# All Seminars

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Title: Connections between Classical and Umbral Moonshine
Defense: Dissertation
Speaker: Sarah Trebat-Leder of Emory University
Contact: Sarah Trebat-Leder, sarah.trebat-leder@emory.edu
Date: 2018-04-02 at 2:00PM
Venue: W302
Abstract:
Both results of this dissertation involve finding unexpected connections between the classical theory of monstrous moonshine and the newer umbral moonshine. In our first result, we use generalized Borcherds products to associate to each pure A-type Niemeier lattice a conjugacy class g of the monster group and give rise to identities relating dimensions of representations from umbral moonshine to values of McKay-Thompson series. Our second result focuses on the Mathieu group M23. While it inherits a moonshine from being a subgroup of M24, we find a new and simpler moonshine for M23 such that the graded traces are, up to constant terms, identical to the monstrous moonshine Hauptmoduln.
Title: Maass forms and modular forms: applications to class numbers and partitions
Defense: Dissertation
Speaker: Olivia Beckwith of Emory University
Contact: Olivia Beckwith, olivia.dorothea.beckwith@emory.edu
Date: 2018-04-02 at 3:00PM
Venue: W302
Abstract:
This dissertation is about arithmetic information encoded by analytic characteristics (such as Fourier coefficients) of classical modular forms and a real-analytic generalization of modular forms called harmonic Maass forms. For example, I use the theory of harmonic Maass forms to extend and refine a theorem of Wiles on class number divisibility. I also prove asymptotic bounds for Rankin-Selberg shifted convolution L-functions in shift aspect. In partition theory, I use the circle method to describe the expected distribution of parts of integer partitions over residue classes, and I use effective estimates for partition functions to obtain simple formulas for functions arising in group theory.
Title: Patching and local-global principles for gerbes with an application to homogeneous spaces
Defense: Dissertation
Speaker: Bastian Haase of Emory University
Contact: Bastian Haase, bastian.haase@emory.edu
Date: 2018-04-02 at 4:00PM
Venue: W302
Abstract:
Starting in 2009, Harbater and Hartmann introduced a new patching setup for semi-global fields, establishing a patching framework for vector spaces, central simple algebras, quadratic forms and other algebraic structures. In subsequent work with Krashen, the patching framework was refined and extended to torsors and certain Galois cohomology groups. After describing this framework, we will discuss an extension of the patching equivalence to bitorsors and gerbes. Building up on these results, we then proceed to derive a characterisation of a local- global principle for gerbes and bitorsors in terms of factorization. These results can be expressed in the form of a Mayer-Vietoris sequence in non-abelian hypercohomology with values in the crossed-module $G->Aut(G)$. After proving the local-global principle for certain bitorsors and gerbes using the characterization mentioned above, we conclude with an application on rational points for homogeneous spaces.
Title: Truncated Singular Value Decomposition Approximation for Structured Matrices via Kronecker Product Summation Decomposition
Defense: Dissertation
Speaker: Clarissa Garvey of Emory University
Contact: James Nagy, jnagy@emory.edu
Date: 2018-04-02 at 4:30PM
Venue: W301
Abstract:
Singular value decompositions are a particularly attractive matrix factorization for ill-posed problems because singular value magnitudes reveal information about the relative importance of data in the matrix. However, computing a singular value decomposition is typically computationally infeasible for large problems, as the cost for traditional methods, such as Lanczos bidiagonalization-based approaches and randomized methods, scales linearly with the number of entries in the matrix times the number of singular values computed. In this work we present two new algorithms and one new hybrid approach for computing the singular value decomposition of matrices cheaply approximable as an ordered Kronecker summation decomposition. Unlike previous work using ordered Kronecker summation decompositions, the factorizations these methods produce are more accurate for certain classes of matrices and have nonnegative singular values. The three proposed methods are also faster, with lower computational and spatial complexity, although also lower accuracy, than traditional methods. Our Kronecker-based methods therefore enable singular value decomposition approximations on larger matrices than traditional methods, while providing more accurate results in many cases than previous Kronecker-based singular value decompositions. We demonstrate the efficacy of these methods on a variety of image deconvolution problems for which the image is modeled as a regular grid of data.
