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Series: Dissertation Defense

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.

Series: PDE Seminar

In a recent work Sideris constructed a finite-parameter family of compactly supported affine solutions to the free boundary compressible Euler equations satisfying the physical vacuum condition. The support of these solutions expands at a linear rate in time. We show that if the adiabatic exponent gamma belongs to the interval(1, 5/3] then these affine motions are globally-in-time nonlinearly stable. If time permits we shall also discuss several classes of global solutions to the compressible Euler-Poisson system. This is a joint work with Juhi Jang.

Series: CDSNS Colloquium

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.

Series: School of Mathematics Colloquium

Associated to a planar cubic graph, there is a closed surface in R^5, as defined by Treumann and Zaslow. R^5 has a canonical geometry, called a contact structure, which is compatible with the surface. The data of how this surface interacts with the geometry recovers interesting data about the graph, notably its chromatic polynomial. This also connects with pseudo-holomorphic curve counts which have boundary on the surface, and by looking at the resulting differential graded algebra coming from symplectic field theory, we obtain new definitions of n-colorings which are strongly non-linear as compared to other known definitions. There are also relationships with SL_2 gauge theory, mathematical physics, symplectic flexibility, and holomorphic contact geometry. During the talk we'll explain the basic ideas behind the various fields above, and why these various concepts connect.

Series: Combinatorics Seminar

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 .

Series: Other Talks

Please check the meeting webpage at http://10to60.math.gatech.edu for program, titles and abstracts.

Series: Research Horizons Seminar

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.

Series: Research Horizons Seminar

Four
dimensions is unique in many ways. For example $n$-dimensional
Euclidean space has a unique smooth structure if and only if $n$ is not
equal to four. In other words, there is only one way to understand
smooth functions on $R^n$ if and only if
$n$ is not 4. There are many other way that smooth structures on
4-dimensional manifolds behave in surprising ways. In this talk I will
discuss this and I will sketch the beautiful interplay of ideas (you got
algebra, analysis and topology, a little something
for everyone!) that go into proving $R^4$ has more that one smooth
structure (actually it has uncountably many different smooth structures
but that that would take longer to explain).

Series: CDSNS Colloquium

A special class of dynamical systems that we will focus on are substitutions. This class of systems provides a variety of ergodic theoretic behavior and is connected to self-similar interval exchange transformations. During this talk we will explore rigidity sequences for these systems. A sequence $\left( n_m \right)$ is a rigidity sequence for the dynamical system $(X,T,\mu)$ if $\mu(T^{n_m}A\cap A)\rightarrow \mu(A)$ for all positive measure sets $A$. We will discuss the structure of rigidity sequences for substitutions that are rank-one and substitutions that have constant length. This is joint work with Jon Fickenscher.

Series: Job Candidate Talk

We study the convergence rate of the least squares estimator (LSE) in a regression model with possibly heavy-tailed errors. Despite its importance in practical applications, theoretical understanding of this problem has been limited. We first show that from a worst-case perspective, the convergence rate of the LSE in a general non-parametric regression model is given by the maximum of the Gaussian regression rate and the noise rate induced by the errors. In the more difficult statistical model where the errors only have a second moment, we further show that the sizes of the 'localized envelopes' of the model give a sharp interpolation for the convergence rate of the LSE between the worst-case rate and the (optimal) parametric rate. These results indicate both certain positive and negative aspects of the LSE as an estimation procedure in a heavy-tailed regression setting. The key technical innovation is a new multiplier inequality that sharply controls the size of the multiplier empirical process associated with the LSE, which also finds applications in shape-restricted and sparse linear regression problems.