# 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 Mon02/20/20124:00pm ColloquiumExploiting Sparsity and Low-Dimensional Structure: Techniques and ApplicationsEwout van den Berg, Stanford UniversityContact: James Nagy, nagy@mathcs.emory.eduVenue: Mathematics and Science Center, Room W201Download printable flyer (PDF, 40.4 kB)Show abstractThe apparent complexity of signals and data often belies a considerable amount of underlying structure that is characteristic of a particular source or application. A good example of such hidden structure is found in natural images, which are known to have an approximate sparse representation when expressed in certain wavelet bases. Knowledge of structure like this is highly valuable and can be utilized for various purposes such as data compression, denoising, signal reconstruction, or classification. Sparse signal structure plays a crucial role in compressed sensing. In particular, the main theoretical results underlying this field show that sparse signals can be stably recovered from a limited number of linear measurements by finding the least one-norm solution to a particular set of linear equations. The same technique can be used for other sparse recovery problems, and extensions to different types of low-dimensional structure, such a low-rank matrices, have been successfully applied and analyzed. In this talk, I will describe a specialized algorithm for solving an important class of sparse recovery problems in which a non-smooth convex objective is minimized subject to a two-norm constraint on a residual term. This class includes, as a special case, the one-norm minimization problem arising in compressed sensing. I will then outline the design of a silicon photomultiplier chip that takes advantage of spatial and temporal sparsity in photon arrival, thereby enabling a high spatio-temporal resolution while greatly reducing circuit complexity. As a third topic I will show how low-rank structure can be used to solve heterogeneous image registration and volume reconstruction problems arising in cryo-electron microscopy/tomography. Mon02/20/20123:00pm Seminar: Algebra and Number TheoryEssential dimension of central simple algebrasAnthony Ruozzi, UCLAContact: R. Parimala, parimala@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 37.8 kB)Show abstractThe essential dimension of an algebraic group $G$ measures the least number of parameters" needed to define a $G$-torsor. I will give a brief survey of how we can compute this number and how it relates to the structure of central simple algebras. Fri02/17/20124:00pm Seminar: CombinatoricsRamsey Numbers for Cycles RevisitedTomasz Luczak, Adam Mickiewicz University and Emory UniversityContact: Dwight Duffus, dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 105 kB) Thu02/16/20123:00pm ColloquiumUnderstanding Non-rigid Registration: Some Theory and ApplicationsHemant D. Tagare, Yale UniversityContact: James Nagy, nagy@mathcs.emory.eduVenue: EUH Annex - N120Download printable flyer (PDF, 39.4 kB)Show abstractAlgorithms for rigid and non-rigid registration are widely used, but a conceptual and theoretical framework for understanding non-rigid registration has been scarce. Research presented in this talk suggests that viewing registration algorithms from a geometric point-of-view provides a basis for understanding registration. In this framework, intra- and inter-modality registration appear on common ground and the fundamental role of the geometric volume form becomes clear. Different volume forms give registration algorithms with different properties. Four properties of registration objective functions are identified as useful, and a unique volume form is shown to impart these properties. Experimental results confirm that the theoretical results hold in practice, even the presence of noise in the images. Numerical techniques as well as applications to registration of Cardiac and Brain MRI will also be presented. Tue02/14/20124:00pm Seminar: Analysis and Differential GeometryPoint Allocations and Transport AlgorithmsAnastasia Svishcheva, Emory UniversityContact: Vladimir Oliker, oliker@mathcs.emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 35.5 kB)Show abstractThe work by Gershon Wolansky and Dmitry Meisler discussing surprising connections between the Monge-Kantorovich optimal transport and the problem of optimal partition of the state space will be presented. Mon02/13/20124:00pm ColloquiumMotion and spatial regularization designs in motion-compensated image reconstruction (MCIR): application to simultaneous PET-MRSe Young Chun, University of MichiganContact: James Nagy, nagy@mathcs.emory.eduVenue: Mathematics and Science Center, Room W201Download printable flyer (PDF, 43.4 kB)Show abstractOften medical imaging systems cannot capture ideal quality images due to their innate acquisition speeds and patient motion. Gating methods have been investigated to reduce motion artifacts, but can suffer from insufficient measurements that result in