Seminars and Colloquiums
for the week of September 22, 2008
Speakers:
Professor Horst Behncke, University of Osnabruck , Monday
Matija Cencelj University of Ljubljana, Monday
Professor Jie Xiong, Monday
Professor Carl Sundberg, Wednesday
Mr. Chad Giusti, University of Oregon, Wednesday
Professor Michael Holst, Univ of CA, San Diego, Friday
Monday, September 22, 2008
DIFFERENTIAL EQUATIONS AND COMPUTATIONAL MATHEMATICS SEMINAR
TIME: 3:35 – 4:25 p.m.
ROOM: Ayres 309A
SPEAKER: Professor Horst Behncke, University of Osnabruck
TITLE: “Spectral Theory of Higher Order Difference Operators“
ABSTRACT: In the last 20 years it has become increasingly clear that second order difference operators ( Jacobi matrices ) and Sturm-Liouville operators have closely related theories. This makes it plausible to extend the results of a joint paper by Behncke, Hinton, and Remling ( J. Differential Equations, 2001) on differential operators of order 2n to the difference equation setting. Some further results on fourth order difference operators with unbounded coefficients are also shown. Again they resemble closely their differential equation counterparts.
TOPOLOGY SEMINAR
TIME: 1:25 – 2:15 p.m.
ROOM: Ayres 214
SPEAKER: Matija Cencelj (University of Ljubljana)
TITLE: “Coarse cohomological dimension of expanders with large girth”
ABSTRACT: Expanders are highly connected sparse graphs of great interest in computer science, in areas ranging from parallel computation to complexity theory, from cryptography to coding theory, and, most recently, computational group theory. The (anti-Cech) coarse homological dimension of the families of graphs with large girth is proved to be 1 and the coarse cohomological dimension is 2. The main significance of this result is that expanders with large girth have infinite asymptotic dimension, but finite coarse (co-)homological dimension. (Joint work with A. N. Dranishnikov and A. Vavpetic.)
PROBABILITY SEMINAR
TIME: 10:10 – 11:00
ROOM: Ayres 209A
SPEAKER: Professor Jie Xiong
TITLE: “Optimal Stopping with Reward Constraints”
Wednesday, September 24, 2008
ANALYSIS SEMINAR
TIME: 3:35 – 4:30 p.m.
ROOM: Ayres 309B
SPEAKER: Professor Carl Sundberg
TITLE: “Rank-One Perturbations of Self-Adjoint Operators, 4.2”
TOPOLOGY SEMINAR
TIME: 3:35 – 4:25 p.m.
ROOM: Ayres 209A
SPEAKER: Mr. Chad Giusti (University of Oregon)
TITLE: “Plumber's knots and unstable Vassiliev theory”
ABSTRACT: We develop the first complete unstable version of Vassiliev theory.
We choose to bound the complexity of knots by studying plumber's knots, which are piecewise linear with all sticks parallel to the axes. Our first main result gives an algorithm to enumerate such knot types with a bounded number of moves, which we have implemented to determine for example that there are seven components of the space of plumber's knots with five moves, though they all fall into one of three topological types.
We then import Vassiliev's original techniques to this setting, constructing an unstable Vassiliev spectral sequence which is compatible with stabilization. Our second main result is to extend the notion of Vassiliev derivatives to singular knots with singularities other than collections of isolated double-points. This result opens the door to constructing new Vassiliev-style knot invariants and/or seeing the strength of finite-type invariants, if we can understand the behavior of Vassiliev derivatives under stabilization (subdivision of segments).
Friday, September 26, 2008
MATHEMATICS COLLOQUIUM
TIME: 3:35 – 4:25 p.m.
ROOM: Ayres 214
SPEAKER: Professor Michael Holst, Univ of CA, San Diego
HOST: Xiaobing Feng (Contact Prof. Feng for our visitor's schedule.)
TITLE: “Some New Existence and Approximation Results for the Constraints in the Einstein Equations”
ABSTRACT: There is currently tremendous interest in geometric PDE, due in part to activities such as the geometric flow program used recently to solve the Poincare conjecture, and to the NSF-funded LIGO project involving the detection of gravitational wave emission and propogation involving numerical solution of the Einstein equations. In this lecture, we consider the coupled nonlinear elliptic constraints in the Einstein equations. The constraint equations must be solved numerically to produce initial data for gravitational wave simulations, and to enforce the constraints during dynamical simulations. In the first part of the lecture, we consider a thirty-five-year-old open question involving existence of solutions to the constraint equations on space-like hyper-surfaces with arbitrarily prescribed mean extrinsic curvature, and we give a partial answer using a priori estimates and a new type of topological fixed-point argument. In the second part of the lecture, we develop some adaptive numerical methods for which we can prove a number of useful results on convergence, optimality, and scalability. Based on the a priori estimates developed in the first part of the lecture, we establish some critical discrete estimates. We then derive error estimates for Galerkin approximations, and describe a class of nonlinear approximation algorithms based on adaptive finite element methods (AFEM). We establish some new AFEM convergence and optimality results for geometric PDE problems with non-monotone nonlinearities such as the Einsteinconstraints. We finish by illustrating the algorithms with some examplesusing the Finite Element ToolKit (FETK).
This is joint work with Gabriel Nagy and Gantumur Tsogtgerel. (Physical Review Letters, Vol. 100 (2008), No. 16, pp. 161101.1, and http://arxiv.org/abs/0712.0798).
If you are interested in giving or arranging a talk for one of our seminars or colloquiums, please review our calendar.
If you have questions, or a date you would like to confirm, please contact Dr. Steve Wise.
Week of:
Past notices:
Seminars from 2007-2008 academic year
Seminars from 2006-2007 academic year