All Seminars
| Title: A Stochastic Model for Networks at Arise from Conference Scheduling Problems |
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| Defense: Master's Thesis |
| Speaker: Andres Celis of Emory University |
| Contact: Andres Celis, acelis@emory.edu |
| Date: 2015-04-02 at 1:00PM |
| Venue: MSC 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 |
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| Defense: Dissertation |
| Speaker: Megan Cream of Emory University |
| Contact: Megan Cream, mcream@emory.edu |
| Date: 2015-04-02 at 4:00PM |
| Venue: MSC 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 |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Lin Lin of UC Berkeley |
| Contact: Michele Benzi, benzi@mathcs.emory.edu |
| Date: 2015-04-01 at 12:00PM |
| 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 |
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| 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 |
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| 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: MSC 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 |
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| 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 |
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| 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 |
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| 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: MSC 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. |
| Title: Mock theta functions and quantum modular forms |
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| Seminar: Algebra |
| Speaker: Larry Rolen of University of Cologne |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2015-03-17 at 4:00PM |
| Venue: W304 |
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| Abstract: In this talk, I will describe several related recent results related to mock theta functions, which are functions described by the Indian mathematician Ramanujan shortly before his death in 1920. These functions have very recently been understood in a modern framework thanks to the work of Zwegers and Bruinier-Funke. Here, we will revisit the original writings of Ramanujan and look at his original conception of these functions, which gives rise to a surprising picture connecting important objects such as generating functions in combinatorics and quantum modular forms. |
| Title: Interactive Machine Learning Across Domains |
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| Colloquium: N/A |
| Speaker: Lev Reyzin of University of Illinois at Chicago |
| Contact: Vaidy Sunderam, vss@emory.edu |
| Date: 2015-03-16 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Interactive learning algorithms have the power to engage with their data and can overcome many limitations of their passive counterparts. In this talk, I will present new algorithms and new models that I have developed for interactive learning. These results include the development of new pool-based, bandit, and query learners. I will also discuss applications and future research challenges for interactive machine learning settings, focusing on the life sciences. |