All Seminars

Title: Modular linear differential equations
Seminar: Algebra
Speaker: Kiyokazu Nagatomo of Osaka University
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2019-09-03 at 4:00PM
Venue: MSC W303
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Abstract:
The most naive definition of \textit{modular linear differential equations} (MLDEs) would be linear differential equations whose space of solutions are invariant under the weight $k$ slash action of $\Gamma_1=SL_2(\mathbb{Z})$, for some $k$. Then under an analytic condition for coefficients functions and the Wronskians of a~basis of the space of solutions of equations, we have (obvious) expressions of MLDEs as: \[ L(f) \,=\,\mathfrak{d}_k^n(f)+\sum_{i=2}^nP_{2i}\mathfrak{d}_k^{n-i}(f) \] where $P_{2i}$ is a modular form of of weight $2i$ on $SL_2(\mathbb{Z})$ and $\mathfrak{d}_k(f)$ is the \textit{Serre derivative}. (We could replace $\Gamma$ by a Fuchsian subgroup of $SL_2(\mathbb{R})$ and allow the modular forms $P_{2i}$ to be meromorphic.) However, the iterated Serre derivative $\mathfrak{d}_k^n(f)$ (called a ``higher Serre derivation'' because as an operator it preserves modulality) is very complicated since it involves the Eisenstein series $E_2$. MLDEs, of course, can be given in the form % \[ % \mathsf{L}(f) \,=\, D^n(f)+\sum_{i=1}^nQ_iD^i(f)\quad\text{where $D=\frac{1}{2\pi\sqrt{-1}}\frac{d}{d\tau}$.} % \] \[ \mathsf{L}(f) \,=\, D^n(f)+\sum_{i=1}^nQ_iD^i(f) \] where \[ D=\frac{1}{2\pi\sqrt{-1}}\frac{d}{d\tau}. \] Then it is not easy to know if the equation above is an MLDE except the fact that $Q_i$ are quasimodular forms. Very recently, Y.~Sakai and D.~Zagier (my collaborators) found formulas of $\mathsf{L}(f)$ by using the Rankin--Cohen products between $f$ and $g_i$. This is a modular form of weight $2i$, which is a linear function of the differential of~$Q_{j}$. Moreover, there are \textit{inversion formulas} which express $Q_i$ as a linear function of the derivatives of $g_{j}$. The most important fact is that the order $n$ and $n-1$ parts are equal to the so-called higher Serre derivative in the sense of Kaneko and Koike, where the group is $\Gamma_1$. (This holds for any Fuchsian group.) \\ Finally, the most important nature of my talk is that I will use a \textbf{blackboard} instead of \textbf{slides}ss.
Title: Iterative regularization methods for large-scale linear inverse problems
Seminar: Numerical Analysis and Scientific Computing
Speaker: Silvia Gazzola of University of Bath
Contact: James Nagy, jnagy@emory.edu
Date: 2019-08-27 at 2:00PM
Venue: MSC W301
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Abstract:
Inverse problems are ubiquitous in many areas of Science and Engineering and, once discretized, they lead to ill-conditioned linear systems, often of huge dimensions: regularization consists in replacing the original system by a nearby problem with better numerical properties, in order to find a meaningful approximation of its solution. After briefly surveying some standard regularization methods, both iterative (such as many Krylov methods) and direct (such as Tikhonov method), this talk will introduce a recent class of methods that merge an iterative and a direct approach to regularization. In particular, strategies for choosing the regularization parameter and the regularization matrix will be emphasized, eventually leading to the computation of approximate solutions of Tikhonov problems involving a regularization term expressed in some p-norms.
Title: Learning from data through the lens of mathematical models: Bayesian Inverse Problems and Uncertainty Quantification
Seminar: Numerical Analysis and Scientific Computing
Speaker: Umberto Villa, Ph.D. of Washington University in St Louis
Contact: Alessandro Veneziani, ale@mathcs.emory.edu
Date: 2019-06-24 at 2:00PM
Venue: MSC W301
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Abstract:
Recent years have seen rapid growth in the volume of observational and experimental data acquired from physical, biological or engineering systems. A fundamental question in several areas of science, engineering, medicine, and beyond is how to extract insight and knowledge from all of those available data. This process of learning from data is at its core a mathematical inverse problem. That is, given (possibly noisy) data and a (possibly uncertain) forward model describing the map from parameters to data, we seek to reconstruct or infer the parameters that characterize the model. Inverse problems are often ill-posed, i.e. their solution may not exist or may not be unique or may be unstable to perturbation in the data. Simply put, there may not be enough information in the data to fully determine the model parameters. In these cases, uncertainty is a fundamental feature of the inverse problem. The goal then is to both reconstruct the model parameters and quantify the uncertainty in such reconstruction. The ability to quantify these uncertainties is crucial to reliably predict the future behavior of the physical, biological or engineering systems, and to make informed decisions under uncertainty. This talk will illustrate the mathematical concepts and computational tools necessary for the solution of inverse problems in a deterministic and probabilistic (Bayesian) framework. Examples of inverse problems arising in imaging, geoscience, material engineering, and other fields of science will be presented. https://engineering.wustl.edu/Profiles/Pages/Umberto-Villa.aspx
Title: Discretize-Optimize Methods for Residual Neural Networks
Seminar: Numerical Analysis and Scientific Computing
Speaker: Derek Onken of Emory University
Contact: Lars Ruthotto, lruthotto@emory.edu
Date: 2019-04-26 at 2:00PM
Venue: MSC W301
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Abstract:
Neural networks (discrete universal approximators) demonstrate impressive performance in myriad tasks. Specifically, Residual Neural Networks (ResNets) have won numerous image classification contests since they were introduced a few years ago. Deep learning centers around the addition of more and more layers (and thus parameters) to these networks in efforts to improve performance. In this talk, we interpret ResNets as a discretization of an ordinary differential equation (ODE). This viewpoint exposes the similarity between the learning problem and problems of optimal control of the ODE. We use a discretize-optimize approach for training the weights of the ResNet and study the impact of the particular discretization strategy on the network performance. Varying the discretization of the features and parameters allows us to determine if the improved accuracy from deeper architectures stems from the larger number of parameters or more layers.
