All Seminars
| Title: Using Mathematics to Aid in the Evaluation of Robotic Systems |
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| Evans Hall Lecture: Mathematics/Robotic Systems |
| Speaker: Mili Shah, PhD of Associate Professor, Department of Mathematics, The Cooper Union |
| Contact: Kristen Schroeder, kfschro@emory.edu |
| Date: 2019-04-25 at 4:00PM |
| Venue: MSC: E208 |
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| Abstract: Robotic systems use a variety of sensors to perform tasks that are assigned to them. In order to use data collected from these sensors, they must be registered with respect to a common coordinate frame. This talk will begin with an overview of the mathematical tools needed to tackle a registration problem and will conclude with current research and open problems related to the mathematics of evaluating robotic systems. |
| Title: Large girth approximate Steiner triple systems |
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| Seminar: Combinatorics |
| Speaker: Lutz Warnke of The Georgia Institute of Technology |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2019-04-24 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: One can define the girth of a graph to be the minimum g such that there is a set of g vertices that spans at least g edges. This definition can be extended to the setting of Steiner triple systems by defining the girth to be the smallest g at least 4 for which there is a set of g vertices that spans at least g - 2 triples. In this talk we discuss a natural randomized algorithm that produces an approximate Steiner triple system of arbitrarily large girth, i.e., with (1/6-o(1)) n^2 triples, answering a question of Erdos from 1973 (that was independently also asked by Lefmann, Phelps, and Rodl in 1993, and Ellis and Linial in 2013). Joint work with Tom Bohman: https://arxiv.org/abs/1808.01065 |
| Title: Monstrous moonshine, elliptic curves and vertex algebras |
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| Defense: Dissertation |
| Speaker: Victor Manuel Aricheta of Emory University |
| Contact: Victor Aricheta, victormanuel.aricheta@emory.edu |
| Date: 2019-04-16 at 10:00AM |
| Venue: MSC E408 |
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| Abstract: In mathematics, moonshine refers to the unexpected connection between group theory and number theory. Monstrous moonshine is the first and best understood example of moonshine. It links the monster sporadic group to the modular j-function, a fact that we now know is explained by the presence of a certain algebraic structure called the moonshine module. Our comprehension of monstrous moonshine remains incomplete however, and in this talk we discuss several aspects of it that remain mysterious.\\ \\ First we investigate a theorem and an observation of Ogg in 1975 that foreshadowed monstrous moonshine. In particular we generalize his theorem on supersingular j-invariants to supersingular elliptic curves with level structure. Ogg observed—which we now know is partly explained by monstrous moonshine—that the level one case yields a characterization of the primes dividing the order of the monster. Here we show that the corresponding analyses for higher levels give analogous characterizations of the primes dividing the orders of other sporadic simple groups (e.g. baby monster, Fischer's largest group). More generally we characterize, in terms of supersingular elliptic curves with level, the primes arising as orders of Fricke elements in centralizer subgroups of the monster. This situates Ogg's theorem and observation in a broader setting.\\ \\ Second we build on the study of Duncan, Griffin and Ono concerning the moonshine module. They studied its homogeneous subspaces, and discovered in particular that the moonshine module exhibits a curious property: its homogeneous subspaces tend to a multiple of the regular representation of the monster. We prove that an analogous result holds for any vertex operator algebra satisfying certain hypotheses, for which the moonshine module is the first natural example. |
| Title: Constructing Picard curves with complex multiplication using the Chinese Remainder Theorem |
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| Seminar: Algebra |
| Speaker: Sonny Arora of Emory University |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-04-16 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: For cryptographic protocols whose security relies on the di?culty of the discrete log problem of the underlying group, one often wants to ?nd a group whose order is divisible by a large prime. One option for the group is the group of points of an elliptic curve over a ?nite ?eld, or more generally, the group of points on the Jacobian of a curve over a ?nite ?eld. To find curves over a finite field whose Jacobian has number of points divisible by a large prime, it suffices to construct curves whose Jacobian is ordinary and has complex multiplication (CM) by a given field K. Working with higher genus curves allows one to work over smaller fields than the elliptic curve case. I will present a new algorithm to construct a particular class of genus 3 curves, called Picard curves, whose Jacobian is ordinary with CM by a given field CM field K. This is joint work with Kirsten Eisentraeger. |
| Title: Computational challenges in ice sheet modeling |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Mauro Perego of Sandia National Laboratories, NM |
| Contact: Sofia Guzzetti, sofia.guzzetti@emory.edu |
| Date: 2019-04-12 at 2:00PM |
| Venue: MSC W301 |
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| Abstract: |
| Title: a-Numbers of curves in Artin–Schreier covers |
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| Seminar: Algebra |
