# 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 Thu11/16/20172:30pm Defense: Honors ThesisApplication of Global Optimization to Image RegistrationHuiying Zhu, Emory UniversityVenue: Mathematics and Science Center, Room E408Download printable flyer (PDF, 38.9 kB)Show abstractGiven two images, image registration aims to transform an image into a given reference image so that the two images look alike. This technique is vital in many applications, such as medical imaging and astronomy. Finding the best transform can be phrased as solving a mathematical optimization problem. Due to the non-convexity of the objective function, commonly employed optimization techniques often generate local minimizers, limiting the accuracy of the registration. This thesis evaluates the applicability of a global optimization method, called as DDNCID, for image registration. Direct application of DDNCID in image registration could cause minimizers to be infeasible. Thus, a focus of this thesis is to add a bound constraint by imposing a barrier function into the objective function to extend DDNCID. Tue11/14/20175:00pm Seminar: AlgebraOn semi-simplicity of tensor products in positive characteristicsVikraman Balaji, Chennai Mathematical InstituteContact: John Duncan, john.duncan@emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 43.2 kB)Show abstractWe work over an algebraically closed field k of characteristic p greater than 0. In 1994, Serre showed that if semi-simple representations V_i of a group \Gamma are such that \sum ( dim(V_i) - 1 ) less than p, then their tensor product is semi-simple. In the late nineties, Serre generalized this theorem comprehensively to the case where \Gamma is a subgroup of G(k), for G a reductive group, and answered the question of “complete reducibility” of \Gamma in G (Seminaire Bourbaki, 2003). In 2014, Deligne generalized the results of Serre (of 1994) to the case when the V_i are semi-simple representations of a group scheme \mathfrak{G}. In my talk I will present the recent work of mine with Deligne and Parameswaran where we consider the case when \mathfrak{G} is a subgroup scheme of a reductive group G and generalize the results of Serre and Deligne. A key result is a structure theorem on “doubly saturated” subgroup schemes \mathfrak{G} of reductive groups G. As an application, we obtain an analogue of classical Luna's etale slice theorem in positive characteristics. Tue11/14/20174:00pm Seminar: AlgebraAn arithmetic count of the lines on a cubic surface.Kirsten Wickelgren, Georgia Institute of TechnologyContact: John Duncan, john.duncan@emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 39.3 kB)Show abstractA celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. Over the real numbers, it is a lovely observation of Finashin–Kharlamov and Okonek–Teleman that while the number of real lines depends on the surface, a certain signed count of lines is always 3. We extend this count to an arbitrary field k using an Euler number in A1-homotopy theory. The resulting count is valued in the Grothendieck-Witt group of non-degenerate symmetric bilinear forms. This is joint work with Jesse Kass. Mon11/13/20174:00pm Seminar: CombinatoricsRamsey Properties of Random Graphs and HypergraphsAndrzej Dudek, Western Michigan UniversityContact: Dwight Duffus, dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W302Download printable flyer (PDF, 106 kB) Fri11/10/20172:00pm Seminar: Numerical Analysis and Scientific ComputingAccelerated Diffeomorphisms for Motion Estimation and Segmentation from VideoGanesh Sundaramoorthi, KAUSTContact: James Nagy, nagy@mathcs.emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 45.6 kB)Show abstractAccelerated optimization methods have gained wide applicability within the machine learning and optimization communities. They are known for leading to optimal convergence rates among schemes that use only use gradient (first order) information in the convex case. In the non-convex case, they appear to provide robustness to shallow local minima. The intuitive idea is that by considering a particle with mass that moves in an energy landscape, the particle will gain momentum and surpass shallow local minimum and settle in in more wider, deeper local extrema in the energy landscape. Although these techniques have been widely used, it was only within the last few years that theoretical attempts have been made to understand them and put them in a mathematical framework. Recent work has shown that accelerated methods may be formulated with variational principles, although in finite dimensions. Motivated by the success of accelerated methods in finite dimensional problems, we formulate optimization problems on infinite dimensional manifolds of diffeomorphisms using a generalization of this approach. The talk will mainly be about the mathematical formulation and some simple examples to illustrate the advantages of this approach. We note very large speed-ups in optical flow computation compared with standard approaches, and robustness to local minimum. Finally, we outline considerations for generalizing this approach to video data and applications in motion-based object segmentation, which require one to optimize diffeomorphisms not just defined on the image domain, but evolving regions of interest that encompass the domain of each of the objects in the scene.