Title: Counting Problems for Elliptic Curves over a Fixed Finite Field
Seminar: Algebra
Speaker: Nathan Kaplan of UC Irvine
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2018-03-27 at 4:00PM
Venue: W304
Abstract:
Let $E$ be an elliptic curve defined over a finite field $\mathbb{F}_q$. Hasses theorem says that $\#E(\mathbb{F}_q) = q + 1 - t_E$ where $|t_E| \le 2\sqrt{q}$. Deuring uses the theory of complex multiplication to express the number of isomorphism classes of curves with a fixed value of $t_E$ in terms of sums of ideal class numbers of orders in quadratic imaginary fields. Birch shows that as $q$ goes to infinity the normalized values of these point counts converge to the Sato-Tate distribution by applying the Selberg Trace Formula. \\ In this talk we discuss finer counting questions for elliptic curves over $\mathbb{F}_q$. For example, what is the probability that the number of rational points is divisible by $5$? What is the probability that the group of rational points is cyclic? If we choose a curve at random, and then pick a random point on that curve, what is the probability that the order of the point is odd? We study the distribution of rational point counts for elliptic curves containing a specified subgroup, giving exact formulas for moments in terms of traces of Hecke operators. We will also discuss some open problems. This is joint with work Ian Petrow (ETH Zurich).
Title: On large multipartite subgraphs of H-free graphs
Seminar: Combinatorics
Speaker: Jan Volec of McGill Univeristy
Contact: Dwight Duffus, dwight@mathcs.emory.edu
Date: 2018-03-26 at 4:00PM
Venue: W303
Abstract:
A long-standing conjecture of Erd\H{o}s states that any n-vertex triangle-free graph can be made bipartite by deleting at most $n^2/25$ edges. In this talk, we study how many edges need to be removed from an H-free graph for a general graph H. By generalizing a result of Sudakov for 4-colorable graphs H, we show that if H is 6-colorable then G can be made bipartite by deleting at most $4n^2/25+O(n)$ edges. In the case of $H=K_6$, we actually prove the exact bound $4n^2/25$ and show that this amount is needed only in the case G is a complete 5-partite graph with balanced parts. As one of the steps in the proof, we use a strengthening of a result of $F\ddot{u}redi$ on stable version of $Tur\acute{a}n's$ theorem. This is a joint work with P. Hu, B. $Lidick\acute{y}$, T. Martins-Lopez and S. Norin.
Title: Estimating bilinear forms via extrapolation
Seminar: Numerical Analysis and Scientific Computing
Speaker: Marilena Mitrouli of National and Kapodistrian University of Athens
Contact: Michele Benzi, benzi@mathcs.emory.edu
Date: 2018-03-21 at 4:00PM
Venue: W301
Abstract:
A spectrum of applications arising from Statistics, Machine Learning, Network Analysis require computation of bilinear forms $x^Tf(A)y$, where $A$ is a diagonalizable matrix and $x$, $y$ are given vectors. In this work we are interested in efficiently computing bilinear forms primarily due to their importance in several contexts. For large scale computation problems it is preferable to achieve approximations of bilinear forms without exploiting the whole matrix function. For this purpose an extrapolation procedure has been developed, attaining the approximation of bilinear forms with one, two or three term estimates in a complexity of square order. The extrapolation procedure gives us the flexibility to define the moments of a matrix $A$ either directly or through the polarization identity. The presented approach is characterized by easy applicable formulae of low complexity that can be implemented in vectorized form.
