# Seminars archive

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 Upcoming Seminars Tue02/27/20184:00pm Colloquium: AlgebraCounting points, counting fields, and heights on stacksJordan Ellenberg, University of Wisconsin-MadisonContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 59.6 kB)Show abstractThe basic objects of algebraic number theory are number fields, and the basic invariant of a number field is its discriminant, which in some sense measures its arithmetic complexity. A basic finiteness result is that there are only finitely many degree-$d$ number fields of discriminant at most $X$; more generally, for any fixed global field $K$, there are only finitely many degree-$d$ extensions $L/K$ whose discriminant has norm at most $X$. (The classical case is where $K = \mathbb{Q}$.) \\ When a set is finite, we greedily ask if we can compute its cardinality. Write $N_d(K,X)$ for the number of degree-$d$ extensions of $K$ with discriminant at most $d$. A folklore conjecture holds that $N_d(K,X)$ is on order $c_d X$. In the case $K = \mathbb{Q}$, this is easy for $d=2$, a theorem of Davenport and Heilbronn for $d=3$, a much harder theorem of Bhargava for $d=4$ and 5, and completely out of reach for $d > 5$. More generally, one can ask about extensions with a specified Galois group $G$; in this case, a conjecture of Malle holds that the asymptotic growth is on order $X^a (\log X)^b$ for specified constants $a,b$. \\ I'll talk about two recent results on this old problem: \\ 1) (joint with TriThang Tran and Craig Westerland) We prove that $N_d(\mathbb{F}_q(t),X)) < c_{\epsilon} X^{1+\epsilon}$ for all $d$, and similarly prove Malle’s conjecture up to epsilon" — this is much more than is known in the number field case, and relies on a new upper bound for the cohomology of Hurwitz spaces coming from quantum shuffle algebras: https://arxiv.org/abs/1701.04541 \\ 2) (joint with Matt Satriano and David Zureick-Brown) The form of Malle's conjecture is very reminiscent of the Batyrev-Manin conjecture, which says that the number of rational points of height at most $X$ on a Batyrev-Manin variety also grows like $X^a (\log X)^b$ for specified constants $a,b$. What’s more, an extension of $\mathbb{Q}$ with Galois group $G$ is a rational point on a Deligne--Mumford stack called $BG$, the classifying stack of $G$. A natural reaction is to say “the two conjectures is the same; to count number fields is just to count points on the stack BG with bounded height?” The problem: there is no definition of the height of a rational point on a stack. I'll explain what we think the right definition is, and explain how it suggests a heuristic which has both the Malle conjecture and the Batyrev--Manin conjecture as special cases. Thu03/01/20183:00pm Defense: DissertationOn Cycles, Chorded Cycles, and Degree ConditionsAriel Keller, Emory UniversityContact: Ariel Keller, ariel.keller@emory.eduVenue: MSC N301Download printable flyer (PDF, 50.4 kB)Show abstractSufficient 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. Tue03/27/20184:00pm Seminar: AlgebraTitle to be announcedNathan Kaplan, UC IrvineContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 19.2 kB) Mon04/02/20184:00pm Defense: DissertationPatching and local-global principles for gerbes with an application to homogeneous spacesBastian Haase, Emory UniversityContact: Bastian Haase, bastian.haase@emory.eduVenue: Mathematics and Science Center, Room W302Download printable flyer (PDF, 44.9 kB)Show abstractStarting 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. Tue04/03/20184:00pm Seminar: AlgebraTitle to be announcedJennifer Berg, RiceContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 19.4 kB) Thu04/05/20184:00pm ColloquiumTitle to be announcedSherry Li, Lawrence Berkeley National LabContact: Lar Ruthotto, lruthotto@emory.eduVenue: Mathematics and Science Center, Room W201Download printable flyer (PDF, 19.3 kB) Thu04/12/20184:00pm Colloquium: AlgebraTitle to be announcedK. Soundararajan, Stanford UniversityContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 19.7 kB) Tue04/17/20184:00pm Seminar: AlgebraTitle to be announcedBrandon William, UC BerkeleyContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 19.1 kB) Tue04/24/20184:00pm Seminar: AlgebraTitle to be announcedFrank Thorne, University of South CarolinaContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 19.3 kB) Past Seminars Fri09/22/20172:00pm Seminar: Numerical Analysis and Scientific ComputingA new tensor framework - theory and applicationsDr. Misha Kilmer, Tufts UniversityContact: James Nagy, jnagy@emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 39.6 kB)Show abstractTensors (aka multiway arrays) can