# All Seminars

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
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
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
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
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
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
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
Title: Elliptic Curves and Moonshine
Defense: Dissertation
Speaker: Maryam Khaqan of Emory University
Contact: Maryam Khaqan, maryam.khaqan@emory.edu
Date: 2021-04-16 at 12:00PM
Venue: https://tinyurl.com/MKhaqan
Abstract:
In this talk, I will describe the main results of my doctoral dissertation. The first main result of my thesis is a characterization of all infinite-dimensional graded modules for the Thompson group whose graded traces are certain weight 3/2 weakly holomorphic modular forms satisfying special properties. This characterization serves as an example of moonshine for the Thompson group.\\ \\ I will begin the talk by giving a brief history of moonshine, describing some of the existing examples of the phenomenon in the literature, and discussing how my work fits into the story. I will then demonstrate how we can use the aforementioned Thompson-modules to study geometric invariants (e.g., rank, p-Selmer groups, and Tate—Shafarevich groups) of certain families of elliptic curves. In particular, this serves as an example of using moonshine answer questions in number theory.\\ \\ Meeting ID: 984 6807 4730 Passcode: 196884
Title: Randomization in Numerical Linear Algebra (RandNLA)
Seminar: Numerical Analysis and Scientific Computing
Speaker: Petros Drineas of Purdue University
Contact: Yuanzhe Xi, yxi26@emory.edu
Date: 2021-04-09 at 1:30PM
Venue: https://emory.zoom.us/j/95900585494
Abstract:
The introduction of randomization in the design and analysis of algorithms for matrix computations (such as matrix multiplication, regression, the Singular Value Decomposition (SVD), etc.) over the past 20 years provided a new paradigm and a complementary perspective to traditional numerical linear algebra approaches. These novel approaches were motivated by technological developments in many areas of scientific research that permit the automatic generation of Big Data, which are often modeled as matrices. In this talk, we will primarily focus on how such approaches can be used to design fast solvers for least-squares problems, ridge-regression problems, and even linear programs.
Title: Top-k Extreme Contextual Bandits with Arm Hierarchy
Seminar: Numerical Analysis and Scientific Computing
Speaker: Lexing Ying of Stanford University
Contact: Yuanzhe Xi, yxi26@emory.edu
Date: 2021-03-26 at 1:30PM
Venue: https://emory.zoom.us/j/95900585494
Abstract:
Contextual bandit is an online decision making framework that has found many applications in recommendation systems and search tasks. In this talk, we consider the extreme contextual bandit problem where the enormous number of arms poses the main theoretical and algorithmic challenges. This setting is particularly relevant to the mission of Amazon Search but yet rather under-explored in the literature. To address the large arm space, we introduce two new techniques. The first is an extension of the inverse gap weighting to the multiple arm feedback case, where the optimal regret bound is shown under realizability condition. The second is an arm space hierarchy that exploits the potential similarity between the arms. By combining these two techniques, we develop an extreme top-k contextual bandit algorithm that scales logarithmically in terms of the number of arms. Joint work with Rajat Sen, Alexander Rakhlin, Rahul Kidambi, Dean Foster, Daniel Hill, and Inderjit Dhillon.
Title: Isogenies of Elliptic Curves and Arithmetical Structures on Graphs
Defense: Dissertation
Speaker: Tomer Reiter of Emory University
Contact: Tomer Reiter, tomer.reiter@emory.edu
Date: 2021-03-19 at 11:00AM
Venue: https://emory.zoom.us/j/92804829998?pwd=OFpZcWdlS2lrUFRLbDZQNklxZ3IwQT09
Abstract:
In this defense, we prove two results that come from studying curves. The first is a classification result for elliptic curves. Let $\mathbf{Q}(2^{\infty})$ be the compositum of all quadratic extensions of $\mathbf{Q}$. Torsion subgroups of rational elliptic curves base changed to $\mathbf{Q}(2^{\infty})$ were classified by Laska, Lorenz, and Fujita. Recently, Daniels, Lozano-Robledo, Najman, and Sutherland classified torsion subgroups of rational elliptic curves base changed to $\mathbf{Q} (3^{\infty})$, the compositum of all cubic extensions of $\mathbf{Q}$. We classify cyclic isogenies of rational elliptic curves base changed to $\mathbf{Q}(2^{\infty})$, for all but finitely many elliptic curves over $\mathbf{Q}(2^{\infty})$.\\ \\ Next, we turn to arithmetical structures, which Lorenzini introduced to model degenerations of curves. Let $G$ be a connected undirected graph on $n$ vertices with no loops but possibly multiedges. Given an arithmetical structure $(\textbf{r}, \textbf{d})$ on $G$, we describe a construction which associates to it a graph $G'$ on $n-1$ vertices and an arithmetical structure $(\textbf{r}', \textbf{d}')$ on $G'$. By iterating this construction, we derive an upper bound for the number of arithmetical structures on $G$ depending only on the number of vertices and edges of $G$. In the specific case of complete graphs, possibly with multiedges, we refine and compare our upper bounds to those arising from counting unit fraction representations.