Title: Using Mathematics to Aid in the Evaluation of Robotic Systems
Evans Hall Lecture: Mathematics/Robotic Systems
Speaker: Mili Shah, PhD of Associate Professor, Department of Mathematics, The Cooper Union
Contact: Kristen Schroeder, kfschro@emory.edu
Date: 2019-04-25 at 4:00PM
Venue: MSC: E208
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Abstract:
Robotic systems use a variety of sensors to perform tasks that are assigned to them. In order to use data collected from these sensors, they must be registered with respect to a common coordinate frame. This talk will begin with an overview of the mathematical tools needed to tackle a registration problem and will conclude with current research and open problems related to the mathematics of evaluating robotic systems.
Title: Large girth approximate Steiner triple systems
Seminar: Combinatorics
Speaker: Lutz Warnke of The Georgia Institute of Technology
Contact: Dwight Duffus, dwightduffus@emory.edu
Date: 2019-04-24 at 4:00PM
Venue: MSC W301
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Abstract:
One can define the girth of a graph to be the minimum g such that there is a set of g vertices that spans at least g edges. This definition can be extended to the setting of Steiner triple systems by defining the girth to be the smallest g at least 4 for which there is a set of g vertices that spans at least g - 2 triples. In this talk we discuss a natural randomized algorithm that produces an approximate Steiner triple system of arbitrarily large girth, i.e., with (1/6-o(1)) n^2 triples, answering a question of Erdos from 1973 (that was independently also asked by Lefmann, Phelps, and Rodl in 1993, and Ellis and Linial in 2013). Joint work with Tom Bohman: https://arxiv.org/abs/1808.01065
Title: Monstrous moonshine, elliptic curves and vertex algebras
Defense: Dissertation
Speaker: Victor Manuel Aricheta of Emory University
Contact: Victor Aricheta, victormanuel.aricheta@emory.edu
Date: 2019-04-16 at 10:00AM
Venue: MSC E408
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Abstract:
In mathematics, moonshine refers to the unexpected connection between group theory and number theory. Monstrous moonshine is the first and best understood example of moonshine. It links the monster sporadic group to the modular j-function, a fact that we now know is explained by the presence of a certain algebraic structure called the moonshine module. Our comprehension of monstrous moonshine remains incomplete however, and in this talk we discuss several aspects of it that remain mysterious.\\ \\ First we investigate a theorem and an observation of Ogg in 1975 that foreshadowed monstrous moonshine. In particular we generalize his theorem on supersingular j-invariants to supersingular elliptic curves with level structure. Ogg observed—which we now know is partly explained by monstrous moonshine—that the level one case yields a characterization of the primes dividing the order of the monster. Here we show that the corresponding analyses for higher levels give analogous characterizations of the primes dividing the orders of other sporadic simple groups (e.g. baby monster, Fischer's largest group). More generally we characterize, in terms of supersingular elliptic curves with level, the primes arising as orders of Fricke elements in centralizer subgroups of the monster. This situates Ogg's theorem and observation in a broader setting.\\ \\ Second we build on the study of Duncan, Griffin and Ono concerning the moonshine module. They studied its homogeneous subspaces, and discovered in particular that the moonshine module exhibits a curious property: its homogeneous subspaces tend to a multiple of the regular representation of the monster. We prove that an analogous result holds for any vertex operator algebra satisfying certain hypotheses, for which the moonshine module is the first natural example.
Title: Constructing Picard curves with complex multiplication using the Chinese Remainder Theorem
Seminar: Algebra
Speaker: Sonny Arora of Emory University
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2019-04-16 at 4:00PM
Venue: MSC W201
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Abstract:
For cryptographic protocols whose security relies on the di?culty of the discrete log problem of the underlying group, one often wants to ?nd a group whose order is divisible by a large prime. One option for the group is the group of points of an elliptic curve over a ?nite ?eld, or more generally, the group of points on the Jacobian of a curve over a ?nite ?eld. To find curves over a finite field whose Jacobian has number of points divisible by a large prime, it suffices to construct curves whose Jacobian is ordinary and has complex multiplication (CM) by a given field K. Working with higher genus curves allows one to work over smaller fields than the elliptic curve case. I will present a new algorithm to construct a particular class of genus 3 curves, called Picard curves, whose Jacobian is ordinary with CM by a given field CM field K. This is joint work with Kirsten Eisentraeger.
Title: Computational challenges in ice sheet modeling
Seminar: Numerical Analysis and Scientific Computing
Speaker: Mauro Perego of Sandia National Laboratories, NM
Contact: Sofia Guzzetti, sofia.guzzetti@emory.edu
Date: 2019-04-12 at 2:00PM
Venue: MSC W301
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Title: a-Numbers of curves in Artin–Schreier covers
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.