# 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. Mon03/05/20184:00pm Defense: DissertationOn Spanning Trees with few Branch VerticesWarren Shull, Emory UniversityContact: Warren Shull, warren.edward.shull@emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 53.8 kB)Show abstractHamiltonian 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. 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 Tue04/15/201410:30am Undergraduate Thesis Defense: Computer ScienceSupport Vector Machine Classification of Resting State fMRI Datasets Using Dynamic Network ClustersHyo Yul Byun, Emory UniversityVenue: Mathematics and Science Center, Room E408Download printable flyer (PDF, 25.5 kB) Mon04/14/20144:00pm Seminar: CombinatoricsQuasirandom Discrete Structures and Powers of Hamilton CyclesHiep Han, Emory UniversityContact: Dwight Duffus, dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 36.8 kB)Show abstractThe aim of the talk is to give a gentle introduction into the topic of quasirandom discrete structures, putting emphasis on linear quasirandom hypergraphs and subsets of integers with small linear bias. We then continue with the study of the extremal behaviour of sparse pseudorandom graphs, a problem which has attracted the attention of many researchers in recent years. In particular, we shall discuss how to find powers of Hamilton cycles in sufficiently pseudorandom graphs. Fri04/04/20143:00pm Defense: DissertationAdaptive Approaches to Utility Computing for Scientific ApplicationsJaroslaw Slawinski, Emory UniversityContact: Jaroslaw Slawinski, jaross@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 41.3 kB)Show abstractCoupling scientific applications to heterogeneous computational targets requires specialized expertise and enormous manual effort. To simplify the deployment process, we propose a novel adaptive approach that helps execute unmodified applications on raw computational resources. Our method is based on situation-specific “adapter” middleware that builds up target capabilities to fulfill application requirements, avoiding homogenization that may conceal platform-specific features. We investigate three dimensions of adaptation: performance, execution paradigm, and software deployment and propose the ADAPT framework as a methodology and a toolkit that automates execution-related tasks. For parallel applications, ADAPT matches logical communication patterns to physical interconnect topology and improves execution performance by reducing use of long-distance connections. In a proof-of-concept demonstration of application–platform paradigm transformation, ADAPT enables execution of unmodified MPI applications on the Map–Reduce Platform as a Service cloud by recreating and emulating missing MPI capabilities. To facilitate software deployment, ADAPT automatically provisions resources by applying soft-install adapters that dynamically transform target capabilities to satisfy application requirements. As a result of these types of transformations, a broader spectrum of resources can smoothly execute scientific applications, which brings the notion of utility computing closer to reality. Fri04/04/201410:00am Seminar: CS Undergraduate Honors Thesis DefenseLive-Coding in Introductory Computer Science EducationAmy Shannon, Contact: Valerie Summet, valerie@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 37 kB)Show abstractLive-coding, an active learning technique in which students create code solutions during class through group discussion, is an under-used method in computer science education. However, this technique may produce greater learning gains than traditional lectures while requiring less time and effort from the instructor. We begin with a discussion of active learning techniques in STEM disciplines and then present a study to evaluate this instructional method in introductory Computer Science courses. While our results were inconclusive, we discuss several interesting and positive trends related to our live-coding results that deserve further investigation. Wed04/02/20146:00pm Some People Have All The LuckSkip Garibaldi / Lawrence Mower, Emory and UCLA / Palm Beach PostContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: MSC E208Download printable flyer (PDF, 193 kB)Show abstractWinning a prize of at least $600 in the lottery is a remarkable thing — for a typical scratcher ticket the odds are worse than 1-in-1200 and 1-in-9000 is a more typical figure. Some people have won several of these large prizes, and clearly they are very lucky or they buy a ton of lottery tickets. When we investigated records of all claimed lottery prizes, we discovered that some people had won hundreds of these prizes! Such people seem to be not just lucky, but suspiciously lucky. We will explain what we thought they might have been up to, what mathematics says about it, and what further investigations revealed. This talk is about joint work with Philip B. Stark. Skip Garibaldi is associate director of UCLA's Institute for Pure and Applied Mathematics and a professor in Emory University's Department of Mathematics & Computer Science. His previous work on the lottery received the Lester R. Ford Award and is the subject of a chapter in the popular book “Brain Trust". Millions of people have seen him talk about math on 20/20, CNN, and Fox & Friends, and he is featured in a museum exhibit about mathematics currently on display