All Seminars

Title: Beating flops, communication and synchronization in sparse factorizations
Colloquium: Computational Mathematics
Speaker: Sherry Li of Lawrence Berkeley National Lab
Contact: Lars Ruthotto, lruthotto@emory.edu
Date: 2018-04-05 at 3:00PM
Venue: MSC E 208
Download Flyer
Abstract:
Multiphysics and multiscale simulations often need to solve discretized sparse algebraic systems that are highly indefinite, nonsymmetric and extremely ill-conditioned. For such problems, factorization based algorithms are often at the core of the solvers toolchain. Compared to pure iterative methods, the higher computation and communication costs in factorization methods present serious hurdles to utilizing extreme-scale hardware. I will present several research vignettes aimed at reducing those costs. By incorporating data-sparse low-rank structures, such as hierarchical matrix algebra, we can obtain lower arithmetic complexity as well as robust preconditioner. By replicating small amount of data in sparse factorization, we can avoid communication with provablly lower communication complexity. By means of asynchronous, custermized broadcast/reduction, we can reduce the dominating latency cost in sparse triangular solution. The effectiveness of these techniques will be demonstrated with our open source software STRUMPACK and SuperLU.

Bio:
Sherry Li is a Senior Scientist at Lawrence Berkeley National Laboratory. She has worked on diverse problems in high performance scientific computations, including parallel computing, sparse matrix computations, high precision arithmetic, and combinatorial scientific computing She has (co)authored over 100 publications. She is the lead developer of SuperLU sparse direct solver library, and has contributed to several other widely-used mathematical libraries, including ARPREC, LAPACK, STRUMPACK, and XBLAS. She received Ph.D. in Computer Science from UC Berkeley in 1996. She is a SIAM Fellow and an ACM Senior Member.
Title: Efficient, stable, and reliable solvers for the steady incompressible Navier-Stokes equations in computational hemodynamics
Defense: Dissertation
Speaker: Alexander Fuller Viguerie of Emory University
Contact: Alexander Fuller Viguerie, aviguer@emory.edu
Date: 2018-04-04 at 9:00AM
Venue: MSC E406
Download Flyer
Abstract:
In recent years, improvements in medical imaging and image-reconstruction algorithms have led to increased interest in the use of Computational Fluid Dynamics (CFD) as a clinical tool in hemodynamics. While such methods have long been employed in the design of medical devices and in basic medical research, many of the techniques commonly employed in these contexts are not ideal in the clinical setting. In particular, in clinical settings typically one is faced with more demanding turnaround times for simulations, less powerful computational resources, and noisy, incomplete, or missing data.\\ \\ In this thesis, we discuss these challenges and introduce CFD methods which are more practical for direct clinical application. Frequently in these settings, the variable of interest is the temporal average of some time-periodic quantity, such as wall shear-stress, over a cardiac cycle. In these cases, the standard procedure is to perform an unsteady simulation over several cardiac cycles and then to take the time average of the last one. Here, we propose to instead surrogate the unsteady time-averaged solution with the solution of a steady-state problem, allowing us to compute it directly. This approach, if properly applied, can dramatically lower computational cost as we show here; however in many respects the steady problem is arguably more difficult numerically than its unsteady counterpart.\\ \\ We will address these difficulties and propose effective workarounds. In particular, we aim to develop methods for steady solvers that are \textit{efficient}, \textit{stable}, and \textit{reliable}. Roughly speaking, this work is divided into three parts, with each part focusing on one of these aspects. Concerning efficiency, we extend the inexact algebraic factorization approach popular for the unsteady problem into the steady setting. We will address the issue of stability by taking inspiration from nonlinear filtering techniques used in turbulence modeling to develop stabilization techniques for the steady problem. Finally, we will develop and validate methods for assigning boundary conditions in data-deficient settings while maintaining reliability. Throughout each section, we will provide both theoretical and numerical justification for our methods.
