All Seminars

Title: Variations on a theme of Shinzel and Wójcik
Seminar: Algebra and Number Theory
Speaker: Matthew Just of Emory University
Contact: David Zureick-Brown, dzureic@emory.edu
Date: 2021-10-19 at 4:00PM
Venue: MSC W301
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Abstract:
Let $\alpha$ and $\beta$ be rational numbers not equal to 0 or $\pm 1$. How does the order of $\alpha$ (mod $p$) compare to the order of $\beta$ (mod $p$) as $p$ varies? A result of Shinzel and W\'ojcik states that there are infinitely many primes $p$ for which the order of $\alpha$ (mod $p$) is equal to the order of $\beta$ (mod $p$). In this talk, we discuss the problem of determining whether there are infinitely many primes $p$ for which the order of $\alpha$ (mod $p$) is strictly greater than the order of $\beta$ (mod $p$). This is joint work with Paul Pollack.
Title: Inference, Computation, and Games
Seminar: Numerical Analysis and Scientific Computing
Speaker: Florian Schaefer of Georgia Institute of Technology
Contact: Yuanzhe Xi, yxi26@emory.edu
Date: 2021-09-24 at 12:30PM
Venue: MSC W201
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Abstract:

In this talk, we develop algorithms for numerical computation, based on ideas from competitive games and statistical inference.

In the first part, we propose competitive gradient descent (CGD) as a natural generalization of gradient descent to saddle point problems and general sum games. Whereas gradient descent minimizes a local linear approximation at each step, CGD uses the Nash equilibrium of a local bilinear approximation. Explicitly accounting for agent-interaction significantly improves the convergence properties, as demonstrated in applications to GANs, reinforcement learning, and computer graphics.

In the second part, we show that the conditional near-independence properties of smooth Gaussian processes imply the near-sparsity of Cholesky factors of their dense covariance matrices. We use this insight to derive simple, fast solvers with state-of-the-art complexity vs. accuracy guarantees for general elliptic differential- and integral equations. Our methods come with rigorous error estimates, are easy to parallelize, and show good performance in practice.

Title: Clusters and semistable models of hyperelliptic curves
Seminar: Algebra and Number Theory
Speaker: Jeffrey Yelton of Emory University
Contact: David Zureick-Brown, dzureic@emory.edu
Date: 2021-09-21 at 4:00PM
Venue: MSC W301
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Abstract:
For every hyperelliptic curve $C$ given by an equation of the form $y^2 = f(x)$ over a discretely-valued field of mixed characteristic $(0, p)$, there exists (after possibly extending the ground field) a model of $C$ which is \emph{semistable} -- that is, a model whose special fiber (i.e. the reduction over the residue field) consists of reduced components and has at worst very mild singularities. When $p$ is not $2$, the structure of such a special fiber is determined entirely by the distances (under the discrete valuation) between the roots of $f$, which we call the \emph{cluster data} associated to $f$. When $p = 2$, however, the cluster data no longer tell the whole story about the components of the special fiber of a semistable model of $C$, and constructing a semistable model becomes much more complicated. I will give an overview of how to construct``nice" semistable models for hyperelliptic curves over residue characteristic not $2$ and then describe recent results (from joint work with Leonardo Fiore) on semistable models in the residue characteristic $2$ situation.
Title: Steklov-Poincaré analysis of the basic three-domain stent problem
Seminar: Numerical Analysis and Scientific Computing
Speaker: Irving Martinez of Emory University
Contact: Yuanzhe Xi, yxi26@emory.edu
Date: 2021-09-17 at 12:30PM
Venue: MSC W201
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Abstract:
The Steklov-Poincaré problem was previously considered in the artery lumen and wall setting with a single interface. Here the analysis is expanded to incorporate solute behavior in the presence of a fixed-volume, solid, simple stent. In this geometry, a third domain is added to the two-domain structure of artery wall and lumen. Through this intersecting domain volume setting there are three interfaces: lumen-wall, stent-lumen, and wall-stent. Steady-state incompressible Navier-Stokes equations are used to explain the behavior of blood through the lumen, while advection-diffusion dynamics are considered for the solute mechanics across the lumen, wall, and stent. Having a fixed blood velocity value, Steklov-Poincaré decomposition of the advection-diffusion equations is applied locally to each of the interfaces. To unify these instances on a global scale, their overall intersection is explored in a smaller manifold, reducing the problem to one previously solved by Quarteroni, Veneziani, and Zunino. Through finite element analysis (FEM), the solution is discretized and found to be convergent. Finally, computational simulations with one, three, and five stent rings, placed between the volumes of inner and outer cylindrical meshes, were performed using NGSolve, confirming the convergence of the solution and its relation to the coarseness of the mesh.
Title: Turán density of cliques of order five in 3-uniform hypergraphs with quasirandom links
Seminar: Combinatorics
Speaker: Mathias Schacht of The University of Hamburg and Yale University
Contact: Dwight Duffus, dwightduffus@emory.edu
Date: 2021-09-17 at 3:00PM
Venue: MSC E408
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Abstract:
We show that 3-uniform hypergraphs with the property that all vertices have a quasirandom link graph with density bigger than 1/3 contain a clique on five vertices. This result is asymptotically best possible. This is joint work with S. Berger, S. Piga, Chr. Reiher and V. R$\ddot{\mathrm{o}}$dl.
Title: The Interplay of Curvature and Control in Dynamical Systems
Seminar: Numerical Analysis and Scientific Computing
Speaker: Romeil Sandhu of Strony Brook University
Contact: Lars Ruthotto, lruthotto@emory.edu
Date: 2021-09-10 at 12:30PM
Venue: N301
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Abstract:
This talk will focus on recent advances in geometry and control with a specific emphasis on how curvature (a measure of “flatness” in Riemannian geometry) is intimately tied to rate functions with applications in areas of inverse problems in autonomy, systems biology, to seemingly disparate areas in economics. In the first part of this talk, we will motivate our discussion with a broader thematic result due to Lott, Villani, and Sturm whereby one form of curvature, namely Ricci curvature, is intimately connected to Boltzmann entropy. In turn, we reexamine the open problem of developing Ricci curvature over discrete metric spaces and how such advances that leverage coarse geometry can be employed to exploit (network) functionality. For example, by placing a probability structure on a graph as opposed to dealing directly with the discrete space, the graph can be treated as a Riemannian manifold for which there exists a richness of tools and advantages that will be discussed. Other (combinatorial) discretizations will be introduced and their use in the context of control towards biological systems. From this and through the lens of Riemannian geometry, we will then pivot towards inverse problems in imaging for the second half of the talk. Here, we will show that stability of classical 3D shape inversion from a 2D scene is intimately tied to a form of curvature. Time permitting, we will close with a few problems in economics. Such disparate applications are presented with the intent to highlight the richness of interplay between Riemannian geometry and control and as such, this talk is designed to be accessible to a general audience with an interest in any of the above domains with a general interest in dynamical systems.

