All Seminars
| Title: Odd order transcendental obstructions to the Hasse principle on general K3 surfaces |
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| Seminar: Algebra |
| Speaker: Jennifer Berg of Rice University |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2018-04-03 at 4:00PM |
| Venue: W304 |
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| Abstract: Varieties that fail to have rational points despite having local points for each prime are said to fail the Hasse principle. A systematic tool accounting for these failures is called the Brauer-Manin obstruction, which uses [subsets of] the Brauer group, Br X, to preclude the existence of rational points on a variety X. After fixing numerical invariants such as dimension, it is natural to ask which birational classes of varieties fail the Hasse principle, and moreover whether the Brauer group (or certain distinguished subsets) always explains this failure. In this talk, we will focus on K3 surfaces (e.g. a double cover of the plane branched along a smooth sextic curve) which are relatively simple surfaces in terms of geometric complexity, but whose arithmetic is more mysterious. For example, in 2014, Skorobogatov asked whether any odd torsion in the Brauer group of a K3 surface could obstruct the Hasse principle. We answer this question in the affirmative; via a purely geometric approach, we construct a 3-torsion transcendental Brauer class on a degree 2 K3 surface which obstructs the Hasse principle. Moreover, we do this without needing to explicitly write down a central simple algebra. This is joint work with Tony Varilly-Alvarado. |
| Title: Deterministic and stochastic acceleration techniques for Richardson-type iterations |
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| Defense: Dissertation |
| Speaker: Massimiliano Lupo Pasini of Emory University |
| Contact: Massimiliano Lupo Pasini, massimiliano.lupo.pasini@emory.edu |
| Date: 2018-04-02 at 1:00PM |
| Venue: W306 |
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| Abstract: The next generation of computational science applications will require numerical solvers that are both reliable and capable of high performance on projected exascale platforms. In order to meet these goals, solvers must be resilient to soft and hard system failures, provide high concurrency on heterogeneous hardware configurations, and retain numerical accuracy and efficiency. This work focuses on the solution of large sparse systems of linear equations, for example of the kind arising from the discretization of partial differential equations (PDEs). Specifically, the goal is to investigate alternative approaches to existing solvers (such as preconditioned Krylov subspace or multigrid methods). To do so, we consider stochastic and deterministic accelerations of relaxation schemes. On the one hand, starting from a convergent splitting of the coefficient matrix, we analyze various types of Monte Carlo acceleration schemes applied to the original preconditioned Richardson (stationary) iteration. These methods are expected to have considerable potential for resiliency to faults when implemented on massively parallel machines. In this framework, we have identified classes of problems and preconditioners that guarantee convergence. On the other hand, we consider Anderson-type accelerations to increase efficiency and improve the convergence rate with respect to one level fixed point schemes. In particular, we focus on a recently introduced method called Alternating Anderson-Richardson (AAR). We provide theoretical results to explain the advantages of AAR over other similar schemes presented in literature and we show numerical results where AAR is competitive against restarted versions of the generalized minimum residual method (GMRES) for problems of different nature and different preconditioning techniques. |
| Title: The Quantum McKay Correspondence: Classifying "Finite Subgroups" of a Quantum Group with Graphs |
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| Defense: Master's Thesis |
| Speaker: Paul Vienhage of Emory University |
| Contact: Paul Vienhage, |
| Date: 2018-04-02 at 1:00PM |
| Venue: MSC E406 |
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| Abstract: The McKay Correspondence classifies finite subgroups of the rotation group of 3-space via graphs. In this talk we explore a quantum version of this correspondence. Specifically, we will cover the needed background on category theory, vertex operator algebras, and quantum groups to explain a powerful technique used by Kirillov and Ostrik to develop a quantum analog to the McKay correspondence. |
| Title: Connections between Classical and Umbral Moonshine |
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| Defense: Dissertation |
| Speaker: Sarah Trebat-Leder of Emory University |
| Contact: Sarah Trebat-Leder, sarah.trebat-leder@emory.edu |
| Date: 2018-04-02 at 2:00PM |
| Venue: W302 |
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| Abstract: Both results of this dissertation involve finding unexpected connections between the classical theory of monstrous moonshine and the newer umbral moonshine. In our first result, we use generalized Borcherds products to associate to each pure A-type Niemeier lattice a conjugacy class g of the monster group and give rise to identities relating dimensions of representations from umbral moonshine to values of McKay-Thompson series. Our second result focuses on the Mathieu group M23. While it inherits a moonshine from being a subgroup of M24, we find a new and simpler moonshine for M23 such that the graded traces are, up to constant terms, identical to the monstrous moonshine Hauptmoduln. |
| Title: Maass forms and modular forms: applications to class numbers and partitions |
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| Defense: Dissertation |
| Speaker: Olivia Beckwith of Emory University |
| Contact: Olivia Beckwith, olivia.dorothea.beckwith@emory.edu |
| Date: 2018-04-02 at 3:00PM |
| Venue: W302 |
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| Abstract: This dissertation is about arithmetic information encoded by analytic characteristics (such as Fourier coefficients) of classical modular forms and a real-analytic generalization of modular forms called harmonic Maass forms. For example, I use the theory of harmonic Maass forms to extend and refine a theorem of Wiles on class number divisibility. I also prove asymptotic bounds for Rankin-Selberg shifted convolution L-functions in shift aspect. In partition theory, I use the circle method to describe the expected distribution of parts of integer partitions over residue classes, and I use effective estimates for partition functions to obtain simple formulas for functions arising in group theory. |
