Seminars and Colloquia by Series

Absence of shocks in Euler-Maxwell system for two-fluid models in plasma

Series
School of Mathematics Colloquium
Time
Thursday, April 23, 2015 - 11:01 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yan Guo – Brown University
As the cornerstone of two-fluid models in plasma theory, Euler-Maxwell (Euler-Poisson) system describes the dynamics of compressible ion and electron fluids interacting with their own self-consistent electromagnetic field. It is also the origin of many famous dispersive PDE such as KdV, NLS, Zakharov, ...etc. The electromagnetic interaction produces plasma frequencies which enhance the dispersive effect, so that smooth initial data with small amplitude will persist forever for the Euler-Maxwell system, suppressing any possible shock formation. This is in stark contrast to the classical Euler system for a compressible neutral fluid, for which shock waves will develop even for small smooth initial data. A survey along this direction for various two-fluid models will be given during this talk.

The classifying space of the stable mapping class group is an infinite loop space

Series
Geometry Topology Student Seminar
Time
Wednesday, April 22, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonathan Paprocki – Georgia Tech

Please Note: For Prof. Wickelgren's Stable Homotopy Theory class

Harer's homology stability theorem states that the homology of the mapping class group for oriented surfaces of genus g with n boundary components is independent of g for low degrees, increasing with g. Therefore the (co)homology of the mapping class group stabilizes. In this talk, we present Tillmann's result that the classifying space of the stable mapping class group is homotopic to an infinite loop space. The string category of a space X roughly consists of objects given by disjoint unions of loops in X, with morphisms given by cobordisms between collections of loops. Sending X to the loop space of the realization of the nerve of the string category of X is a homotopy functor from Top to the category of infinite loop spaces. Applying this construction for X=pt obtains the result. This result is an important component of the proof of Mumford's conjecture stating that the rational cohomology of the stable mapping class group is generated by certain tautological classes.

Matrix weighted function spaces and the Carleson Embedding Theorem

Series
Analysis Seminar
Time
Wednesday, April 22, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Amalia Culiuc – Brown University
We will prove a recent version of the weighted Carleson Embedding Theorem for vector-valued function spaces with matrix weights. Time permitting, we will discuss the applications of this theorem to estimates on well-localized operators. This result relies heavily on the work of Kelly Bickel and Brett Wick and is joint with Sergei Treil.

Stability and long time dynamics of Hamiltonian PDEs

Series
Research Horizons Seminar
Time
Wednesday, April 22, 2015 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Zhiwu Lin – Georgia Tech
Many physical models without dissipation can be written in a Hamiltonian form. For example, nonlinear Schrodinger equation for superfluids and Bose-Einstein condensate, water waves and their model equations (KDV, BBM, KP, Boussinesq systems...), Euler equations for inviscid fluids, ideal MHD for plasmas in fusion devices, Vlasov models for collisionless plasmas and galaxies, Yang-Mills equation in gauge field theory etc. There exist coherent structures (solitons, steady states, traveling waves, standing waves etc) which play an important role on the long time dynamics of these models. First, I will describe a general framework to study linear stability (instability) when the energy functional is bounded from below. For the models with indefinite energy functional (such as full water waves), approaches to find instability criteria will be mentioned. The implication of linear instability (stability) for nonlinear dynamics will be also briefly discussed.

Arithmetic Combinatorics and Character Sums

Series
Combinatorics Seminar
Time
Tuesday, April 21, 2015 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Brandon Hanson – University of Toronto
Characters are a central tool for understanding arithmetic. For example, the most familiar character is the Legendre symbol, which detects the quadratic residues. In this talk I will present a few ideas as to how character sums may be useful in arithmetic combinatorics and vice versa. Traditionally, estimates for character sums have been used to count arithmetic configurations of interest to the combinatorialist. More recently, arithmetic combinatorics has proved useful in the estimation of certain character sums. Many character sums are easy to estimate provided they have enough summands - this is sometimes called the square-root barrier and is a natural obstruction. I will show how the sum-product phenomenon can be leveraged to push past this barrier.

What is and how to compute efficiently the Markovian Joint Spectral Radius?

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 20, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Antonio Cicone – L'Aquila, Italy
Given a finite set of matrices F, the Markovian Joint Spectral Radius represents the maximal rate of growth of products of matrices in F when the matrices are multiplied each other following some Markovian law. This quantity is important, for instance, in the study of the so called zero stability of variable stepsize BDF methods for the numerical integration of ordinary differential equations. Recently Kozyakin, based on a work by Dai, showed that, given a set F of N matrices of dimension d and a graph G, which represents the admissible products, it is possibile to compute the Markovian Joint Spectral Radius of the couple (F,G) as the classical Joint Spectral Radius of a new set of N matrices of dimension N*d, which are produced as a particular lifting of the matrices in F. Clearly by this approach the exact evaluation or the simple approximation of the Markovian Joint Spectral Radius becomes a challenge even for reasonably small values of N and d. In this talk we briefly review the theory of the Joint Spectral Radius, and we introduce the Markovian Joint Spectral Radius. Furthermore we address the question whether it is possible to reduce the exact calculation computational complexity of the Markovian Joint Spectral Radius. We show that the problem can be recast as the computation of N polytope norms in dimension d. We conclude the presentation with some numerical examples. This talk is based on a joint work with Nicola Guglielmi from the University of L'Aquila, Italy, and Vladimir Yu. Protasov from the Moscow State University, Russia.