be instrumental in revealing latent correlations residing in high dimensional spaces. Despite their applicability to a broad range of applications in machine learning, speech recognition, and imaging, inconsistencies between tensor and matrix algebra have been complicating their broader utility. Researchers seeking to overcome those discrepancies have introduced several different candidate extensions, each introducing unique advantages and challenges. In this talk, we review some of the common tensor definitions, discuss their limitations, and introduce our tensor product framework which permits the elegant extension of linear algebraic concepts and algorithms to tensors. Following introduction of fundamental tensor operations, we discuss in further depth tensor decompositions and in particular the tensor SVD (t-SVD) and its randomized variant, which can be computed efficiently in parallel. We present details of the t-SVD, theoretical results, and provide numerical results that show the promise of our approach for compression and analysis of operators and datasets, highlighting examples such as facial recognition and model reduction. Thu09/21/20174:00pm Seminar: AlgebraUnifying relaxed notions of modular formsMartin Raum, Chalmers Technical University, Gothenburg, SwedenContact: John Duncan, john.duncan@emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 41 kB)Show abstractElliptic modular forms are functions on the complex upper half plane that are invariant under a certain action of the special linear group with integer entries. Their history comprises close to two centuries of amazing discoveries and application: The proof of Fermat's Last Theorem is probably the most famous; The theory of theta functions is among its most frequently employed parts.\\ \\During the past decade it has been à la mode to study relaxed notions of modularity. Relevant keywords that we will discuss are mock modular forms and higher order modular forms. We have witnessed their application, equally stunning as surprising, to conformal field theory, string theory, combinatorics, and many more areas.\\ \\In this talk, we suggest a change of perspective on such generalizations. Most of the novel variants of modular forms (with one prominent exception) can be viewed as components of vector-valued modular forms. This unification draws its charm from the past and the future. On the one hand, we integrate results by Kuga and Shimura that hitherto seemed almost forgotten. On the other hand, we can point out connections, for example, between mock modular forms and so-called iterated integrals that have not yet been noticed. Experts will be pleased to have in the future a Petersson pairing'' for mixed mock modular forms at their disposal.\\ \\This is joint work with Michael Mertens Fri09/15/20172:00pm Seminar: Numerical Analysis and Scientific ComputingOn the Birkhoff--von Neumann decomposition and its use in solving sparse linear systemsDr. Bora Ucar, CNRS and ENS Lyon, France (visiting GaTech this year)Contact: Michele Benzi, benzi@mathcs.emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 42.1 kB)Show abstractThe Birkhoff--von Neumann decomposition expresses a doubly stochastic matrix as a convex combination of permutation matrices. This talk will be an introduction to this decomposition. We are going to see its use in solving sparse linear systems, and investigate some algorithmic and combinatorial problems associated with it. This talk contains results from joint work with Michele Benzi (Emory Univ., Atlanta), Fanny Dufosse (Inria, France), Kamer Kaya (Sabanci Univ, Turkey), and Ioannis Panagiotas (ENS Lyon, France). Thu09/14/20174:00pm Seminar: AlgebraLinked Fields of Characteristic 2 and their u-Invariant.Dr. Adam Chapman, Tel-Hai Academic College, Israel.Contact: Dr. John Duncan, john.duncan@emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 39.6 kB)Show abstractThe u-invariant of a field is the maximal dimension of a nonsingular anisotropic quadratic form over that field, whose order in the Witt group of the field is finite. By a classical theorem of Elman and Lam, the u-invariant of a linked field of characteristic different from 2 can be either 0,1,2,4 or 8. The analogous question in the case of characteristic 2 remained open for a long time.. We will discuss the proof of the equivalent statement in characteristic 2, recently obtained in a joint work by Andrew Dolphin and the speaker. Tue08/29/20174:00pm Seminar: AlgebraSpectrum of singularities, exponential sums and the irreducibility of polynomials in two variablesJorge Jimenez Urroz, U. Politecnica, CatalunyaContact: John Duncan, john.duncan@emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 