low SNR images. Motion-compensated image reconstruction (MCIR) methods based on statistical iterative image reconstruction have been studied to improve image quality by using all collected data and motion information so that high SNR images are reconstructed without motion artifacts. This talk presents two different regularization designs for MCIR. First of all, we investigated methods for motion regularization. The usual choice for a motion regularizer in MCIR has been an elastic regularizer. Recently, there has been much research on regularizing nonrigid deformations with diffeomorphic motion priors. Conventional methods that enforce deformations to be locally invertible require high computational complexity and large memory. We developed a sufficient condition that guarantees the local invertibility of B-spline deformation and proposed a simple regularizer based on that sufficient condition. This motion regularizer was applied to the motion correction using prototype simultaneous PET-MR. We estimated deformable motion from simultaneously acquired tagged MR volumes and incorporated it into the system matrix of unregularized OSEM algorithm for list-mode PET. We demonstrated the improvement of image quality and detection task performance with deformable phantom, rabbit, and small non-human primate studies. Secondly, we studied the spatial resolution properties of MCIR methods and shows that nonrigid local motion can lead to non-uniform and anisotropic spatial resolution for conventional quadratic regularizers. This undesirable property is akin to the known effects of interactions between heteroscedastic loglikelihoods (e.g. Poisson likelihood) and quadratic regularizers, and can cause non-uniform bias in small or narrow structures such as small lesions or rings of reconstructed images and lead to quantification errors. We proposed novel spatial regularization design methods for three different MCIR methods that account for known nonrigid motion. We develop MCIR regularization designs that provide approximately uniform and isotropic spatial resolution and that match a user-specified target spatial resolution. 2D PET simulations demonstrate the performance and benefits of the proposed spatial regularization design methods. Mon02/13/20123:00pm ColloquiumA geometric multigrid preconditioner for microFE analysis for bone structures based on a pointer-less octreePeter Arbenz, ETH Zurich, SwitzerlandContact: Michele Benzi, benzi@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 40.3 kB)Show abstractThe state of the art method to predict bone stiffness is micro finite element (microFE) analysis based on high-resolution computed tomography (CT). Modern parallel solvers enable simulations with billions of degrees of freedom. In this talk we present a solver that works directly on the CT-image and exploits the geometric properties given by the 3D-pixel. The data is stored in a pointer-less octree. The tree data structure provides different resolutions of the image that is used for the design of a geometric multigrid preconditioner. It makes possible the use of matrix-free implementations on all levels. This new solver reduces the memory footprint by more than a factor of 10 compared to a solver that uses an algebraic multigrid (AMG) preconditioner. It allows to solve much bigger problems and scales excellently on a Cray XT-5 Supercomputer. Fri02/10/20123:00pm ColloquiumScalable and Elastic Data Management in the CloudAmr El Abbadi, University of California, Santa BarbaraContact: Li Xiong, lxiong@emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 43.2 kB)Show abstractOver the past two decades, database and systems researchers have made significant advances in the development of algorithms and techniques to provide data management solutions that carefully balance the three major requirements when dealing with critical data: high availability, reliability, and data consistency. However, over the past few years the data requirements, in terms of data availability and system scalability, from Internet scale enterprises that provide services and cater to millions of users has been unprecedented. Cloud computing has emerged as an extremely successful paradigm for deploying Internet and Web-based applications. Scalability, elasticity, pay-per-use pricing, and autonomic control of large-scale operations are the major reasons for the successful widespread adoption of cloud infrastructures. In this talk, we analyze the design choices that allowed modern scalable data management systems to achieve orders of magnitude higher levels of scalability compared to traditional databases. With this understanding, we highlight some design principles for data management systems that can be used to augment existing databases with new cloud features such as scalability, elasticity, and autonomy. We then analyze several state of the art systems and discuss our proposed system, G-Store, which provides transactional guarantees on data granules formed on-demand while being efficient and scalable.