Spin Bundles

Series
Geometry Topology Student Seminar
Time
Monday, April 20, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Shane Scott – Georgia Tech
Spin bundles give the geometric data necessary for the description of fermions in physical theories. Not all manifolds admit appropriate spin structures, and the study of spin-geometry interacts with K-theory. We will discuss spin bundles, their associated spectra, and Atiyah-Bott-Shapiro's K orientation of MSpin--the spectrum classifying spin-cobordism.

Tracking Control for Neuromuscular Electrical Stimulation

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 20, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Michael Malisoff – Louisiana State University

Please Note: Speaker’s Biography:Michael Malisoff received his PhD in 2000 from the Department of Mathematics at Rutgers University in New Brunswick, NJ. In 2001, he joined the faculty of the Department of Mathematics at Louisiana State University in Baton Rouge (LSU), where he is now the Roy Paul Daniels Professor #3 in theLSU College of Science. His main research has been on controller design and analysis for nonlinear control systems with time delays and uncertainty and their applications in engineering. One of his projects is joint with the Georgia Tech Savannah Robotics team, and helped develop marine robotic methods to help understand the environmental impacts of oil spills. His more than 100 publications include a Springer monograph on constructive Lyapunov methods. His awards include the First Place Student Best Paper Award at the 1999 IEEE Conference on Decision and Control, two three-year NationalScience Foundation Mathematical Sciences Priority Area grants, and 9 Best Presentation awards in American Control Conference sessions. He is an associate editor for IEEE Transactions on Automatic Control and for SIAM Journal on Control and Optimization.

We present a new tracking controller for neuromuscular electrical stimulation, which is an emerging technology that can artificially stimulateskeletal muscles to help restore functionality to human limbs. We use a musculoskeletal model for a human using a leg extension machine. The novelty of our work is that we prove that the tracking error globally asymptotically and locally exponentially converges to zero for any positive input delay andfor a general class of possible reference trajectories that must be tracked, coupled with our ability to satisfy a state constraint. The state constraint is that for a seated subject, the human knee cannot be bent more than plus or minus 90 degrees from the straight down position. Also, our controller only requires sampled measurements of the states instead of continuousmeasurements and allows perturbed sampling schedules, which can be important for practical applications where continuous measurement of the states is not possible. Our work is based on a new method for constructing predictor maps for a large class of nonlinear time-varying systems, which is of independent interest. Prediction is a key method for delay compensation that uses dynamic control to compensate for arbitrarily long input delays. Reference: Karafyllis, I., M. Malisoff, M. de Queiroz, M. Krstic, and R. Yang, "Predictor-based tracking for neuromuscular electrical stimulation," International Journal of Robust and Nonlinear Control, to appear. doi: 10.1002/rnc.3211

Pressure Ulcers and Applied Mathematics

Series
Applied and Computational Mathematics Seminar
Time
Friday, April 17, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Stephen Sprigle – Schools of Industrial Design and Applied Physiology, Georgia Tech
The Rehabilitation Engineering and Applied Research Lab (REARLab) performs both experimental research and product development activities focused on persons with disabilities. The REARLab seeks collaboration from the School of Mathematics on 2 current projects. This session will introduce wheelchair seating with respect to pressure ulcer formation and present two projects whose data analysis would benefit from applied mathematics. 3D Tissue Deformation- Sitting induces deformation of the buttocks tissues. Tissue deformation has been identified as the underlying cause of tissue damage resulting from external loading. The REARLab has been collecting multi-planar images of the seated buttocks using MRI. This data clearly shows marked differences between persons, as expected. We are interested in characterizing tissue deformation as a combination of displacement and distortion. Some tissues- such as muscle- displace (translate within the sagittal, coronal and transverse planes) and distort (change shape). Other tissue such as skin and subcutaneous fat, simple distorts. We seek a mathematical means to characterize tissue deformation that reflects its multi-planar nature. Categorizing Weight-shifting behaviors - many wheelchair users have limitations to their motor and/or sensory systems resulting in a risk of pressure ulcers. Pressure ulcers occur when localized loading on the skin causes ischemia and necrosis. In an attempt to reduce risk of pressure ulcer occurrence, wheelchair users are taught to perform weight-shifts. Weight shifts are movements that re-distribute loads off the buttocks for short periods of time. The REARLab is measuring weight shifting behaviors of wheelchair users during their everyday lives. We seek a means to classify patterns of behavior and relate certain patterns to healthy outcomes versus other patterns that result in unhealthy outcomes.

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