39.9 kB)Show abstractIn order to control the spectrum of singularities of the generalized Riemann function, we need to find sharp bounds for certain kind of Gauss sums with frequencies on polynomials. This can be achieved by Weil bounds on completely irreducible algebraic curves, which lead us to prove some theorems on irreducibility of polynomials in two variables. We will prove a general theorem in this field. An example of the theorem is the absolute irreducibility of p(x)-p(y)+1, for any p(x) with integer coefficients. Thu08/10/20172:00pm Defense: DissertationTopics in Tropical and Analytic GeometryCharles Morrissey, Emory UniversityContact: Charles Morrissey, cjmorr3@emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 36 kB)Show abstractIn this thesis, the author proves theorems on the existence and mapping properties of tropical stacks that arise from result concerning toric Artin stacks. The author also provides a generalization, using the same ideas from the toric Artin stacks, of recent work involving analytic stacks and their tropicalizations. The author also proves results on the notion of a tropical jet space. Fri07/07/201710:00am DefenseApplication of the DIKW Model in Malaria Systems Biology: From NGS Data to Disease Progression InsightJung-Ting Chien, Emory UniversityContact: Jung-Ting Chien, jchien2@emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 45.9 kB)Show abstractThe data, information, knowledge and wisdom (DIKW) model has been widely used in data science fields to generate a comprehensive view of each domain. It provides a hierarchical representation of the understanding of the domain knowledge; the DIKW model can reveal insights in systems biology by integrating different types of –omics data to form a comprehensive understanding.\\ \\The foundation of systems biology is mining genomics data with machine learning. As the use of high-throughput, next-generation sequencing (NGS) applications grows, research in genomics enters the “big data” era. NGS applications can be divided into two major categories, short-read and long-read techniques, which are based on the principle differences in generating “reads”. A “read” is the fundamental element of genomic information. Short-read applications have been widely applied in several fields of genomics research, while long-read applications just came to market in 2011. Long-read applications have shown the potential to handle several areas of genomic questions. However, obtaining a well-defined genome still has a number of challenges in malaria systems biology research, and these challenges block researchers’ understanding the mechanism of the malaria disease progression.\\ \\To tackle these challenges, we built a novel long-read NGS pipeline with third party modules and modified them to solve complicated Plasmodium genome assembly questions. These techniques provided a solution where traditional, short-read technologies could not because of the Plasmodium genome’s highly repetitive nature. We also implemented infrastructure to solve data management difficulties and developed several novel and robust pipelines to process and analyze the data. We host this pipeline along with other third party applications for data quality control, generic data visualization and data management tools. Our pipeline is also scalable and flexible to combine different technologies (long reads and short reads) to assemble the Plasmodium genome and conduct downstream annotations.\\ \\This dissertation describes an overview of –omics research in the big data era and reveals the possibility of applying DIKW models through mining genomics data. A detailed discussion on how to apply our platform to solve questions, including multiple Plasmodium genome assemblies and annotations, and an initial discussion of applying machine learning approaches in a host-pathogen transcriptome analysis and its data mining applications are also provided. Mon05/22/20174:00pm Seminar: Computer SciencePerfect Secrecy vs. Computational Security in Private Key Encryption SchemesSteven La Fleur, Emory UniversityVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 37.8 kB)Show abstractAs evidenced by recent events, privacy and security of data is increasingly important. There is a lot of interest in the ability to securely encrypt and send messages between two parties in such a way that any potential eavesdropper will be unable to read the message. But what does "security" of an encryption scheme mean, and how do we measure how secure a given scheme is?