| Title: Patching and local-global principles for gerbes with an application to homogeneous spaces |
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| Defense: Dissertation |
| Speaker: Bastian Haase of Emory University |
| Contact: Bastian Haase, bastian.haase@emory.edu |
| Date: 2018-04-02 at 4:00PM |
| Venue: W302 |
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| Abstract: Starting 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. |
| Title: Truncated Singular Value Decomposition Approximation for Structured Matrices via Kronecker Product Summation Decomposition |
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| Defense: Dissertation |
| Speaker: Clarissa Garvey of Emory University |
| Contact: James Nagy, jnagy@emory.edu |
| Date: 2018-04-02 at 4:30PM |
| Venue: MSC W301 |
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| Abstract: Singular value decompositions are a particularly attractive matrix factorization for ill-posed problems because singular value magnitudes reveal information about the relative importance of data in the matrix. However, computing a singular value decomposition is typically computationally infeasible for large problems, as the cost for traditional methods, such as Lanczos bidiagonalization-based approaches and randomized methods, scales linearly with the number of entries in the matrix times the number of singular values computed. In this work we present two new algorithms and one new hybrid approach for computing the singular value decomposition of matrices cheaply approximable as an ordered Kronecker summation decomposition. Unlike previous work using ordered Kronecker summation decompositions, the factorizations these methods produce are more accurate for certain classes of matrices and have nonnegative singular values. The three proposed methods are also faster, with lower computational and spatial complexity, although also lower accuracy, than traditional methods. Our Kronecker-based methods therefore enable singular value decomposition approximations on larger matrices than traditional methods, while providing more accurate results in many cases than previous Kronecker-based singular value decompositions. We demonstrate the efficacy of these methods on a variety of image deconvolution problems for which the image is modeled as a regular grid of data. |
| Title: Counting Problems for Elliptic Curves over a Fixed Finite Field |
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| Seminar: Algebra |
| Speaker: Nathan Kaplan of UC Irvine |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2018-03-27 at 4:00PM |
| Venue: W304 |
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| Abstract: Let $E$ be an elliptic curve defined over a finite field $\mathbb{F}_q$. Hasse’s theorem says that $\#E(\mathbb{F}_q) = q + 1 - t_E$ where $|t_E| \le 2\sqrt{q}$. Deuring uses the theory of complex multiplication to express the number of isomorphism classes of curves with a fixed value of $t_E$ in terms of sums of ideal class numbers of orders in quadratic imaginary fields. Birch shows that as $q$ goes to infinity the normalized values of these point counts converge to the Sato-Tate distribution by applying the Selberg Trace Formula. \\ In this talk we discuss finer counting questions for elliptic curves over $\mathbb{F}_q$. For example, what is the probability that the number of rational points is divisible by $5$? What is the probability that the group of rational points is cyclic? If we choose a curve at random, and then pick a random point on that curve, what is the probability that the order of the point is odd? We study the distribution of rational point counts for elliptic curves containing a specified subgroup, giving exact formulas for moments in terms of traces of Hecke operators. We will also discuss some open problems. This is joint with work Ian Petrow (ETH Zurich). |
| Title: On large multipartite subgraphs of H-free graphs |
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| Seminar: Combinatorics |
| Speaker: Jan Volec of McGill Univeristy |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2018-03-26 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: A long-standing conjecture of Erd\H{o}s states that any n-vertex triangle-free graph can be made bipartite by deleting at most $n^2/25$ edges. In this talk, we study how many edges need to be removed from an H-free graph for a general graph H. By generalizing a result of Sudakov for 4-colorable graphs H, we show that if H is 6-colorable then G can be made bipartite by deleting at most $4n^2/25+O(n)$ edges. In the case of $H=K_6$, we actually prove the exact bound $4n^2/25$ and show that this amount is needed only in the case G is a complete 5-partite graph with balanced parts. As one of the steps in the proof, we use a strengthening of a result of $F\ddot{u}redi$ on stable version of $Tur\acute{a}n's$ theorem. This is a joint work with P. Hu, B. $Lidick\acute{y}$, T. Martins-Lopez and S. Norin. |
| Title: Estimating bilinear forms via extrapolation |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Marilena Mitrouli of National and Kapodistrian University of Athens |
| Contact: Michele Benzi, benzi@mathcs.emory.edu |
| Date: 2018-03-21 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: A spectrum of applications arising from Statistics, Machine Learning, Network Analysis require computation of bilinear forms $x^Tf(A)y$, where $A$ is a diagonalizable matrix and $x$, $y$ are given vectors. In this work we are interested in efficiently computing bilinear forms primarily due to their importance in several contexts. For large scale computation problems it is preferable to achieve approximations of bilinear forms without exploiting the whole matrix function. For this purpose an extrapolation procedure has been developed, attaining the approximation of bilinear forms with one, two or three term estimates in a complexity of square order. The extrapolation procedure gives us the flexibility to define the moments of a matrix $A$ either directly or through the polarization identity. The presented approach is characterized by easy applicable formulae of low complexity that can be implemented in vectorized form. |