All Seminars
| 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. |
| Title: On the relativistic Landau equation |
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| Colloquium: Analysis and Differential Geometry |
| Speaker: Maja Taskovic of University of Pennsylvania |
| Contact: David Borthwick, dborthw@emory.edu |
| Date: 2019-02-28 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: In kinetic theory, a large system of particles is described by the particle density function. The Landau equation, derived by Landau in 1936, is one such example. It models a dilute hot plasma with fast moving particles that interact via Coulomb interactions. This model does not include the effects of Einstein’s theory of special relativity. However, when particle velocities are close to the speed of light, which happens frequently in a hot plasma, then relativistic effects become important. These effects are captured by the relativistic Landau equation, which was derived by Budker and Beliaev in 1956. We study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. The difficulty of the problem lies in the extreme complexity of the kernel in the relativistic collision operator. We present a new decomposition of such kernel. This is then used to prove the global Entropy dissipation estimate, the propagation of any polynomial moment for a weak solution, and the existence of a true weak solution for a large class of initial data. This is joint work with Robert M. Strain. |
| Title: A new approach to bounding $L$-functions |
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| Seminar: Algebra |
| Speaker: Jesse Thorner of Stanford |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-02-26 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: An $L$-function is a type of generating function with multiplicative structure which arises from either an arithmetic-geometric object (like a number field, elliptic curve, abelian variety) or an automorphic form. The Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ is the prototypical example of an $L$-function. While $L$-functions might appear to be an esoteric and special topic in number theory, time and again it has turned out that the crux of a problem lies in the theory of these functions. Many equidistribution problems in number theory rely on one's ability to accurately bound the size of $L$-functions; optimal bounds arise from the (unproven!) Riemann Hypothesis for $\zeta(s)$ and its extensions to other $L$-functions. I will discuss some motivating equidistribution problems along with recent work (joint with K. Soundararajan) which produces new bounds for $L$-functions by proving a suitable "statistical approximation" to the (extended) Riemann Hypothesis. |
| Title: A pair-degree condition for Hamiltonian cycles in 3-graphs |
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| Seminar: Combinatorics |
| Speaker: Bjarne Schuelke of The University of Hamburg |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2019-02-25 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: For graphs, the fundamental theorem by Dirac ensuring the existence of Hamiltonian cycles in a graph G with large minimum degree was generalised by Chvatal to a characterisation of those degree sequences that force a graph to have a Hamiltonian cycle. After a Dirac-like result was proved for 3-uniform hypergraphs by Rodl, Rucinski, and Szemeredi, we will discuss a first step towards a more general characterisation of pair degree matrices of 3-uniform hypergraphs that force Hamiltonicity. The presented result can be seen as a 3-uniform analogue of a result on graphs by Posa that is more general than Dirac's and is generalised by Chvatal's theorem. In particular we will prove that for each c > 0 there exists an n such that the following holds: If H is a 3-uniform hypergraph with vertex set {1,...,n} and d(i,j) > min { (i+j)/2, n/2 } + cn holds for all pairs of vertices, then H contains a tight Hamilton cycle. |
| Title: Filtering techniques for eigenvalue problems |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Yousef Saad of University of Minnesota |
| Contact: Yuanzhe Xi, yxi26@emory.edu |
| Date: 2019-02-22 at 2:00PM |
| Venue: MSC W301 |
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| Abstract: The solution of large symmetric eigenvalue problems is central to applications ranging from electronic structure calculations to the study of vibrations in mechanical systems. A few of these applications require the computation of a large number of eigenvalues and associated eigenvectors. For example, when dealing with excited states in quantum mechanics, it is not uncommon to seek a few tens of thousands of eigenvalues of matrices of sizes in the tens of millions. In such situations it is imperative to resort to `spectrum slicing' strategies, i.e., strategies that extract slices of the spectrum independently. The presentation will discuss a few techniques in this category, namely those based on a combination of filtering (polynomial, rational) and standard projection methods (Lanczos, subspace iteration). Filtering consists of computing eigenvalues and vectors of a matrix of the form $B=f(A)$, where $f$ is typically a polynomial or rational function. With the mapping $f$ the wanted eigenvalues of the original matrix are transformed in such a way that they become easier to extract. This particular area blends ideas from approximation theory with standard matrix algorithms. The presentation will emphasize rational filtering and will discuss some recent work on nonlinear eigenvalue problems. |
| Title: Turan-type problems for bipartite graphs and hypergraphs |
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| Colloquium: Combinatorics |
| Speaker: Liana Yepremyan of University of Oxford |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2019-02-22 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: A central problem of extremal combinatorics is to determine the Turan number of a given graph or a hypergraph F, i.e. the maximum number of edges in an r-uniform hypergraph on n vertices that does not contain a copy of F. For graphs, the asymptotic answer to this question is given by the celebrated Erdos-Stone theorem. However, it is wide open for bipartite graphs and for hypergraphs. For bipartite graphs, a beautiful conjecture of Erdos and Simonovits says that every rational number between 1 and 2 must appear as an exponent of some Turan number. Despite decades of effort, there are only few families for which this conjecture is known to be true. We will discuss some recent progress on this as well as related so-called supersaturation problems. For hypergraphs, since the problem was introduced over sixty years ago, it has only been solved for relatively few hypergraphs F. Many of these results were found very recently by means of the stability method, which has brought new life to research in a challenging area. We will discuss a variation of this method utilizing the Lagrangian function (we call it local stability method) which gives a generic unified approach for obtaining exact Turan numbers from asymptotic results and allowed us to enlarge the list of known Turan numbers of hypergraphs, in particular solving a conjecture of Frankl and Furedi from the 80's. Various parts of the work are joint with Sergey Norin, Adam Bene Watts, Tao Jiang and Jie Ma. |