Georgia Institute of Technology College of Sciences

Let P be a graph with a vertex v such that P-v is a forest and let Q be an outerplanar graph. In 1993 Paul Seymour asked if every two-connected graph of sufficiently large path-width contains P or Q as a minor.mDefine g(H) as the minimum number for which there exists a positive integer p(H) such that every g(H)-connected H-minor-free graph has path-width at most p(H). Then g(H) = 0 if and only if H is a forest and there is no graph H with g(H) = 1, because path-width of a graph G is the maximum of the path-widths of its connected components.Let A be the graph that consists of a cycle (a_1,a_2,a_3,a_4,a_5,a_6,a_1) and extra edges a_1a_3, a_3a_5, a_5a_1. Let C_{3,2} be a graph of 2 disjoint triangles. In 2014 Marshall and Wood conjectured that a graph H does not have K_{4}, K_{2,3}, C_{3,2} or A as a minor if and only if g(H) >= 2. In this thesis we answer Paul Seymour's question in the affirmative and prove Marshall and Wood's conjecture, as well as extend the result to three-connected and four-connected graphs of large path-width. We introduce ``cascades", our main tool, and prove that in any tree-decomposition with no duplicate bags of bounded width of a graph of big path-width there is an ``injective" cascade of large height. Then we prove that every 2-connected graph of big path-width and bounded tree-width admits a tree-decomposition of bounded width and a cascade with linkages that are minimal. We analyze those minimal linkages and prove that there are essentially only two types of linkage. Then we convert the two types of linkage into the two families of graphs P and Q. In this process we have to choose the ``right'' tree decomposition to deal with special cases like a long cycle. Similar techniques are used for three-connected and four-connected graphs with high path-width.

Over recent years, a great deal of analytical studies and modeling simulations have been brought together to identify the key signatures that allow dynamically similar nonlinear systems from diverse origins to be united into a single class. Among these key structures are bifurcations of homoclinic and heteroclinic connections of saddle equilibria and periodic orbits. Such homoclinic structures are the primary cause for high sensitivity and instability of deterministic chaos in various systems. Development of effective, intelligent and yet simple algorithms and tools is an imperative task for studies of complex dynamics in generic nonlinear systems. The core of our approach is the reduction of the time evolution of a characteristic observable in a system to its symbolic representation to conjugate or differentiate between similar behaviors. Of our particular consideration are the Lorenz-like systems and systems with spiral chaos due to the Shilnikov saddle-focus. The proposed approach and tools will let one detect homoclinic and heteroclinic orbits, and carry out state of the art studies homoclinic bifurcations in parameterized systems of diverse origins.

For a fixed graph $G$, let $\mathcal{L}_G$ denote the family of Lipschitz functions $f:V(G) \rightarrow \mathbb{R}$ such that $0 = \sum_u f(u)$. The \emph{spread} of $G$ is denoted $c(G) := \frac{1}{|V(G)|} \max_{f \in \mathcal{L}_G} \sum_u f(u)^2$ and the subgaussian constant is $e^{\sigma_G^2} := \sup_{t > 0} \max_{f \in \mathcal{L}_G} \left( \frac{1}{|V(G)|} \sum_u e^{t f(u)} \right)^{2/t^2}$. Motivation of these parameters comes from their relationship with the isoperimetric number of a graph (given a number $t$, find a set $W \subset V(G)$ such that $2|W| \geq |V(G)|$ that minimizes $i(G,t) := |\{u : d(u, W) \leq t \}|$). While the connection to the isoperimetric number is interesting, the spread and subgaussian constant have not been any easier to understand. In this talk, we will present results that describe the functions $f$ achieving the optimal values. As a corollary to these results, we will resolve two conjectures (one false, one true) about these parameters. The conjectures that we resolve are the following. We denote the Cartesian product of $G$ with itself $d$ times as $G^d$. Alon, Boppana, and Spencer proved that the set $\{u: f(u) < k\}$ for extremal function $f$ for the spread of $G^d$ gives a value that is asymptotically close to the isoperimetric number when $d, t$ grow at specific rates and $k=0$; and they conjectured that the value is exactly correct for large $d$ and $k,t$ in ``appropriate ranges.'' The conjecture was proven true for hypercubes by Harper and the discrete torus of even order by Bollob\'{a}s and Leader. Bobkov, Houdr\'{e}, and Tetali constructed a function over a cycle that they conjectured to be optimal for the subgaussian constant, and it was proven correct for cycles of even length by Sammer and Tetali. This work appears in the manuscript https://arxiv.org/abs/1705.09725 .

The Institute for Defense Analyses - Center for Computing Sciences is a nonprofit research center that works closely with the NSA. Our center has around 60 researchers (roughly 30 mathematicians and 30 computer scientists) that work on interesting and hard problems. The plan for the seminar is to begin with a short mathematics talk on a project that was completed at IDA-CCS and declassified, then tell you a little about what we do, and end with your questions. The math that we will discuss involves symbolic dynamics and automata theory. Specifically we will develop a metric on the space of regular languages using topological entropy. This work was completed during a summer SCAMP at IDA-CCS. SCAMP is a summer program where researchers from academia (professors and students), the national labs, and the intelligence community come to IDA-CCS to work on the agency's hard problems for 11 weeks.