All Seminars

Title: Counting Elliptic Curves Over Number Fields
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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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
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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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.
Title: Jordan Decompositions of Tensors
Seminar: Algebra
Speaker: Luke Oeding of Auburn University
Contact: David Zureick-Brown, david.zureick-brown@emory.edu
Date: 2022-09-21 at 2:30PM
Venue: MSC E208
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The Jordan normal form for similar matrices is a powerful classification tool as it provides a test to determine which matrices are similar (in the same orbit), and whether one orbit contains another or not. We expand on an idea of Vinberg to take a tensor space and the natural Lie algebra which acts on it and embed them into an auxiliary algebra. Viewed as endomorphisms of this algebra we associate adjoint operators to tensors. We show that the group actions on the tensor space and on the adjoint operators are consistent, which endows the tensor with a Jordan decomposition. We utilize aspects of the Jordan decomposition to study orbit separation and classification in examples that are relevant for quantum information. My talk will contain many examples and open questions.
Title: Short hands-on course on CUQIpy - a new Python platform for computational uncertainty quantification in inverse problems
Seminar: Computational and Data Enabled Science
Speaker: Jakob Sauer Jørgensen of Technical University Denmark
Contact: Julianne Chung, julianne.mei-lynn.chung@emory.edu
Date: 2022-09-15 at 8:30AM
Venue: MSC N301
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CUQIpy (pronounced "cookie pie") is a new computational modelling environment in Python that uses UQ (Bayesian statistics and sampling) to access and quantify the uncertainties in solutions to inverse problems. The overall goal of the software package is to allow both expert and non-expert (without deep knowledge of statistics and UQ) users to perform UQ related analysis of their inverse problem while focusing on the modelling aspects. To achieve this goal the package utilizes state-of-the-art tools and methods in statistics and scientific computing specifically tuned to the ill-posed and often large-scale nature of inverse problems to make the UQ feasible. The training course will be very hands-on with Jupyter notebook exercises demonstrating basic and more advanced functionality of CUQIpy. No installation is necessary, as exercises will be run on our online platform accessed through a normal browser, so participants should just bring a laptop with wifi access to join the zoom meeting and the online platform (instructions will be given during the course).
Title: Short hands-on course on CUQIpy - a new Python platform for computational uncertainty quantification in inverse problems
Seminar: Computational and Data Enabled Science
Speaker: Jakob Sauer Jørgensen of Technical University Denmark
Contact: Julianne Chung, julianne.mei-lynn.chung@emory.edu
Date: 2022-09-15 at 10:00AM
Venue: MSC W301
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Abstract:
CUQIpy (pronounced "cookie pie") is a new computational modelling environment in Python that uses UQ (Bayesian statistics and sampling) to access and quantify the uncertainties in solutions to inverse problems. The overall goal of the software package is to allow both expert and non-expert (without deep knowledge of statistics and UQ) users to perform UQ related analysis of their inverse problem while focusing on the modelling aspects. To achieve this goal the package utilizes state-of-the-art tools and methods in statistics and scientific computing specifically tuned to the ill-posed and often large-scale nature of inverse problems to make the UQ feasible. The training course will be very hands-on with Jupyter notebook exercises demonstrating basic and more advanced functionality of CUQIpy. No installation is necessary, as exercises will be run on our online platform accessed through a normal browser, so participants should just bring a laptop with wifi access to join the zoom meeting and the online platform (instructions will be given during the course).
Title: On deriving the Vlasov equation and its Hamiltonian structure
Seminar: Analysis and Differential Geometry
Speaker: Joseph Miller of University of Texas at Austin
Contact: Maja Taskovic, maja.taskovic@emory.edu
Date: 2022-09-15 at 4:00PM
Venue: MSC W301
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The Vlasov equation is a nonlinear PDE used to model plasmas in physics. It can be rigorously derived from Newton's laws of motion for many particles via empirical measures, or by a hierarchy of equations called the BBGKY hierarchy in a mean-field limit. The Vlasov equation itself contains geometric information, called a Hamiltonian structure, which is shared by the finite particle dynamics. In this talk, I will explain how to rigorously derive one Hamiltonian structure from the other. This is joint work with Andrea R. Nahmod, Natasa Pavlovic, Matt Rosenzweig, and Gigliola Staffilani.
Title: Reduced order modelling as enabler for optimization and digital twins
Seminar: Computational and Data Enabled Science
Speaker: Marco Tezzele of UT Austin
Contact: Alessandro Veneziani, avenez2@emory.edu
Date: 2022-09-08 at 10:00AM
Venue: MSC W301
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We present data-driven reduced order models with a focus on reduction in parameter space to fight the curse of dimensionality in design optimization. We show two extensions of the Active Subspaces (AS) technique: a kernel version exploiting an intermediate mapping to a higher dimensional space, and a local approach in which a clustering induced by a global active subspace is used for regression and classification tasks. Parameter space reduction methods can also be used within a multi-fidelity nonlinear autoregressive scheme to improve the approximation accuracy of high-dimensional functions, using only high-fidelity data. Finally, we integrate AS into the genetic algorithm to enhance the convergence during the optimization of high-dimensional quantities of interest. These methods, together with non-intrusive reduced order models based on proper orthogonal decomposition, are applied to the structural optimization of cruise ships and shape optimization of a combatant hull. The last part of the talk will be devoted to an ongoing work on digital twins and adaptive planning strategies in a Bayesian setting.