at Exploration Place in Wichita. Lawrence Mower is an investigative reporter with The Palm Beach Post. He joined The Post in 2013, after working for the Las Vegas Review-Journal, where his yearlong investigation into Las Vegas police shootings sparked a Department of Justice investigation and led to reforms in policy and oversight. The five-part series was awarded by the National Headliner Awards, Investigative Reporters and Editors, and the ACLU of Nevada, and in 2012 he was named Nevada's Outstanding Journalist by the Nevada Press Association. He is a 2006 graduate of the University of Nevada, Las Vegas. Wed04/02/20144:00pm Defense: DissertationLinear Preserver Problems and Cohomological InvariantsHernando Bermudez, Emory UniversityContact: Hernando Bermudez, hbermud@emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 38.2 kB)Show abstractLet G be a simple linear algebraic group over a field F. In this work we prove several results about G and it's representations.. In particular we determine the stabilizer of a polynomial f on an irreducible representation V of G for several interesting pairs (V,f). We also prove that in most cases if f is a polynomial whose stabilizer has identity component G then there is a correspondence between similarity classes of twisted forms of f and twisted forms of G. In a different direction we determine the group of normalized degree 3 cohomological invariants for most G which are neither simply connected nor adjoint. Wed04/02/20144:00pm SeminarValidation of an open source framework for the simulation of blood flow in rigid and deformable vesselsAnnalisa Quaini, University of HoustonVenue: Mathematics and Science Center, Room W302Download printable flyer (PDF, 36.7 kB)Show abstractWe discuss the validation of an open source framework for the solution of problems arising in hemodynamics. The framework is assessed through experimental data for fluid flow in an idealized medical device with rigid boundaries and a numerical benchmark for flow in compliant vessels. The core of the framework is an open source parallel finite element library that features several algorithms for fluid and fluid-structure interaction problems. A detailed account of the methods is provided. Tue04/01/20144:00pm Seminar: AlgebraNumber Theory of MoonshineSarah Trebat-Leder, Emory UniversityContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W302Download printable flyer (PDF, 45.4 kB)Show abstractThe classical theory of monstrous moonshine describes the unexpected connection between the representation theory of the monster group$M$, the largest of the simple sporadic groups, and certain modular functions, called Hauptmodln. For example, the$n$th Fourier coefficient of Klein's$j(\tau)$function is the dimension of the grade$n$part of a special infinite dimensional representation$V$of the monster group. Similar phenomena have been shown to hold for the Matthieu group$M_{24}$, but instead of modular functions, mock modular forms must be used. This has been conjecturally generalized even further, to umbral moonshine, which associates to each of 23 Niemeier lattices a finite group, infinite dimensional representation, and mock modular form. We use generalized Borcherds products to relate monstrous moonshine and umbral moonshine. Namely, we show that twisted traces of classical moonshine functions are equal to coefficients of umbral mock modular forms. We also show that certain umbral coefficients have$p$-adic properties. Tue04/01/20142:30pm Undergraduate Honors Thesis Defense: Algebra3F2-hypergeometric functions and supersingular elliptic curvesSarah Pitman, Emory UniversityContact: Ken Ono, ono@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 39.8 kB)Show abstractHere we explore elliptic curves, specifically supersingular elliptic curves, and their relationship to hypergeometric functions. We begin with some background on elliptic curves, supersingularity, hypergeometric functions, and then use work of El-Guindy, Ono, Kaneko, Zagier, and Monks to extend results. In recent work, Monks described the supersingular locus of families of elliptic curves in terms of 2F1-hypergeometric functions. We “lift" his work to the level of 3F2-hypergeometric functions by means of classical transformation laws and a theorem of Clausen. Mon03/31/201411:30am Defense: DissertationCombinatorial Objects at the Interface of$q$-series and Modular FormsMarie Jameson, Emory UniversityContact: Marie Jameson, mjames7@emory.eduVenue: Mathematics and Science Center, Room E406Download printable flyer (PDF, 42.1 kB)Show abstractIn this work, the author proves various results related to$q$-series and modular forms by employing a broad range of tools from analytic number theory, combinatorics, the theory of modular forms, and algebraic number theory. More specifically, the circle method, the connection between modular forms and elliptic curves, continued fractions, period polynomials, and several other tools from the theory of modular forms are used here. These allow the author to prove a number of results related to$q\$-series and partition functions, modular forms, period polynomials, and certain quadratic polynomials of a fixed discriminant. This includes a proof of the Alder-Andrews conjecture on certain restricted partition functions, and a resolution of a speculation of Don Zagier regarding the Eichler integrals of a distinguished class of modular forms.