Georgia Institute of Technology College of Sciences

A cubic graph is one where every vertex has degree three. A linear forest is a disjoint union of paths. It is known that the edge set of every cubic graph can be partitioned into two linear forests where each path is short (of constant size). A conjecture of Wormald asks for such a partition where the two forests are isomorphic (we no longer insist on having short paths, although that is also an open question). Note that this also can be phrased as an edge-colouring question. Is it possible to colour the edge set of a cubic graph by red and blue such that the two monochromatic components induce isomorphic linear forests? Recently we proved this for all connected graphs on a sufficiently large number of vertices. I will talk about the result and give some idea of the proof method. This is joint work with Gal Kronenberg, Shoham Letzter and Alexey Pokrovskiy.

The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no inclusion-maximal clique is monochromatic (ignoring isolated vertices). 

For the binomial random graph G_{n,p} the clique chromatic number has been studied in a number of works since 2016, but for sparse edge-probabilities in the range n^{-2/5} \ll p \ll 1 even the order of magnitude remained a technical challenge.

Resolving open problems of Alon and Krivelevich as well as Lichev, Mitsche and Warnke, we determine the clique chromatic number of the binomial random graph G_{n,p} in most of the missing regime: we show that it is of order (\log n)/p for edge-probabilities n^{-2/5+\eps} \ll p \ll n^{-1/3} and n^{-1/3+\eps} \ll p \ll 1, for any constant \eps > 0.

Perhaps surprisingly for a result about random graphs, a key ingredient in the proof is an application of the probabilistic method (that hinges on careful counting and density arguments).

This talk is based on joint work with Lutz Warnke.

 In 1996, C.~Heil, J.~Ramanatha, and P.~Topiwala conjectured that the (finite) set $\mathcal{G}(g, \Lambda)=\{e^{2\pi i b_k \cdot}g(\cdot - a_k)\}_{k=1}^N$ is linearly independent for any  non-zero square integrable function $g$ and  subset $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2.$ This problem is now known as the HRT Conjecture, and is still largely unresolved. 

In this talk,  I will then introduce an inductive approach to investigate the conjecture, by attempting to answer the following question. Suppose the HRT conjecture is true for a function $g$ and a fixed set of $N$ points $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2.$ For what other point $(a, b)\in \mathbb{R}^2\setminus \Lambda$ will the HRT remain true for the same function $g$ and the new set of $N+1$ points $\Lambda'=\Lambda \cup \{(a, b)\}$?  I will report on a recent joint work with V.~Oussa in which we use this approach to prove the conjecture when the initial configuration  $\Lambda=\{(a_k, b_k)\}_{k=1}^N $  is either a subset of the unit lattice $\mathbb{Z}^2$ or a subset of a line $L$.   

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.