\\ \\Bio: Ganesh Sundaramoorthi received the PhD in Electrical and Computer Engineering from Georgia Institute of Technology, Atlanta, USA, and BS in Computer Engineering and BS Mathematics from the same institution in 2003. He was then a postdoctoral researcher in the Computer Science Department at the University of California, Los Angeles between 2008 and 2010. In 2011, he was appointed Assistant Professor of Electrical Engineering and Assistant Professor of Applied Mathematics and Computational Science at King Abdullah University of Science and Technology (KAUST). His research interests include computer vision and its mathematical foundations with recent interest in shape and motion analysis, video analysis, invariant representations for visual tasks, and applications. He was an area chair for IEEE ICCV 2017 and IEEE CVPR 2018. Tue11/07/20174:00pm Seminar: AlgebraCongruences from quaternion algebrasKimball Martin, University of OklahomaContact: John Duncan, john.duncan@emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 38.4 kB)Show abstractStudying congruences between modular forms is a prosperous avenue in number theory. One approach to obtaining congruences involves computations on the Jacobian (Mazur, Ribet, ...). For instance, Mazur uses the Jacobian to determine when there is a weight 2 cusp form of prime level congruent to an Eisenstein series, which has various applications. We will explore another approach to obtaining congruences of modular forms using the arithmetic of quaternion algebras and the Jacquet-Langlands correspondence. This will lead to (1) generalizations work of Mazur and Ribet on weight 2 Eisenstein congruences, and (2) a phenomenon of many mod 2 congruences between weight k cusp forms. Mon11/06/20174:00pm Seminar: CombinatoricsExtremal number of configurations in a gridMarcelo Sales, University of Sao PauloContact: Dwight Duffus, dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W302Download attached abstract (PDF, 103 kB)Download printable flyer (PDF, 40.5 kB)Show abstractA configuration is a finite set of points with no three collinear. Two configurations have the same order type if there exists a bijection between these two configurations that preserves the orientation of every ordered triple. A configuration A contains a copy of a configuration B some subset of A has the same order type of B and we denote by B \subset A. For a configuration B and an integer m, the extremal number ex(m,B)= max {|A| : B is not a subset of A, A \subset [m]^2} is the maximum size of a subset of the grid $[m]^2$ without a copy of $B$. We discuss some bounds on this function for general B. Fri11/03/20172:00pm Seminar: Numerical Analysis and Scientific ComputingInsights from computational fluid dynamic modelling for aortic arch pathologiesMassimiliano Marrocco-Trischitta, San Donato Hospital in Milan, ItalyContact: Adrien Lefieux, adrien.lefieux@emory.eduVenue: Mathematics and Science Center, Room W301Download attached abstract (PDF, 51.7 kB)Download printable flyer (PDF, 40.9 kB)Show abstractObjectives. To assess whether the geometrical and hemodynamic reappraisal of the Ishimaru’s Aortic Arch Map according to Aortic Arch Classification in Type I, II, and III, may provide valuable information regarding the suitability for thoracic endovascular aortic repair (TEVAR), and the risk of aortic dissection. Methods. Anonymized thoracic computed tomography scans of healthy aortas were reviewed, and stratified according to the Aortic Arch Classification. Twenty patients of each Type of Arch were selected. Further processing allowed calculation of angulation and tortuosity of each proximal landing zones. Data were described indicating both proximal landing zone and Type of Arch (e.g. 0/I). Also, among these 60 CT angiography scans, 15 were selected, 5 per Type of Arch, for further analysis. Computational fluid dynamics were performed to compute displacement forces, exerted by pulsatile blood flow on the aortic wall in the defined landing areas. Equivalent surface tractions were computed dividing the displacement forces magnitude of each proximal landing zone by the corresponding area. The three-dimensional orientation (x,y,z) of displacement forces was described as an upward (z direction), and a sideways component (x-y plane). Thu11/02/20174:00pm Defense: DissertationImproving Question Answering by Bridging Linguistic Structures and Statistical LearningTomasz Jurczyk, Emory UniversityVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 38.9 kB)Show abstractQuestion answering (QA) has lately gained lots of interest from both academic and industrial research. No matter the question, search engine users expect the machines to provide answers instantaneously, even without searching through relevant websites.\\ \\While a significant portion of these questions ask for concise and well known facts, more complex questions do exist and they often require dedicated approaches to provide robust and accurate systems.\\ \\This thesis explores linguistically-oriented approaches for both factoid and non-factoid question answering and applications to cross-genre tasks. The contributions include new annotation schemes for the question answering oriented corpora, extracting linguistic structures and performing matching, and early exploration of applications to conversation dialog tasks. Tue10/24/20175:00pm Seminar: AlgebraSurvey on recent results on maximal tori of algebraic groupsPhilippe Gille, CNRS, LyonContact: John Duncan, john.duncan@emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 37.9 kB)Show abstractPrasad and Rapinchuk investigated the isopectrality problem for certain Riemannian varieties by analysing in what extent a semisimple algebraic group defined over a number field is determined by its maximal tori. We shall report advances on this topic by Chernousov/Rapinchuk/Rapinchuk, Bayer-Fluckiger/Lee/Parimala and others by discussing the case of non-archimedean fields and local-global principles.