Title: Deligne's Exceptional Series and Modular Linear Differential Equations
Type: Master's Defense
Speaker: Robert Dicks of Emory University
Contact: Robert Dicks, robert.julian.dicks@emory.edu
Date: 2018-04-03 at 1:00PM
Venue: White Hall 200
Download Flyer
Abstract:
In 1988, Mathur, Mukhi, and Sen studied rational conformal field theories in terms of differential equations satisfied by their characters. These differential equations are modular invariant, and the solutions they obtain for order 2 equations have relationships with certain Lie algebras. In fact, the Lie algebras in the Deligne Exceptional series appear, whose study is motivated by uniformities which appear in their representation theory. This thesis studies the Deligne Exceptional Series from these two perspectives, and gives a sequence of finite groups which has analogies with the Deligne series.
Title: Eulerian series, zeta functions and the arithmetic of partitions
Defense: Dissertation
Speaker: Robert Schneider of Emory University
Contact: Robert Schneider, robert.schneider@emory.edu
Date: 2018-04-03 at 3:00PM
Venue: PAIS 280
Download Flyer
Abstract:
In this talk we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory, $q$-series, algebra, modular forms and analytic number theory. We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and explore new classes of partition-theoretic zeta functions and Dirichlet series --- as well as ``Eulerian'' $q$-hypergeometric series --- enjoying many interesting relations. We find a number of theorems of classical number theory and analysis arise as particular cases of extremely general combinatorial structure laws.\\ \\ Among our applications, we prove explicit formulas for the coefficients of the $q$-bracket of Bloch-Okounkov, a partition-theoretic operator from statistical physics related to quasi-modular forms; we prove partition formulas for arithmetic densities of certain subsets of the integers, giving $q$-series formulas to evaluate the Riemann zeta function; we study $q$-hypergeometric series related to quantum modular forms and the ``strange'' function of Kontsevich; and we show how Ramanujan's odd-order mock theta functions (and, more generally, the universal mock theta function $g_3$ of Gordon-McIntosh) arise from the reciprocal of the Jacobi triple product via the $q$-bracket operator, connecting also to unimodal sequences in combinatorics and quantum modular-like phenomena.
Title: Odd order transcendental obstructions to the Hasse principle on general K3 surfaces
Seminar: Algebra
Speaker: Jennifer Berg of Rice University
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2018-04-03 at 4:00PM
Venue: W304
Download Flyer
Abstract:
Varieties that fail to have rational points despite having local points for each prime are said to fail the Hasse principle. A systematic tool accounting for these failures is called the Brauer-Manin obstruction, which uses [subsets of] the Brauer group, Br X, to preclude the existence of rational points on a variety X. After fixing numerical invariants such as dimension, it is natural to ask which birational classes of varieties fail the Hasse principle, and moreover whether the Brauer group (or certain distinguished subsets) always explains this failure. In this talk, we will focus on K3 surfaces (e.g. a double cover of the plane branched along a smooth sextic curve) which are relatively simple surfaces in terms of geometric complexity, but whose arithmetic is more mysterious. For example, in 2014, Skorobogatov asked whether any odd torsion in the Brauer group of a K3 surface could obstruct the Hasse principle. We answer this question in the affirmative; via a purely geometric approach, we construct a 3-torsion transcendental Brauer class on a degree 2 K3 surface which obstructs the Hasse principle. Moreover, we do this without needing to explicitly write down a central simple algebra. This is joint work with Tony Varilly-Alvarado.
Title: Deterministic and stochastic acceleration techniques for Richardson-type iterations
Defense: Dissertation
Speaker: Massimiliano Lupo Pasini of Emory University
Contact: Massimiliano Lupo Pasini, massimiliano.lupo.pasini@emory.edu
Date: 2018-04-02 at 1:00PM
Venue: W306
Download Flyer
Abstract:
The next generation of computational science applications will require numerical solvers that are both reliable and capable of high performance on projected exascale platforms. In order to meet these goals, solvers must be resilient to soft and hard system failures, provide high concurrency on heterogeneous hardware configurations, and retain numerical accuracy and efficiency. This work focuses on the solution of large sparse systems of linear equations, for example of the kind arising from the discretization of partial differential equations (PDEs). Specifically, the goal is to investigate alternative approaches to existing solvers (such as preconditioned Krylov subspace or multigrid methods). To do so, we consider stochastic and deterministic accelerations of relaxation schemes. On the one hand, starting from a convergent splitting of the coefficient matrix, we analyze various types of Monte Carlo acceleration schemes applied to the original preconditioned Richardson (stationary) iteration. These methods are expected to have considerable potential for resiliency to faults when implemented on massively parallel machines. In this framework, we have identified classes of problems and preconditioners that guarantee convergence. On the other hand, we consider Anderson-type accelerations to increase efficiency and improve the convergence rate with respect to one level fixed point schemes. In particular, we focus on a recently introduced method called Alternating Anderson-Richardson (AAR). We provide theoretical results to explain the advantages of AAR over other similar schemes presented in literature and we show numerical results where AAR is competitive against restarted versions of the generalized minimum residual method (GMRES) for problems of different nature and different preconditioning techniques.
