All Seminars
| Title: Neural networks investigation of bifurcating phenomena in fluid-dynamics |
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| Seminar: CODES@Emory |
| Speaker: Federico Pichi of EPFL Lausanne |
| Contact: Alessandro Veneziani, ale@mathcs.emory.edu |
| Date: 2022-10-27 at 10:00AM |
| Venue: MSC W301 |
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| Abstract: Bifurcating phenomena, i.e. sudden changes in the qualitative behavior of the system linked to the non-uniqueness of the solution naturally arise in several fields. Since the reconstruction of bifurcation diagrams requires a many-query context, which is usually unaffordable using high-fidelity simulations, we propose a combination of Reduced Order Models (ROMs) and Machine Learning techniques to reduce the computational burden associated with the investigation of such complex phenomena.This work aims to show the applicability of the Reduced Basis (RB) model reduction and Artificial Neural Network (ANN), utilizing the POD-NN approach and its physics-informed variant [2, 1], to analyze multi-parameter bifurcating applications in fluid-dynamics. We considered the Navier-Stokes equations for a viscous, steady, and incompressible flow: (i) in a planar straight channel with a narrow inlet of varying width and (ii) in a triangular parametrized lid-driven cavity. Within this context, we present a new empirical strategy to employ the RB and ANN coefficients for a non-intrusive detection of the bifurcation points [3]. Finally, we introduce a newly developed ROM methodology based on Graph Neural Network, with powerful applications to general parametrized PDEs and branches classification when dealing with bifurcating phenomena [4]. References: [1] W. Chen, Q. Wang, J. S. Hesthaven, and C. Zhang. Physics-informed machine learning for reduced-order modeling of nonlinear problems. Journal of Computational Physics, 446:110666, 2021. [2] J. S. Hesthaven and S. Ubbiali. Non-intrusive reduced order modeling of nonlinear problems using neural networks. Journal of Computational Physics, 363:55–78, 2018. [3] F. Pichi, F. Ballarin, G. Rozza, and J. S. Hesthaven. Artificial neural network for bifurcating phenomena modelled by nonlinear parametrized PDEs. arXiv:2109.10765, 2021. [4] F. Pichi, B. Moya, and J. S. Hesthaven. A convolutional graph neural network approach to model order reduction for nonlinear parametrized PDEs. In preparation, 2022. |
| Title: Overconvergent differential operators acting on Hilbert modular forms |
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| Seminar: Algebra |
| Speaker: Jon Aycock of University of California, San Diego |
| Contact: David Zureick-Brown, david.zureick-brown@emory.edu |
| Date: 2022-10-25 at 4:00PM |
| Venue: MSC N304 |
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| Abstract: In 1978, Katz gave a construction of the $p$-adic $L$-function of a CM field by using a $p$-adic analog of the Maass--Shimura operators acting on $p$-adic Hilbert modular forms. However, this $p$-adic Maass--Shimura operator is only defined over the ordinary locus, which restricted Katz's choice of $p$ to one that splits in the CM field. In 2021, Andreatta and Iovita extended Katz's construction to all $p$ for quadratic imaginary fields using overconvergent differential operators constructed by Harron--Xiao and Urban, which act on elliptic modular forms. Here we give a construction of such overconvergent differential operators which act on Hilbert modular forms. |
| Title: Stability, Optimality, and Fairness in Federated learning |
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| Seminar: CODES@Emory |
| Speaker: Kate Donahue of Cornell University |
| Contact: Elizabeth Newman, elizabeth.newman@emory.edu |
| Date: 2022-10-20 at 10:00AM |
| Venue: MSC W301 |
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| Abstract: Federated learning is a distributed learning paradigm where multiple agents, each only with access to local data, jointly learn a global model. There has recently been an explosion of research aiming not only to improve the accuracy rates of federated learning, but also provide certain guarantees around social good properties such as total error or fairness. In this talk, I describe two papers analyzing federated learning through the lens of cooperative game theory (both joint with Jon Kleinberg). In the first paper, we discuss fairness in federated learning, which relates to how error rates differ between federating agents. In this work, we consider two notions of fairness: egalitarian fairness (which aims to bound how dissimilar error rates can be) and proportional fairness (which aims to reward players for contributing more data). For egalitarian fairness, we obtain a tight multiplicative bound on how widely error rates can diverge between agents federating together. For proportional fairness, we show that sub-proportional error (relative to the number of data points contributed) is guaranteed for any individually rational federating coalition. The second paper explores optimality in federated learning with respect to an objective of minimizing the average error rate among federating agents. In this work, we provide and prove the correctness of an efficient algorithm to calculate an optimal (error minimizing) arrangement of players. Building on this, we give the first constant-factor bound on the performance gap between stability and optimality, proving that the total error of the worst stable solution can be no higher than 9 times the total error of an optimal solution (Price of Anarchy bound of 9). Relevant Links: https://arxiv.org/abs/2010.00753, https://arxiv.org/abs/2106.09580, https://arxiv.org/abs/2112.00818 |
