# All Seminars

Title: Subspace configurations and low degree points on curves
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Seminar: Analysis and Differential Geometry
Speaker: Ricardo Alonso of Texas A$\&$M at Qatar