Title: Secants, Gorenstein ideals, and stable complexes
Seminar: Algebra
Speaker: Brooke Ullery of Emory University
Contact: David Zureick-Brown, david.zureick-brown@emory.edu
Date: 2022-09-06 at 4:00PM
Venue: MSC N304
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As Bertram describes in his thesis, certain rank two vector bundles on curves can be parametrized by a projective space into which the curve itself embeds. The idea is that the least so-called ``stable" bundles correspond to points on the embedded curve, the next most unstable lie on the first secant variety, then on the second secant variety, and so on. In this talk, I'll describe joint work with Bertram in which we generalize this to $\mathbb{P}^2$ by replacing vector bundles with complexes of vector bundles. This leads to a surprising connection between height three Gorenstein ideals, secant varieties (and their generalizations), and stability conditions on the bounded derived category of coherent sheaves on $\mathbb{P}^2$.
Title: Effects of elastic shear modulus on soil liquefaction modelling and effective stress analysis
Seminar: Algebra
Speaker: Jimena Tempestti of Emory University
Contact: Alessandro Veneziani, avenez2@emory.edu
Date: 2022-09-01 at 10:00AM
Venue: MSC W301
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This research builds upon a well-established constitutive model for fully coupled effective stress analysis of liquefaction problems, the Stress Density Model (SDM). Recently, SDM has been calibrated based on semi-empirical relationships between liquefaction resistance and penetration resistance, and in this approach, SDM requires only CPT data as input. As traditionally SDM has been used with somewhat degraded initial elastic modulus, this study investigates in particular, the influence of the elastic parameter on the performance and calibration of the model. Elastic parameters have a complex role in liquefaction modelling because they simultaneously affect the dynamic response of the system and stress-strain behavior of the soil. When investigated these changes, specific attention is given to the development of parameters for a generic sand with the ability of the model to concurrently simulate the effects of density and confining stress on the liquefaction resistance and their effects on the rate of strain development during cyclic mobility. First, the initial shear modulus G0 was obtained from literature for 41 clean sands tested at different densities and confining stresses. From each test, a value of A (i.e. SDM material parameter that defines the material constant in the relationship for the elastic shear modulus) was back-calculated. The shear modulus degradation curves for clean sands were scrutinised to quantify how the initial value of A changes at small strains so that a proper link between SDM and experimental data is established. Subsequently, the average value and dispersion of the data were computed. Secondly, a separate series of undrained cyclic laboratory tests on clean sand published in the literature were compiled and investigated to quantify the strain development during cyclic mobility (an aspect indirectly related to the elastic parameter A). Effects of relative density Dr and CSR (Cyclic Stress Ratio) on the strain-rate development were evaluated. As only Dr was found to correlate to the rate of deformation, mathematical expressions were developed to describe the effect of Dr on the deformation rate, both before and after achieving the selected liquefaction triggering criterion. Incorporating the relationships resulting from the laboratory data scrutiny required a minor modification in SDM, as the identified initial value of A constraints excessively the development of shear strain during cyclic mobility, particularly for dense soils and relative low CSRs. This was the principal reason why in the original SDM, a degraded value for A was used. In this thesis, different alternatives were studied including to use A as a variable, which is justified in principle, as A is strictly speaking strain-dependent. At small strains, at the beginning of the simulation, the value of A was set as suggested by laboratory data, allowing more rigorous modelling of the elastic shear stiffness. Then, as the effective stress path approaches and enters cyclic mobility and the deformation increases, the value of A was degraded to allow for the development of larger strains. Three types of representative relationships for sand were considered to evaluate the modification introduced in SDM and its calibration: (i) effects of soil density on the liquefaction resistance; (ii) effects of overburden stress on the liquefaction resistance; and (iii) effects of soil density on strain development during liquefaction and cyclic mobility. Once the simulation results were satisfactory at the element level, the modified model was evaluated in 1D effective stress analyses using the software FLAC. Two sites from Christchurch that liquefied during the 2010 Darfield (Mw=7.1) and 2011 Christchurch (Mw=6.2) earthquakes were the subject of 1D analyses. The input motion (deconvoluted record from the 2010 Darfield earthquake) was scaled to produce two levels of liquefaction response. The results were not entirely satisfactory, as a few anomalies (high dilation peaks) were noted in the acceleration time histories of the modified version. A detailed explanation of a plausible source of these peculiarities is provided regarding the interaction of the modified model and the selected numerical platform.
Title: Algebraic Relations Between Solutions of Order One Differential Equations
Seminar: Algebra
Speaker: Taylor Dupuy of Mathematics at University of Vermont
Contact: David, david.zureick-brown@emory.edu
Date: 2022-08-02 at 4:00PM
Venue: MSC W301
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Abstract: