models. Such heteroclinics are key to the generation of chaos and large scale transport in astrodynamical systems.

This talk focuses on Eulerian graphs whose arcs are directed and labelled in a group. Each circuit yields a word over the group, and we say that a circuit is non-zero if this word does not evaluate to 0. We give a precise min-max theorem for the following problem. Given a vertex $v$, what is the maximum number of non-zero circuits in a circuit decomposition where each circuit begins and ends at $v$? This is joint work with Jim Geelen and Paul Wollan. Our main motivation is a surprising connection with vertex-minors which is due to Bouchet and Kotzig.

Tensegrities are mechanical structures that include cable-like elements that are strong and lightweight relative to rigid rods yet support only extensile stress. From suspension bridges to the musculoskeletal system to individual biological cells, humanity makes excellent use of tensegrities, yet the sharply nonlinear response of cables presents serious challenges to analytical theory. Here we consider large tensegrity structures with randomly placed cables (and struts) overlaid on a regular rigid backbone whose corresponding system of inequalities is reduced via analytic theory to an exact graph theory. We identify a novel coordination number that controls two rigidity percolation transitions: one in which global interactions between cables first support external loads and one in which the structure becomes fully rigid.  We show that even the addition of a few cables strongly modifies conventional rigidity percolation, both by modifying the sharpness of the transition and by introducing avalanche effects in which a single constraint can eliminate multiple floppy modes. 

Also ONLINE: https://gatech.zoom.us/j/99313032175

MCMC and variational inference are two competing paradigms for the problem of sampling from a given probability distribution. In this talk, I'll show how they can work together to give the first polynomial-time sampling algorithm for approximately low-rank Ising models. Sampling was previously known when all eigenvalues of the interaction matrix fit in an interval of length 1; however, a single outlier can cause Glauber dynamics to mix torpidly. Our result covers the case when all but O(1) eigenvalues lie in an interval of length 1. To deal with positive eigenvalues, we use a temperature-based heuristic for MCMC called simulated tempering, while to deal with negative eigenvalues, we define a nonconvex variational problem over Ising models, solved using SGD. Our result has applications to sampling Hopfield networks with a fixed number of patterns, Bayesian clustering models with low-dimensional contexts, and antiferromagnetic/ferromagnetic Ising model on expander graphs.