All Seminars

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Title: An epsilon improvement to the asymptotic density of k-critical graphs
Defense: Dissertation
Speaker: Victor Larsen of Emory University
Contact: Victor Larsen, vlarsen@emory.edu
Date: 2015-04-08 at 4:30PM
Venue: W303
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Abstract:
Given a graph $G$ the \emph{chromatic number}, denoted $\chi(G)$, is smallest number of colors necessary to color $V(G)$ such that no adjacent vertices receive the same color. A graph $G$ is $k$-critical if $\chi(G)=k$ but every proper subgraph has chromatic number less than $k$. As $k$-critical graphs can be viewed as minimal examples of graphs with chromatic number $k$, it is natural to ask how small such a graph can be. Let $f_k(n)$ denote the minimum number of edges in a $k$-critical graph on $n$ vertices. The Ore construction, used to build larger $k$-critical graphs, implies that \[f_k(n+k-1)\leq f_k(n)+(k-1)\left(\frac{k}{2}-\frac{1}{k-1}\right).\] A recent paper by Kostochka and Yancey provides a lower bound for $f_k(n)$ which implies that the asymptotic density $\phi_k:=\lim_{n\to\infty}f_k(n)/n = \frac{k}{2}-\frac{1}{k-1}$. In this work, we use the method of discharging to prove a lower bound on the number of edges which includes structural information about the graph. This lower bound shows that the asymptotic density of a $k$-critical graph can be increased by $\epsilon>0$ by restricting to $(K_{k-2})$-free $k$-critical graphs.\\ \\ We also prove that the graphs constructible from the Ore construction and $K_k$, called \emph{$k$-Ore} graphs, are precisely the graphs which attain Kostochka and Yancey's bound. Moreover, we also provide results regarding subgraphs which must exist in $k$-Ore graphs. For the discharging argument, carried out in two stages, we also prove results regarding the density of nearly-bipartite subgraphs in $k$-critical graphs.
Title: Cloud Computing: Just a buzzword or already a reality?
Colloquium: DEPARTMENT
Speaker: Gongbing Hong of William and Mary
Contact: Ken Mandelberg, KM@mathcs.emory.edu
Date: 2015-04-03 at 3:00PM
Venue: W303
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Abstract:
In this talk, we will give a brief introduction to Cloud Computing. First, we will provide a few definitions for cloud computing and discuss the big ideas of this new computing paradigm. Then we will talk about the services provided by cloud computing as seen today along with examples as offered by big players in the industry. We will also talk about how clouds are being deployed and the future of the clouds. At the end, we will discuss the impacts, issues, and the challenges of cloud computing.
Title: A Stochastic Model for Networks at Arise from Conference Scheduling Problems
Defense: Master's Thesis
Speaker: Andres Celis of Emory University
Contact: Andres Celis, acelis@emory.edu
Date: 2015-04-02 at 1:00PM
Venue: E408
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Abstract:
A conference scheduling problem may be viewed as an undirected graph whose vertices correspond to the events of the conference, and whose edges correspond to constraints that prohibit two events from being scheduled at the same time. In this thesis we propose and analyze a new random graph model inspired by a series of experimental observations on datasets from the industry, as well as conversations with a conference scheduler.\\ \\ Our models differs from existing random graph models in the following: \begin{enumerate} \item We find that graph models with independently chosen edges do not result in degree distributions found in conference scheduling problems. Thus our model's edges are statistically dependent on each other. \item Our model introduces new vertices into the graph as time evolves. The existing models that do this have small, bounded clique and chromatic numbers and are trivial to color, which is not an accurate representation of scheduling problems. We show that the expected clique number of our model has a lower bound of $\Omega(T^{1/4}/(\log T)^{3/4})$. We also argue that the expected clique and chromatic numbers of our model are upper bounded by $O(T^{1/4})$ and $O(T^{2/5})$, respectively. \end{enumerate}
Title: On Chorded Cycles
Defense: Dissertation
Speaker: Megan Cream of Emory University
Contact: Megan Cream, mcream@emory.edu
Date: 2015-04-02 at 4:00PM
Venue: W303
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Abstract:
Historically, there have been many results concerning sufficient conditions for implying certain sets of cycles in graphs. My thesis aims to extend many of these well known results to similar results on sets of {\it chorded} (and sometimes even {\it doubly chorded}) cycles. In particular, we consider the minimum degree, $\delta(G)$ and a Ore-type degree sum condition, $\sigma_2(G)$ of a graph $G$, sufficient to guarantee the existence of $k$ vertex disjoint chorded cycles, often containing specified elements of the graph, such as certain vertices or edges. Further, we extend a result on vertex disjoint cycles and chorded cycles to an analogous result on vertex disjoint cycles and {\it doubly} chorded cycles. We define a new graph property called chorded pancyclicity, and investigate a density condition and forbidden subgraphs in claw-free graphs that imply this new property. Specifically, we forbid certain paths and triangles with pendant paths. This is joint work with Dongqin Cheng, Ralph Faudree, Ron Gould, and Kazuhide Hirohata.
