Georgia Institute of Technology College of Sciences

Thomassen famously showed that every planar graph is 5-choosable, and that every planar graph of girth at least five is 3-choosable.  These theorems are best possible for uniform list assignments: Voigt gave a construction of a planar graph that is not 4-choosable, and of a planar graph of girth four that is not 3-choosable. In this talk, I will introduce the concept of a local girth list assignment: a list assignment wherein the list size of each vertex depends not on the girth of the graph, but only on the length of the shortest cycle in which the vertex is contained. I will present a local list colouring theorem that unifies the two theorems of Thomassen mentioned above and discuss some of the highlights and difficulties of the proof. This is joint work with Luke Postle.

I will give an introduction to tropical geometry which arises when you take the coordinate-wise logarithm of points in a curve and then take the base of the logarithm to infinity. This gives a combinatorial curve which is basically a bunch of rays starting at the origin. I will also talk a bit about polygons, number theory and geometry.

(Based on paper by Fawzi, Saunderson and Parrilo in 2015) The space of complex-valued functions on a fixed abelian group has an orthonormal basis of group homomorphisms, via the well-known Discrete Fourier Transform. Given any nonnegative function with sparse Fourier support, it turns out that it’s possible to write it as a sum of squares, where the common Fourier support for all squares is not big. This can be used to prove results for the usual degree-based sum-of-squares hierarchy.

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

This talk concentrates on the study of stability of floating objects through mathematical modeling and experimentation. The models are based on standard ideas of center of gravity, center of buoyancy, and Archimedes’ Principle. There will be a discussion of a variety of floating shapes with two-dimensional cross sections for which it is possible to analytically and/or computationally a potential energy landscape in order to identify stable and unstable floating orientations.  I then will compare the analysis and computations to experiments on floating objects designed and created through 3D printing. The talk includes a demonstration of code we have developed for testing the floating configurations for new shapes. I will give a brief overview of the methods involved in 3D printing the objects. 

This research is joint work with Dr. Dan Anderson at GMU and undergraduate students Brandon G. Barreto-Rosa, Joshua D. Calvano, and Lujain Nsair, all of whom  who were part of an undergraduate research program run by the MEGL at GMU. 
 

While gerrymandering has been widely suspected in Georgia for years, it has been difficult to quantify. We generate a large ensemble of randomly generated non-partisan maps that are sampled from a probability distribution which respects the geographical constraints of the redistricting process. Using a Markov chain Monte Carlo process and techniques involving spanning trees, we can quickly generate a robust set of plans.

Based on historical voting data, we compare the Georgia congressional redistricting plan enacted in 2021 with the non-partisan maps. We find that the 2021 plan will likely be highly non-responsive to changing opinions of the electorate, unlike the plans in the ensemble. Using additional spatial analysis, we highlight areas where the map has been redrawn to weaken the influence of Democratic voters.

This talk is based on joint work with Swati Gupta, Gregory Herschlag, Jonathan Mattingly, Dana Randall, and Zhanzhan Zhao.

One of the most interesting and surprising features of manifold topology is the existence of topological 4-manifold that admit infinitely many smooth structures. In these talks I will discuss what is known about these “exotic” smooth structures on open manifolds, starting with R^4 and then moving on to other open 4-manifolds. We will also go over various constructions and open questions about these manifolds.  

The cluster expansion is a classical tool from statistical physics used to understand systems of weakly interacting particles in the high temperature regime of statistical physics models.  It can also be a very useful tool in probabilistic, extremal, and enumerative combinatorics and in the study of large deviations in probability theory.  I will give an introduction to the cluster expansion, present some examples of combinatorial applications, and try to provide some intuition about when the cluster expansion should or should not be a useful tool for a particular problem.

In this talk, I will present the analysis of two astrophysical systems. First, exoplanets (planets orbiting a star that is not our Sun) are thought to sometimes naturally evolve into a state such that its spin axis is significantly tilted from its orbital axis. The most well-known examples of such tilts come from our own Solar System: Uranus with its 98 degree tilt is spinning entirely on its side, while Venus with its 177 degree tilt spins in the opposite direction to its orbit. I show that tilted exoplanets form probabilistically due to encountering a separatrix, and this probability can be analytically calculated using Melnikov's Method. Second, the origin of the binary black holes (BBHs) whose gravitational wave radiation has been detected by the LIGO/VIRGO Collaboration is currently not well-understood. Towards disambiguating among many proposed formation mechanisms, certain studies have computed the distributions of various physical parameters when BBHs form via certain mechanisms. A curious result shows that one such formation mechanism commonly results in black holes tilted on their sides. I show that this can be easily understood by identifying a hidden adiabatic invariant that links the initial and final spin orientations of the BBHs. No astrophysical knowledge is expected; please stop by!