Title: On Spanning Trees with few Branch Vertices
Defense: Dissertation
Speaker: Warren Shull of Emory University
Contact: Warren Shull, warren.edward.shull@emory.edu
Date: 2018-03-05 at 4:00PM
Venue: W301
Abstract:
Hamiltonian paths, which are a special kind of spanning tree, have long been of interest in graph theory and are notoriously hard to compute. One notable feature of a Hamiltonian path is that all its vertices have degree two in the path. In a tree, we call vertices of degree at least three \emph{branch vertices}. If a connected graph has no Hamiltonian path, we can still look for spanning trees that come "close," in particular by having few branch vertices (since a Hamiltonian path would have none). \bigskip A conjecture of Matsuda, Ozeki, and Yamashita posits that, for any positive integer $k$, a connected claw-free $n$-vertex graph $G$ must contain either a spanning tree with at most $k$ branch vertices or an independent set of $2k+3$ vertices whose degrees add up to at most $n-3$. In other words, $G$ has this spanning tree whenever $\sigma_{2k+3}(G)\geq n-2$. We prove this conjecture, which was known to be sharp.
Title: Numerical and Streaming Analyses of Centrality Measures on Graphs
Seminar: Numerical Analysis and Scientific Computing
Speaker: Eisha Nathan of Georgia Institute of Technology
Contact: Michele Benzi, mbenzi@emory.edu
Date: 2018-03-02 at 2:00PM
Venue: W301
Abstract:
Graphs are a natural representation for modeling relationships between entities across many different fields of research. In real-world networks today, new data is constantly being produced, leading to the notion of dynamic graphs. An important query when analyzing graphs is to identify the most important vertices in a graph. Vertex importance is termed as centrality, and centrality scores can be used to provide rankings on the vertices of a graph. Specifically, I focus on Katz Centrality, a linear algebra based metric that measures the affinity between vertices in a graph as a weighted sum of the number of walks between them. Calculating Katz scores involves approximating the solution to a linear system using iterative solvers. Currently, we do not have a sense of how the numerical error from the iterative method in the approximation to the centrality vector affects the identification of the highly ranked vertices. In the first part of the talk, I bridge the two research areas of numerical analysis and network analysis by providing insight into how bounding the error between the approximate and exact solution in the numerical problem affects the correct relative ranking of vertices in the data mining problem. Typically, Katz scores are calculated through linear algebra. In the second section of the talk, I present an new alternative, agglomerative method of calculating personalized Katz scores. The algorithm presented is several orders of magnitude faster than the typical linear algebra approach and is able to return good quality scores of vertices. Finally, updating centrality scores of the vertices in a dynamic graph quickly becomes expensive to recompute from scratch every time the underlying graph is changed, as is the case in evolving networks. I conclude the talk by presenting a new algorithm for updating the Katz scores of vertices in a dynamic graph that is faster than static recomputation while maintaining good quality of the scores returned.
Title: On Cycles, Chorded Cycles, and Degree Conditions
Defense: Dissertation
Speaker: Ariel Keller of Emory University
Contact: Ariel Keller, ariel.keller@emory.edu
Date: 2018-03-01 at 3:00PM
Venue: MSC N301
Abstract:
Sufficient conditions to imply the existence of certain substructures in a graph are of considerable interest in extremal graph theory, and conditions that guarantee a large set of cycles or chorded cycles are a recurring theme. This dissertation explores different degree sum conditions that are sufficient for finding a large set of vertex-disjoint cycles or a large set of vertex-disjoint chorded cycles in a graph. \vskip.1in For an integer $t\ge 1$, let $\sigma_t (G)$ be the smallest sum of degrees of $t$ independent vertices of $G$. We first prove that if a graph $G$ has order at least $7k+1$ and degree sum condition $\sigma_4(G)\ge 8k-3$, with $k\ge 2$, then $G$ contains $k$ vertex-disjoint cycles. Then, we consider an equivalent condition for chorded cycles, proving that if $G$ has order at least $11k+7$ and $\sigma_4(G)\ge 12k-3$, with $k\ge 2$, then $G$ contains $k$ vertex-disjoint chorded cycles. We prove that the degree sum condition in each result is sharp. Finally, we conjecture generalized degree sum conditions on $\sigma_t(G)$ for $t\ge 2$ sufficient to imply that $G$ contains $k$ vertex-disjoint cycles for $k \ge 2$ and $k$ vertex-disjoint chorded cycles for $k \ge 2$. This is joint work with Ronald J. Gould and Kazuhide Hirohata.