\\ \\In this talk we will investigate formal definitions for security of an encryption schemes, and what it means to prove that an encryption scheme is secure using these definitions. We will consider the practical drawbacks of "perfect secrecy" and how the definitions and assumptions made for computational security fix these drawback while still maintaining secrecy from attackers of different strengths.\\ \\The talk is intended for undergraduate students who have taken a course in discrete mathematics for computer science and have a basic understanding of probability, theory of computation and rigorous proof. Wed05/10/20171:00pm Defense: DissertationEfficient and Adaptive Skyline ComputationJinfei Liu, Emory UniversityContact: Jinfei Liu, jliu253@emory.eduVenue: Mathematics and Science Center, Room E406Download printable flyer (PDF, 40.4 kB)Show abstractSkyline, also known as Maxima in computational geometry or Pareto in business management field, is important for many applications involving multi-criteria decision making. The skyline of a set of multi-dimensional data points consists of the points for which no other point exists that is better in at least one dimension and at least as good in every other dimension. Although skyline computation and queries have been extensively studied in both computational geometry and database communities, there are still many challenges need to be fixed, especially in this big data ear. In this dissertation, I present several efficient and adaptive skyline computation algorithms. First, I show a faster output-sensitive skyline computation algorithm which is the state-of-the-art algorithm from the theoretical aspect. Second, traditional skyline computation is inadequate to answer queries that need to analyze not only individual points but also groups of points. To address this gap, I adapt the original skyline definition to the novel group-based skyline (G-Skyline), which represents Pareto optimal groups that are not dominated by other groups. Third, to facilitate skyline queries, I propose a novel concept Skyline Diagram, which given a set of points, partitions the plane into a set of regions, referred to as skyline polyominos. Similar to kth-order Voronoi diagram commonly used to facilitate k nearest neighbor (kNN) queries, any query points in the same skyline polyomino have the same skyline query results. Fri04/28/20171:00pm Seminar: Numerical Analysis and Scientific ComputingNon-backtracking walk centrality for directed networksFrancesca Arrigo, University of StrathclydeContact: Michele Benzi, benzi@mathcs.emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 41.4 kB)Show abstractThe talk is motivated by a practical issue: walk-based centrality measures regard all walks of the same length as being equally important, whereas it is intuitively reasonable to rule out certain classes of walk. We focus here on non-backtracking walks. The theory of zeta functions provides an expression for the generating function of non-backtracking walk counts on a directed network. This expression can be used to produce a centrality measure that eliminates backtracking walks at no cost. The new centrality measure may be interpreted as standard Katz on a modified network, where self loops are added, and where non-reciprocated edges are augmented with negative weights. We also give a multilayer interpretation of the new centrality measure, where (negatively) weighted walks between layers compensate for backtracking walks on the only non-empty layer. We further show that the radius of convergence of the generating function is determined by the spectrum of a three-by-three block matrix involving the original adjacency matrix. This gives a means to choose appropriate values of the attenuation parameter and, in particular, we show that we obtain a larger range of choices for the attenuation parameter than that obtained for standard Katz. By studying the effect of pruning operations on the network (i.e., removing nodes), we show that there is potential for the non-backtracking centrality to be computed more cheaply than Katz for appropriate network structures. Studying the limit as the attenuation parameter approaches its upper bound allows us to propose an eigenvector-based non-backtracking centrality measure in this directed network setting. We illustrate the centrality measure on a synthetic network, where it is shown to eliminate a localization effect present in standard Katz centrality. We also give results for real networks. Finally, we discuss some preliminary results on the non-backtracking version of the total communicability and of some alternating walk-based centrality measures.\\ \\This talk is based on joint work with Prof. Peter Grindrod (University of Oxford, UK), Prof. Desmond J. Higham (University of Strathclyde, UK), and Dr. Vanni Noferini (University of Essex, UK).