# 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 Thu03/26/20153:00pm Seminar: AlgebraInvolutions, odd degree extensions and generic splittingAnne Queguiner-Mathieuem, Universite ParisContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 40.3 kB)Show abstractLet $q$ 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. Tue03/24/20154:00pm Seminar: AlgebraComparison of compactifications of modular curvesAndrew Niles, Holy CrossContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 38.7 kB)Show abstractModular 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. Mon03/23/20154:00pm Seminar: CombinatoricsRecent progress on diamond-free familiesRyan Martin, Iowa State UniversityContact: Dwight Duffus, Dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W302Download printable flyer (PDF, 40 kB)Show abstractIn 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. Thu03/19/20154:00pm Seminar: CombinatoricsThe Li-Yau Inequality and the Geometry of GraphsPaul Horn, The University of DenverContact: Dwight Duffus, Dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 37.7 kB)Show abstractUnderstanding 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. Tue03/17/20154:00pm Seminar: AlgebraMock theta functions and quantum modular formsLarry Rolen, University of CologneContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 37.8 kB)Show abstractIn 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. Mon03/16/20154:00pm ColloquiumInteractive Machine Learning Across DomainsLev Reyzin, University of Illinois at ChicagoContact: Vaidy Sunderam, vss@emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 35.4 kB)Show abstractInteractive 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. Tue03/10/20154:00pm Seminar: Numerical Analysis and Scientific ComputingAlgebraic Preconditioning of Symmetric Indefinite SystemsMiroslav Tuma, Academy of Sciences of the Czech RepublicContact: Michele Benzi, benzi@mathcs.emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 41.4 kB)Show abstractSparse symmetric indefinite linear systems of equations arise in many practical applications. An iterative method is frequently the method of choice to solve such systems but a system transformation called preconditioning is often required for the solver to be effective. In the talk we will deal with development of incomplete factorization algorithms that can be used to compute high quality preconditioners. We will consider both general indefinite systems and saddle-point problems. Our approach is based on the recently adopted limited memory approach (based on the work of Tismenetsky, 1991) that generalizes recent work on incomplete Cholesky factorization preconditioners. A number of new ideas are proposed with the goal of improving the stability, robustness and efficiency of the resulting preconditioner. For general indefinite systems, these include the monitoring of stability as the factorization proceeds and the use of pivot modifications when potential instability is observed. Numerical experiments involving test problems arising from a range of real-world applications are used to demonstrate the effectiveness of our approach and comparisons are made with a state-of-the-art sparse direct solver. The talk will be based on joint work with Jennifer Scott, Rutherford Appleton Laboratory. Tue03/03/20154:00pm Seminar: AlgebraZero-cycles and rational points on rationally connected varieties, after Harpaz and WittenbergJean-Louis Colliot-Thelene, Universite Paris-SudContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 37.6 kB)Show abstractHarpaz and Wittenberg have recently proved a general result on the local-global principle for zero-cycles on rationally connected varieties. There is also a conditional variant for rational points. I shall explain some of the ideas in their paper. Reference : http://arxiv.org/abs/1409.0993 Tue03/03/20154:00pm ColloquiumExtracting medically interpretable concepts from complex health dataJoyce Ho, University of Texas at AustinContact: Vaidy Sunderam, vss@emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 41.4 kB)Show abstractElectronic health records (EHRs) are an increasingly important source of patient information. Efficient analysis of EHRs can help address many healthcare problems by improving clinical decisions, facilitating knowledge discoveries, and enabling the development of cost-effective treatment and management programs. However, EHRs pose many formidable challenges for traditional analytics. The data are collected across diverse populations, consist of heterogeneous and noisy information, and have varying time resolutions. Moreover, healthcare professionals are unaccustomed to interpreting high-dimensional EHRs; they prefer concise medical concepts. Thus, a major question is how to transform EHR into meaningful concepts with modest levels of expert guidance.\\ \\ In this talk, I will discuss two approaches to extract concise, meaningful concepts from certain types of health datasets. First, I will describe a dynamic time series model that tracks a patient's cardiac arrest risk based on physiological measurements (i.e., heart rate, blood pressure, etc.) in an intensive care unit. Our algorithm is inspired by financial econometric and yields interpretability and predictability of a cardiac arrest event. Next, I will present a sparse, nonnegative tensor factorization model to obtain multiple medical concepts with minimal human supervision. Tensor factorization utilizes information in the multiway structure to derive concise latent factors even with limited observations. We applied tensor analysis to real EHRs from the Geisinger Health System to automatically identify relevant medical concepts. Both our models are powerful and data-driven approaches to extract medically interpretable concepts from complex health data. Mon03/02/20154:00pm ColloquiumUnderstanding Information: From Bits to BrainsAvani P. Wildani nee Gadani, The Salk InstituteContact: Vaidy Sunderam, vss@emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 36.4 kB)Show abstractInformation is the currency of the modern era, and there are surprising similarities in data processing and representation between computer systems and neuroscience. In the first half of this talk, I will discuss how to dynamically identify related blocks or files in trace data and use the resulting data groups to make information storage more efficient and robust. From there, I will discuss how the classical systems metrics of reliability, performance, and availability apply to biologically plausible neural networks, including recent work exploring the balance between classification accuracy and robustness. Finally, I will show how computational models from vision can be applied to understand information flow in the visual cortex, and how algebraic topology is a promising method to classify neurons by network function and categorize visual stimuli.