| Title: Subspace configurations and low degree points on curves |
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| Seminar: Algebra |
| Speaker: Borys Kadets of University of Georgia |
| Contact: David Zureick-Brown, david.zureick-brown@emory.edu |
| Date: 2022-10-18 at 4:00PM |
| Venue: MSC N304 |
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| Abstract: The hyperelliptic curve given by the equation $y^2=f(x)$ with coefficients in $\mathbf{Q}$ has an unusual arithmetic property: it admits infinitely many points with coordinates in quadratic extensions of $\mathbf{Q}$ (namely $(a, \sqrt{f(a)})$). Hindry, motivated by arithmetic questions about modular curves, asked if the only curves that possess infinite collections of quadratic points are hyperelliptic and bielliptic; this conjecture was confirmed by Harris and Silverman. I will talk about the general problem of classifying curves that possess infinite collections of degree $d$ points. I will explain how to reduce this classification problem to a study of curves of low genus, and use this reduction to obtain a classification for $d \leq 5$. This relies on analyzing a discrete-geometric object -- the subspace configuration -- attached to curves with infinitely many degree $d$ points. This talk is based on joint work with Isabel Vogt (arXiv:2208.01067). |
| Title: Model order reduction for parametrized optimal control problems: from time-dependency to nonlinearity. |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Maria Strazzullo of Politecnico di Torino, ITALY |
| Contact: Alessandro Veneziani, ale@mathcs.emory.edu |
| Date: 2022-10-17 at 10:00AM |
| Venue: Atwood 360 |
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| Abstract: Parametrized optimal control problems can represent an asset to fill the gap between collected data and partial differential equations in many scientific and industrial applications. Despite their indisputable usefulness, their computational complexity still limits their applicability in many-query and real-time parametric settings, most of all when the problem is time-dependent or nonlinear.\\ \\ We propose reduced order methods as a valid strategy to deal with this issue. The talk focuses on the approaches that provide a low-dimensional framework to accelerate the simulations of the system, maintaining a fair degree of accuracy.\\ \\ The first part of the talk is about the numerical algorithms used to reach this goal. The second part is more related to the applied viewpoint, analyzing the potential of reduced optimal control in many fields, such as bifurcating phenomena and numerical stabilization. |
| Title: A Random Group with Local Data |
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| Seminar: Algebra |
| Speaker: Brandon Alberts of Eastern Michigan University |
| Contact: David Zureick-Brown, david.zureick-brown@emory.edu |
| Date: 2022-10-14 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: The Cohen--Lenstra heuristics describe the distribution of $\ell$-torsion in class groups of quadratic fields as corresponding to the distribution of certain random p-adic matrices. These ideas have been extended to using random groups to predict the distributions of more general unramified extensions in families of number fields (see work by Boston--Bush--Hajir, Liu--Wood, Liu--Wood--Zureick-Brown). Via the Galois correspondence, the distribution of unramified extensions is a specific example of counting number fields with prescribed ramification and bounded discriminant. As of yet, no constructions of random groups have been given in the literature to predict the answers to famous number field counting conjectures such as Malle's conjecture. We construct a "random group with local data" bridging this gap, and use it to describe new heuristic justifications for number field counting questions. |
| Title: Curve classes on conic bundle threefolds and applications to rationality |
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| Seminar: Algebra |
| Speaker: Soumya Sankar of The Ohio State University |
| Contact: David Zureick-Brwon, david.zureick-brown@emory.edu |
| Date: 2022-10-14 at 5:15PM |
| Venue: MSC W301 |
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| Abstract: A variety is k-rational over a field k, if it is birational to projective space over k. From the perspective of rationality, conic bundles are a geometrically rich class of varieties. In this talk, I will discuss the rationality of conic bundle threefolds. The rationality of threefolds is very closely linked to the space of curve classes on them. Indeed, over algebraically closed fields, a rationality criterion for conic bundle threefolds over minimal surfaces has been known since the 80's, due to Shokurov. This criterion is the vanishing of the Intermediate Jacobian obstruction, introduced by Clemens and Griffiths. More recently, Hassett-Tschinkel (over the reals) and Benoist-Wittenberg (over arbitrary fields) introduced a refined obstruction to rationality, namely the Intermediate Jacobian Torsor obstruction. This obstruction has proved to be a powerful tool for threefolds, and its vanishing has been shown to be sufficient for rationality in several cases. In joint work with Sarah Frei, Lena Ji, Bianca Viray and Isabel Vogt, we study curve classes on certain types of conic bundle threefolds over arbitrary fields of odd characteristic. By giving an explicit description of these curve classes, we show that the IJT obstruction is insufficient to characterize rationality. |
| Title: TMulti-Objective Optimization for Best Early Prediction of Extreme Weather Events |
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| Seminar: Computational and Data Enabled Science |
| Speaker: Ariana Brown of Emory University |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2022-10-13 at 10:00AM |
| Venue: MSC W301 |
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| Abstract: In this project, we aim to solve a multi-objective optimization problem regarding the placement, cost, and quality of the meteorological sensing instruments for best early predictions of extreme weather events. This is described by the so-called significance function. Depending on the form of the given data, we proposed two different approaches: a shape-packing strategy and Nonomura’s singular value decomposition strategy. The first one leads us to place the sensors in areas with a high significance value in the domain. The Pareto optimality is then applied to judge the best configuration of types and locations for sensors. The second approach approximates the significance fields over the entire domain of study based on historical data. We further proposed the concept of essential dimension, which is the ”as linearly independent as possible” information seen by a high grade sensor. Essential dimension will answer the cost-quality trade off problem. The users with a significance function at hand can apply shape-packing strategy while those with historical significance data can implement the second approach to best place the sensors in the domain. |
| Title: Multivariate Quantile Function Forecaster |
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| Seminar: Computational and Data Enabled Science Seminar |
| Speaker: Kelvin Kan of Emory University |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2022-10-13 at 10:00AM |
| Venue: MSC W301 |
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| Abstract: We propose Multivariate Quantile Function Forecaster (MQF^2), a global probabilistic forecasting method constructed using a multivariate quantile function and investigate its application to multi-horizon forecasting. Prior approaches are either autoregressive, implicitly capturing the dependency structure across time but exhibiting error accumulation with increasing forecast horizons, or multi-horizon sequence-to-sequence models, which do not exhibit error accumulation, but also do typically not model the dependency structure across time steps. MQF^2 combines the benefits of both approaches, by directly making predictions in the form of a multivariate quantile function, defined as the gradient of a convex function which we parametrize using input-convex neural networks. By design, the quantile function is monotone with respect to the input quantile levels and hence avoids quantile crossing. We provide two options to train MQF^2: with energy score or with maximum likelihood. Experimental results on real-world and synthetic datasets show that our model has comparable performance with state-of-the-art methods in terms of single time step metrics while capturing the time dependency structure. |
| Title: Complexities of the Cytoskeleton: Integration of Scales |
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| Seminar: Computational and Data Enabled Science |
| Speaker: Keisha Cook of Clemson University |
| Contact: Jim Nagy, jnagy@emory.edu |
| Date: 2022-10-07 at 1:00PM |
| Venue: MSC W301 |
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| Abstract: Biological systems are traditionally studied as isolated processes (e.g. regulatory pathways, motor protein dynamics, transport of organelles, etc.). Although more recent approaches have been developed to study whole cell dynamics, integrating knowledge across biological levels remains largely unexplored. In experimental processes, we assume that the state of the system is unknown until we sample it. Many scales are necessary to quantify the dynamics of different processes. These may include a magnitude of measurements, multiple detection intensities, or variation in the magnitude of observations. The interconnection between scales, where events happening at one scale are directly influencing events occurring at other scales, can be accomplished using mathematical tools for integration to connect and predict complex biological outcomes. In this work we focus on building inference methods to study the complexity of the cytoskeleton from one scale to another. |