Title: Approximating spectral densities of large matrices: old and new
Seminar: Numerical Analysis and Scientific Computing
Speaker: Lin Lin of UC Berkeley
Contact: Michele Benzi, benzi@mathcs.emory.edu
Date: 2015-04-01 at 12:00AM
Venue: W306
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Abstract:
In physics, it is sometimes desirable to compute the so-called Density Of States (DOS), also known as the spectral density, of a Hermitian matrix A. The spectral density can be viewed as a probability density distribution that measures the likelihood of finding eigenvalues near some point on the real line. The most straightforward way to obtain this density is to compute all eigenvalues. But this approach is generally costly and wasteful, especially for matrices of large dimension. In many cases, the only affordable operation is matrix-vector multiplication. In this talk I will define the problem of estimating the spectral density carefully, and discuss a few known algorithms based on stochastic sampling, in particular, those using the kernel polynomial method and the Lanczos method, for estimating the spectral density. The accuracy of stochastic algorithms converge as $O(1\sqrt{N_v})$, where $N_v$ is the number of stochastic vectors. I will also talk about recent progresses of stochastic estimation of spectral densities with convergence rate much faster than $O(1\sqrt{N_v})$ using a relatively high degree of polynomials and modest choice of $N_v$. (Joint work with Yousef Saad and Chao Yang)
Title: An Effective Log-Free Zero Density Estimate for Automorphic $L$-functions and the Sato-Tate Conjecture
Seminar: Algebra
Speaker: Jesse Thorner of Emory
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2015-03-31 at 4:00PM
Venue: W304
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Abstract:
The classical techniques used to put primes in intervals of the form $[x,2x]$ are insufficient to put primes in intervals of the form $[x,x+x^{1-\delta}]$ for any $\delta>0$, or to find the least prime in an arithmetic progression $a\bmod q$. Such problems are easily answered assuming the Generalized Riemann Hypothesis, but they can be answered unconditionally using very detailed information about the location and density of zeros of Dirichlet $L$-functions in regions of the critical strip. We will discuss effective results on the distribution of general automorphic $L$-functions in the critical strip and use these distribution results to study generalizations of the aforementioned problems in the context of the Sato-Tate Conjecture.
Title: Involutions, odd degree extensions and generic splitting
Seminar: Algebra
Speaker: Anne Queguiner-Mathieuem of Universite Paris
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2015-03-26 at 3:00PM
Venue: W303
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Abstract:
Let $q$ be a quadratic form over a field $F$, and let $L$ be an odd-degree field extension of $F$. A classical theorem, known as Springer's theorem, asserts that if $q$ is isotropic (resp. hyperbolic) after scalar extension to $L$, it actually is isotropic (resp. hyperbolic) over the base field. One may ask whether a similar result holds for algebras with involution. In the talk, we will survey known results on this question, and explain the relation with the study of isotropy and hyperbolicity over some relevant function fields. New low degree results will also be included.
Title: Comparison of compactifications of modular curves
Seminar: Algebra
Speaker: Andrew Niles of Holy Cross
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2015-03-24 at 4:00PM
Venue: W304
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Abstract:
Modular curves and their compactifications are of fundamental importance in number theory. A key property of modular curves is that they are moduli spaces: their points classify certain geometric objects (elliptic curves equipped with level structure). Similarly, it was shown by Deligne-Rapoport that compactified modular curves may be viewed as moduli spaces for "generalized" elliptic curves equipped with level structure.\\ \\ It was shown by Abramovich-Olsson-Vistoli that modular curves naturally lie inside certain complicated moduli spaces, classifying "twisted stable maps" to certain algebraic stacks. These moduli spaces turn out to be complete, so the closure of a modular curve inside such a moduli space gives a compactification of the modular curve. In this talk I explain how these new compactifications can themselves be viewed as moduli spaces, and I compare them to the "classical" compactified modular curves considered by Deligne-Rapoport.
Title: Recent progress on diamond-free families
Seminar: Combinatorics
Speaker: Ryan Martin of Iowa State University
Contact: Dwight Duffus, Dwight@mathcs.emory.edu
Date: 2015-03-23 at 4:00PM
Venue: W302
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Abstract:
In the Boolean lattice, a diamond is a subposet of four distinct subsets $A, B, C, D$ such that $A \subset B, C$ and $D \supset B, C$. One of the most well-studied problems in extremal poset theory is determining the size of the largest diamond-free family in the $n$-dimensional Boolean lattice. We will discuss some recent progress on this problem.
Title: The Li-Yau Inequality and the Geometry of Graphs
Seminar: Combinatorics
Speaker: Paul Horn of The University of Denver
Contact: Dwight Duffus, Dwight@mathcs.emory.edu
Date: 2015-03-19 at 4:00PM
Venue: W303
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Abstract:
Understanding how local graph parameters, such as degree, are related to global graph properties, such as diameter and the containment of certain subgraphs, is a key aim of extremal graph theory. In the continuous setting of Riemannian manifolds, curvature serves as such a local parameter which is known to provide strong control of global structure. In this talk, we describe a new notion of curvature for graphs which, similar to in the continuous setting, strongly controls global geometric properties of a graph. In particular, it allows us to prove a discrete analogue of the Li-Yau inequality which, in this setting, controls the rate of diffusion of the continuous time random walk on a graph and which can be used to understand many further graph properties.