All Seminars
| 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. |
| Title: Patch Normalizing Regularizer: Reconstruction using only one ground truth image |
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| Seminar: Computational and Data Enabled Science |
| Speaker: Paul Hagemann of TU Berlin |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2022-10-06 at 10:00AM |
| Venue: MSC W301 |
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| Abstract: Reconstructing images from measurements (e.g. sinograms in CT) is a very active research topic. However in many domains, such as medical or material sciences, ground truth data is very hard or costly to obtain. In this talk, we will leverage the idea of patch-based learning for reconstructing images. The regularizer will learn the patch distribution from very few ground truth images by randomly subsampling 6x6 patches and learning their distribution. More specifically, we will use a normalizing flow to learn the patch distribution of the ground truth image, which we call patchNR. In reconstruction, we will minimize a sum of the negative log likelihood of the patches and the data fidelity term. Our method will be compared to other regularization techniques which use little data for CT, material and texture images. Furthermore, an outlook on how our method can be leveraged to perform zero shot superresolution will be given. |
| Title: Smooth limits of plane curves and Markov numbers |
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| Seminar: Algebra |
| Speaker: David Stapleton of The University of Michigan |
| Contact: David Zureick-Brown, david.zureick-brown@emory.edu |
| Date: 2022-10-04 at 4:00PM |
| Venue: MSC N304 |
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| Abstract: When can we guarantee that smooth proper limits of plane curves are still plane curves? Said a different way --- When is the locus of degree d plane curves closed in the (open) moduli space of smooth genus g curves? It is relatively easy to see that if d>1, then d must be prime. Interestingly, this is not sufficient -- Griffin constructed explicit families of quintic plane curves with a smooth limit that is not a quintic plane curve. In this talk we propose the following conjecture: Smooth proper limits of plane curves of degree d are always planar if d is prime and d is not a Markov number. We discuss the motivation and evidence for this conjecture which come from Hacking and Prokhorov's work on Q-Gorenstein limits of the projective plane. |
| Title: About the Lp theory for the non-cutoff Boltzmann equation |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Ricardo Alonso of Texas A$\&$M at Qatar |
| Contact: Maja Taskovic, maja.taskovic@emory.edu |
| Date: 2022-09-29 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: In this talk we discuss different technical elements to obtain a priori estimates for Lp norms of weak solutions to non-cutoff kinetic equations using as example the homogeneous/inhomogeneous Boltzmann equation. Rather than a detailed-proof talk, we point out difficulties and give some intuition related to the main steps of the strategy. In particular, we discuss the localization process of Boltzmann type operators which cover an ample range of operators such as the fractional Laplacian. |
| Title: Counting Elliptic Curves Over Number Fields |
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| Seminar: Algebra |
| Speaker: Tristan Phillips of The University of Arizona |
| Contact: David Zureick-Brown, david.zureick-brown@emory.edu |
| Date: 2022-09-27 at 4:00PM |
| Venue: MSC N304 |
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| Abstract: Let $E$ be an elliptic curve over a number field $K$. The Mordell--Weil Theorem states that the set of rational points $E(K)$ of $E$ forms a finitely generated abelian group. In particular, we may write $E(K) = E(K)_{tors}\oplus \mathbb{Z}^r$, where $E(K)_{tors}$ is a finite torsion group, called the torsion subgroup of $E$, and $r$ is a non-negative integer, called the rank of $E$. In this talk I will discuss some results regarding how frequently elliptic curves with a prescribed torsion subgroup occur, and how one can bound the average analytic rank of elliptic curves over number fields. One of the main ideas behind these results is to use methods from Diophantine geometry to count points of bounded height on modular curves. |
| Title: The Art of Repeatedly Project your Problems |
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| Seminar: Computational and Data Enabled Science |
| Speaker: Matthias Chung of Emory University |
| Contact: TBA |
| Date: 2022-09-22 at 10:00AM |
| Venue: MSC W301 |
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| Abstract: Inference by means of mathematical modeling from a collection of observations remains a crucial tool for scientific discovery and is ubiquitous in application areas such as signal compression, imaging restoration, and supervised machine learning. With ever-increasing model complexities and growing data size, new specially designed methods are urgently needed to recover meaningful quantities of interest. We consider the broad spectrum of linear inverse problems where the aim is to reconstruct quantities with a sparse representation on some vector space; often solved using the (generalized) least absolute shrinkage and selection operator (lasso). The associated optimization problems have received significant attention, in particular in the early 2000s, because of their connection to compressed sensing and the reconstruction of solutions with favorable sparsity properties using augmented Lagrangians, alternating directions and splitting methods. We provide a new perspective on the underlying l1 regularized inverse problem by exploring the generalized lasso problem through variable projection methods. We arrive at our proposed variable projected augmented Lagrangian (vpal) method. We provide numerical examples demonstrating the computational efficiency for various imaging problems. |