Romeil Sandhu is currently an Assistant Professor at Stony Brook University with appointments in Biomedical Informatics and Applied Mathematics & Statistics Departments. He is the recipient of the AFOSR YIP Award for work on 2D3D feedback control and machine learning for autonomous systems and NSF CAREER Award for work on geometric optimization of time-varying networks. Romeil first received his B.S. and M.S, and Ph.D. degrees from the Georgia Institute of Technology. His research interest focuses on the broad area of applied geometry, topology, & control towards the understanding of faltering autonomous agents in an unknown environment where ambiguity often arises.
Title: Angle ranks of Abelian varieties
Seminar: Number Theory
Speaker: David Zureick-Brown of Emory University
Contact: David Zureick-Brown, dzureic@emory.edu
Date: 2021-09-07 at 4:00PM
Venue: MSC W301
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Abstract:
The speaker will discuss an elementary notion -- the angle rank of a polynomial -- and how this relates to the Tate conjecture for Abelian varieties over finite fields.
Title: Initial Guesses for Sequences of Linear Systems in a GPU-accelerated Incompressible Flow Solver
Seminar: Numerical Analysis and Scientific Computing
Speaker: Anthony Austin of Naval Postgraduate School
Contact: Yuanzhe Xi, yxi26@emory.edu
Date: 2021-04-30 at 1:30PM
Venue: https://emory.zoom.us/j/95900585494
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Abstract:
We revisit the projection method of Fischer for generating initial guesses when iteratively solving a sequence of linear systems, showing that it can be implemented efficiently in GPU-accelerated PDE solvers. We specifically consider such a solver for the incompressible Navier--Stokes equations and study the effectiveness of the method at reducing solver iteration counts. Additionally, we propose new methods for generating initial guesses based on stabilized polynomial extrapolation and show that they are generally competitive with projection methods while requiring only half the storage and performing considerably less data movement and communication. Our implementations of these algorithms are freely available as part of the libParanumal collection of GPU-accelerated flow solvers.
Title: Direct Solvers for Elliptic PDEs
Seminar: Numerical Analysis and Scientific Computing
Speaker: Per-Gunnar Martinsson of UT Austin
Contact: Yuanzhe Xi, yxi26@emory.edu
Date: 2021-04-23 at 1:30PM
Venue: https://emory.zoom.us/j/95900585494
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Abstract:
That the linear systems arising upon the discretization of elliptic PDEs can be solved efficiently is well-known, and iterative solvers that often attain linear complexity (multigrid, Krylov methods, etc) have proven very successful. Interestingly, it has recently been demonstrated that it is often possible to directly compute an approximate inverse to the coefficient matrix in linear (or close to linear) time. The talk will describe some recent work in the field and will argue that direct solvers have several advantages, including improved stability and robustness, the ability to solve certain problems that have remained intractable to iterative methods, and dramatic improvements in speed in certain environments.

The talk will in particular focus on methods for solving elliptic PDEs with oscillatory solutions. These are particularly compelling targets for direct solvers, as it is notoriously difficult to attain fast convergence for iterative solvers in this environment. But they also pose additional challenges, as the inherent ill-conditioning of the physics of the problem require very high precision in both discretizing the PDE, and in solving the resulting linear system.
Title: The Hasse norm theorem and a local-global principle for multinorms
Defense: Master's Thesis
Speaker: Yazan Alamoudi of Emory University
Contact: Yazan Alamoudi, yazan.mohammed.alamoudi@emory.edu
Date: 2021-04-20 at 4:00PM
Venue: https://tinyurl.com/YAlamoudi
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Abstract:
In this defense, I will present the main result of the thesis, namely a local-global principle for multinorms from étale algebras associated to dihedral extensions of number fields of degree 2n. More precisely, the étale algebra obtained as the product of n field extensions which are the fixed fields under reflections satisfies the local-global principle for multinorms. A basic ingredient in the proof is the classical Hasse norm theorem and we shall present an outline of the proof of this theorem.\\ \\ Zoom Meeting: https://tinyurl.com/YAlamoudi. Passcode: qbdF0v