| Speaker: Jeremy Booher of University of Arizona |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-04-09 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: Let f : Y -> X be a branched Z/pZ-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic p>0. We investigate the relationship between the a-numbers of Y and X and the ramification of the map f. This is analogous to the relationship between the genus (respectively p-rank) of Y and X given the Riemann-Hurwitz (respectively Deuring--Shafarevich) formula. Except in special situations, the a-number of Y is not determined by the a-number of X and the ramification of the cover, so we instead give bounds on the a-number of Y. We provide examples showing our bounds are sharp. The bounds come from a detailed analysis of the kernel of the Cartier operator. This is joint work with Bryden Cais. |
| Title: On an Eigenvector-Dependent Nonlinear Eigenvalue Problem |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Ren-Cang Li of University of Texas at Arlington |
| Contact: Yuanzhe Xi, yxi26@emory.edu |
| Date: 2019-04-05 at 2:00PM |
| Venue: MSC W301 |
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| Abstract: We first establish existence and uniqueness conditions for the solvability of an algebraic eigenvalue problem with eigenvector nonlinearity. We then present a local and global convergence analysis for a self-consistent field (SCF) iteration for solving the problem. The well-known sin? theorem in the perturbation theory of Hermitian matrices plays a central role. The near-optimality of the local convergence rate of the SCF iteration is demonstrated by examples from the discrete Kohn-Sham eigenvalue problem in electronic structure calculations and the maximization of the trace ratio in the linear discriminant analysis for dimension reduction. This is a joint work with Yunfeng Cai (Peking University), Lei-Hong Zhang (Shanghai University of Finance and Economics), Zhaojun Bai (University of California at Davis). |
| Title: Jacobians of Graphs and Arithmetical Structures |
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| Seminar: Algebra |
| Speaker: Darren Glass of Gettysburg College |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-04-02 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: In analogy to the Jacobian variety associated to a curve, one can define a finite abelian group associated to a graph which we also call the Jacobian of the graph. Several theorems in algebraic geometry translate nicely to the graph theory setting while others do not. In this talk, we will look at this definition and consider further generalizations that were originally motivated by arithmetic geometry but which turn out to have a definition in terms of elementary number theory as well as a number of interesting combinatorial properties. |
| Title: Identifying Differential Equations with Numerical Time evolution |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Sung Ha Kang of Georgia Institute of Technology |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2019-03-29 at 2:00PM |
| Venue: MSC W301 |
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| Abstract: Identifying unknown differential equations from given discrete time dependent data is a challenging problem. A small amount of noise can make the recovery unstable, and nonlinearity and differential equations with varying coefficients add complexity to the problem. We propose a new direction based on the fundamental idea of convergence analysis of numerical partial differential equation schemes. We utilize L1 minimization for efficiency, and a performance guarantee is established based on an incoherence property. The main contribution is to validate and correct the results by Time Evolution Error and Base Element Expansion. The propose method is explored for data with non-periodic boundary conditions, noisy data and PDE with varying coefficient for nonlinear PDE identification. |
| Title: Matrix Computations and Optimization for Spectral Computed Tomography |
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| Defense: Dissertation |
| Speaker: Yunyi Hu of Emory University |
| Contact: Yunyi Hu, yunyi.hu@emory.edu |
| Date: 2019-03-29 at 3:00PM |
| Venue: MSC W201 |
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| Abstract: In the area of image science, the emergence of spectral computed tomography (CT) detectors highlights the concept of quantitative imaging, in which not only reconstructed images are offered, but also weights of different materials that compose the object are provided. For distinct types of detectors and noise, various models and techniques are produced to capture different features. In this thesis, we focus on optimization, preconditioning and model development of spectral CT. For simple energy discriminating detectors, a nonlinear optimization framework is built on a Poisson likelihood estimator and bound constraints. A nonlinear interior-point trust region method is implemented to compute the solution. For energy-windowed spectral CT, a nonlinear least squares approach is proposed to describe the problem and under bound constraints, a two-step method using the projected line search and the trust region approach, incorporated with a stepwise preconditioner, is used to solve the problem. In addition, a weighted least squares formulation is derived from the Gaussian noise assumption and another preconditioner that is based on rank-1 approximation is inserted to obtain robust reconstruction. The Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), along with a projection step, is used to calculate the solution iteratively. Compared with a direct solver, a two-step model is developed using an ancillary variable. With this two-step model, a row-wise computational method is proposed, which further reduces memory requirements and improves solution accuracy. Numerous numerical experiments are conducted to indicate the strength of methods and real-life examples are presented to show possible applications. |