Title: The Quantum McKay Correspondence: Classifying "Finite Subgroups" of a Quantum Group with Graphs
Defense: Master's Thesis
Speaker: Paul Vienhage of Emory University
Contact: Paul Vienhage,
Date: 2018-04-02 at 1:00PM
Venue: MSC E406
Download Flyer
Abstract:
The McKay Correspondence classifies finite subgroups of the rotation group of 3-space via graphs. In this talk we explore a quantum version of this correspondence. Specifically, we will cover the needed background on category theory, vertex operator algebras, and quantum groups to explain a powerful technique used by Kirillov and Ostrik to develop a quantum analog to the McKay correspondence.
Title: Connections between Classical and Umbral Moonshine
Defense: Dissertation
Speaker: Sarah Trebat-Leder of Emory University
Contact: Sarah Trebat-Leder, sarah.trebat-leder@emory.edu
Date: 2018-04-02 at 2:00PM
Venue: W302
Download Flyer
Abstract:
Both results of this dissertation involve finding unexpected connections between the classical theory of monstrous moonshine and the newer umbral moonshine. In our first result, we use generalized Borcherds products to associate to each pure A-type Niemeier lattice a conjugacy class g of the monster group and give rise to identities relating dimensions of representations from umbral moonshine to values of McKay-Thompson series. Our second result focuses on the Mathieu group M23. While it inherits a moonshine from being a subgroup of M24, we find a new and simpler moonshine for M23 such that the graded traces are, up to constant terms, identical to the monstrous moonshine Hauptmoduln.
Title: Maass forms and modular forms: applications to class numbers and partitions
Defense: Dissertation
Speaker: Olivia Beckwith of Emory University
Contact: Olivia Beckwith, olivia.dorothea.beckwith@emory.edu
Date: 2018-04-02 at 3:00PM
Venue: W302
Download Flyer
Abstract:
This dissertation is about arithmetic information encoded by analytic characteristics (such as Fourier coefficients) of classical modular forms and a real-analytic generalization of modular forms called harmonic Maass forms. For example, I use the theory of harmonic Maass forms to extend and refine a theorem of Wiles on class number divisibility. I also prove asymptotic bounds for Rankin-Selberg shifted convolution L-functions in shift aspect. In partition theory, I use the circle method to describe the expected distribution of parts of integer partitions over residue classes, and I use effective estimates for partition functions to obtain simple formulas for functions arising in group theory.
Title: Patching and local-global principles for gerbes with an application to homogeneous spaces
Defense: Dissertation
Speaker: Bastian Haase of Emory University
Contact: Bastian Haase, bastian.haase@emory.edu
Date: 2018-04-02 at 4:00PM
Venue: W302
Download Flyer
Abstract:
Starting in 2009, Harbater and Hartmann introduced a new patching setup for semi-global fields, establishing a patching framework for vector spaces, central simple algebras, quadratic forms and other algebraic structures. In subsequent work with Krashen, the patching framework was refined and extended to torsors and certain Galois cohomology groups. After describing this framework, we will discuss an extension of the patching equivalence to bitorsors and gerbes. Building up on these results, we then proceed to derive a characterisation of a local- global principle for gerbes and bitorsors in terms of factorization. These results can be expressed in the form of a Mayer-Vietoris sequence in non-abelian hypercohomology with values in the crossed-module $G->Aut(G)$. After proving the local-global principle for certain bitorsors and gerbes using the characterization mentioned above, we conclude with an application on rational points for homogeneous spaces.