Georgia Institute of Technology College of Sciences

For graphs with maximum degree $\Delta$, a greedy algorithm shows $\chi(G) \leq \Delta + 1$. Brooks improved this to $\chi(G) \leq \Delta$ when $G$ has no cliques of size $\Delta + 1$, provided $\Delta \geq 3$. If is conjectured that if one forbids other graphs, the bound can be pushed further: for instance, Alon, Krivelevich, and Sudakov conjecture that, for any graph $F$, there is a constant $c(F) > 0$ such that $\chi(G) \leq (c(F) + o(1)) \Delta / \log\Delta$ for all $F$-free graphs $G$ of maximum degree $\Delta$. The only graphs $F$ for which this conjecture has been verified so far---by Alon, Krivelevich, and Sudakov themselves---are the so-called almost bipartite graphs, i.e., graphs that can be made bipartite by removing at most one vertex. Equivalently, a graph is almost bipartite if it is a subgraph of the complete tripartite graph $K_{1,t,t}$ for some $t \in \N$. The best heretofore known upper bound on $c(F)$ for almost bipartite $F$ is due to Davies, Kang, Pirot, and Sereni, who showed that $c(K_{1,t,t}) \leq t$. We prove that in fact $c(F) \leq 4$ for any almost bipartite graph $F$, thus making the bound independent of $F$ in all the known cases of the conjecture. We also establish a more general version of this result in the setting of DP-coloring (also known as correspondence coloring), which we give a gentle introduction to. Finally, we consider consequences of this result in the context of sublinear algorithms.

This is joint work with Anton Bernshteyn and Abhishek Dhawan.

https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09

In this talk we will discuss a shooting method designed for solving two point boundary value problems in a setting where a system has integrals of motion. We will show how it can be applied to obtain certain families of orbits in the circular restricted three body problem. These include transverse ejection/collisions from one primary body to the other, families of periodic orbits, orbits passing through collision, and orbits connecting fixed points to ejections or collisions.

This is joint work with Shane Kepley and Jason Mireles James.

With very minor assumptions, I show that periodic orbits in
an ODE can persist under (singular) perturbations of including a delay
term.  These perturbations change the phase space from finite to
infinite dimensions. The results apply to electrodynamics and give new
approaches to handle state-dependent, small, nested, and distributed
delays.

I will discuss and explain some motivations, the new methods, sketches
of the proofs, and possible applications. I will end the talk giving
some ideas of work in progress and possible future works.

I will talk about some recent work on the stability problem of shear flows and vortices as solutions of the Euler equations in 2D.  Our results include nonlinear stability theorems for monotonic shear  flows and point vortices, as well as linear stability theorems for more general flows. This is joint work with Hao Jia.