\\ \\ Bio: Amr El Abbadi is currently a Professor in the Computer Science Department at the University of California, Santa Barbara. He received his B. Eng. in Computer Science from Alexandria University, Egypt, and received his Ph.D. in Computer Science from Cornell University in August 1987. Prof. El Abbadi is an ACM Fellow. He has served as a journal editor for several database journals, including, currently, The VLDB Journal. He has been Program Chair for multiple database and distributed systems conferences, most recently SIGSPATIAL GIS 2010 and ACM Symposium on Cloud Computing (SoCC) 2011. He has also served as a board member of the VLDB Endowment from 2002—2008. In 2007, Prof. El Abbadi received the UCSB Senate Outstanding Mentorship Award for his excellence in mentoring graduate students. He has published over 250 articles in databases and distributed systems. Thu02/09/20124:00pm ColloquiumAn Image-Based, Parallel Dynamic Meshing Framework for Patient-Specific Medical InterventionsSuzanne M. Shontz, Department of Computer Science and Engineering, Pennsylvania State UniversityContact: James Nagy, nagy@mathcs.emory.eduVenue: Mathematics and Science Center, Room W201Download printable flyer (PDF, 43.3 kB)Show abstractPatient-specific mesh generation is an important issue in computational simulations of the potential effectiveness of a medical device or the progression of a disease in a specific patient. Many radiology applications involve movement by the patient and medical device and require dynamic meshes for use in the associated imaged-based computational simulations. Two important examples include blood clot entrapment by inferior vena cava filters in pulmonary embolism (PE) patients and the evolution of the brain ventricles and cerebrospinal fluid (CSF) in hydrocephalus patients. There are several limitations of existing dynamic meshing algorithms that make them unsuitable for use in radiology applications. For example, the mesh warping algorithms upon which they are based often do not yield the desired volume meshes when applied to a noisy target surface mesh created from medical images of the patient. In addition, the mesh warping algorithms are often not designed to handle the large deformations that the patient's body and the medical device undergo. In particular, existing mesh warping algorithms sometimes yield tangled meshes with inverted elements which are invalid for use by the associated partial differential equation solver in the computational simulation. In this talk, I will describe the image-based, parallel dynamic meshing framework for patient-specific medical interventions, which we are designing. In particular, I will discuss two image-based meshing techninques we have developed for radiology applications. The first algorithm I will describe is a mesh warping technique for virtual implantation of an inferior vena cava (IVC) filter in the venous anatomy of a PE patient. The corresponding embedded geometric models are then used in generation of non-manifold topology meshes of the embedded device and venous anatomy for use in computational fluid dynamics simulations of the blood flow, and ultimately, for improved prevention of the disease. The second algorithm I will discuss is a combined level set/mesh warping technique for use in tracking, and, eventually, for predicting, the evolution of the deforming brain ventricles and CSF in hydrocephalus patients before and after shunt treatment. I will conclude by describing our plans for future research in these areas. Thu02/02/20123:00pm Seminar: Number TheoryQuadratic Forms Representing all Odd Positive IntegersJeremy Rouse, Wake Forest UniversityContact: Zachary A. Kent, kent@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 49.8 kB)Show abstractWe consider the problem of classifying all positive-de nite integer-valued quadratic forms that represent all positive odd integers. Kaplansky considered this problem for ternary forms, giving a list of 23 candidates, and proving that 19 of those represent all positive odds. (Jagy later dealt with a 20th candidate.) Assuming that the remaining three forms represent all positive odds, we prove that an arbitrary, positive-definite quadratic form represents all positive odds if and only if it represents the odd numbers from 1 up to 451. This result is analogous to Bhargava and Hanke's celebrated 290-theorem. In addition, we prove that these three remaining ternaries represent all positive odd integers, assuming the generalized Riemann hypothesis. This result is made possible by a new analytic method for bounding the cusp constants of integer-valued quaternary quadratic forms $Q$ with fundamental discriminant. This method is based on the analytic properties of Rankin-Selberg $L$-functions, and we use it to prove that if $Q$ is a quaternary form with fundamental discriminant, the largest locally represented integer $n$ for which $Q(\vec{x}) = n$ has no integer solutions is $O(D^{2+\epsilon})$.