All Seminars
| 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. |
| Title: Restriction of Scalars, Chabauty's Method, and $\mathbb P^1\smallsetminus \{0,1 \infty\}$. |
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| Seminar: Algebra |
| Speaker: Nicholas Triantafillou of Massachusetts Institute of Technology |
| Contact: David Zurick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-03-26 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: For a number field $K$ and a curve $C/K$, the Chabauty's method is a powerful $p$-adic tool for bounding/enumerating the set $C(K)$. The method typically requires that dimension of the Jacobian $J$ of $C$ is greater than the rank of $J(K)$. Since this condition often fails, especially when $[K:\mathbb Q]$ is large, several techniques have been proposed to augment Chabauty's method. For proper curves, Siksek introduced an analogue of Chabauty's method for the restriction of scalars ${Res}_{K/\mathbb Q} C$ that can succeed when the rank of $J(\mathcal O_{K,S})$ is as large as $[K:\mathbb Q]\cdot (\dim J - 1)$. Using an analogue of Chabauty's method for restrictions of scalars in the non-proper case, we study the power of this approach together with descent for computing $C = (\mathbb P^1\smallsetminus \{0,1,\infty\})(\mathcal O_{K,S})$. As an application, we show that if $3$ splits completely in $K$ then there are no solutions to the unit equation $x + y = 1$ with $x,y \in \mathcal O_{K}^{\times}$. |
| Title: Attacking neural networks with poison frogs: a theoretical look at adversarial examples in machine learning |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Thomas Goldstein of University of Maryland |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2019-03-22 at 2:00PM |
| Venue: MSC W301 |
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| Abstract: Neural networks solve complex computer vision problems with human-like accuracy. However, it has recently been observed that neural nets are easily fooled and manipulated by "adversarial examples," in which an attacker manipulates the network by making tiny changes to its inputs. In this talk, I give a high-level overview of adversarial examples, and then discuss a newer type of attack called "data poisoning," in which a network is manipulated at train time rather than test time. Then, I explore adversarial examples from a theoretical viewpoint and try to answer a fundamental question: "Are adversarial examples inevitable?" Bio: Tom is an Assistant Professor at University of Maryland. His research lies at the intersection of optimization and distributed computing, and targets applications in machine learning and image processing. He designs optimization methods for a wide range of platforms. This includes powerful cluster/cloud computing environments for machine learning and computer vision, in addition to resource limited integrated circuits and FPGAs for real-time signal processing. Before joining the faculty at Maryland, he completed his PhD in Mathematics at UCLA, and was a research scientist at Rice University and Stanford University. He has been the recipient of several awards, including SIAM’s DiPrima Prize, a DARPA Young Faculty Award, and a Sloan Fellowship. |
| Title: Reduced Models and Parallel Computing for Uncertainty Quantification in Cardiovascular Mathematics |
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| Defense: Dissertation |
| Speaker: Sofia Guzzetti of Emory University |
| Contact: Sofia Guzzetti, sofia.guzzetti@emory.edu |
| Date: 2019-03-21 at 10:00AM |
| Venue: MSC E308A |
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| Abstract: Computational fluid dynamics (CFD) has been progressively adopted in the last decade for studying the role of blood flow in the development of arterial diseases. While computational $(in silico)$ investigations - compared to more traditional $in$ $vitro$ and $in$ $vivo$ studies - are generally more flexible and cost-effective, the adoption of CFD for computer-aided clinical trials and surgical planning is still an open challenge. The computational time to accurately and reliably solve mathematical models can be too long for the fast-paced clinical environment - especially in emergency scenarios, and quantifying the reliability of the results comes at an even higher computational cost. Moreover, the $in$ $silico$ analysis of large numbers of patients calls for significant computational resources. Hospitals and healthcare institutions are expected to outsource numerical simulations, which, however, raises concerns about privacy, data protection, and efficiency in terms of cost and performance. In such an articulated and complex scenario, this work addresses the challenges described above by (i) introducing a novel reduced model that guarantees levels of accuracy comparable to those achieved by high-fidelity 3D models, roughly at the same computational cost as the inexpensive yet inaccurate 1D models, by combining the Finite Element Method to describe the main stream dynamics with Spectral Methods to retrieve the transverse components; (ii) designing a new method for uncertainty quantification in large-scale networks that greatly enhances parallelism by performing uncertainty quantification at the subsystem level, and propagating uncertainty information encoded as polynomial chaos coefficients via overlapping domain decomposition techniques; (iii) providing an objective criterion to measure the performance of different parallel architectures based on the user's priorities in terms of budget and tolerance to delay, and reducing the execution time by choosing a task-worker mapping strategy ahead of simulation time, and optimizing the amount of overlap in the domain decomposition phase. |
| Title: Large-Scale Parameter Estimation in Geophysics and Machine Learning |
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| Defense: Dissertation |
| Speaker: Samy Wu Fung of Emory University |
| Contact: Samy Wu Fung, samy.wu@emory.edu |
| Date: 2019-03-20 at 1:00PM |
| Venue: MSC W301 |
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| Abstract: The ability to collect large amounts of data with relative ease has given rise to new opportunities for scientific discovery. It has led to a new class of large-scale parameter estimation problems in geophysics, machine learning, and numerous other applications. Traditionally, parameter estimation aims to infer parameters in a physical model from indirect measurements, where the model is often given by a partial differential equation (PDE). Here, we also associate parameter estimation with machine learning, where rather than having a PDE as the model, we have a hypothesis function, e.g., a neural network, and the parameters of interest correspond to the weights. A common thread in these problems is their massive computational expense. The underlying parameter space in both applications is typically very high-dimensional. This makes the optimization computationally demanding, sometimes intractable, when large amounts of data are available. \\ \\ In this thesis, we address two general approaches to reduce the computational burdens of big-data parameter estimation in geophysics and machine learning. The first approach is an adaptive model reduction scheme that reduces the computational complexity of the model while achieving highly accurate solutions. This approach is tailored to problems in geophysics, where PDEs must be solved numerous times throughout the optimization. The second approach consists of novel parallel/distributed methods that lower the time-to-solution through avoided communication and latency, and can be used in both applications. We exemplarily show the potential of our methods on several geophysics and image classification problems. |
| Title: Modules for subgroups of $M_{24}$ with meromorphic trace functions |
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| Seminar: Algebra |
| Speaker: Lea Beneish of Emory |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-03-19 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: We give vertex operator algebra constructions of infinite-dimensional graded modules for certain subgroups of $M_{24}$. We begin by proving the existence of a module for $M_{24}$, whose trace functions are weight two quasimodular forms. Existence of this module implies certain divisibility conditions on the number of $\mathbb{F}_p$ points on Jacobians of modular curves. We write similar expressions which we show are trace functions for modules of cyclic groups with arbitrary prime order. These expressions can be modified so we can give a vertex operator algebra construction for these modules. However, this modification comes at the expense of any relationship to Jacobians of modular curves. By adding a term to the quasimodular forms, we obtain meromorphic Jacobi forms and prove the existence of a module for $M_{24}$ with these trace functions. For certain subgroups of $M_{24}$, we give a vertex operator algebra construction for a module with these trace functions. In particular, these module constructions give an explicit realization of the relationship between the trace functions and divisibility conditions the number of $\mathbb{F}_p$ points on Jacobians of modular curves. |