Vorträge in der Woche 12.02.2007 bis 18.02.2007
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Montag, 12.02.2007: Constraint Preserving Approximation of Harmonic Maps
Dr. Soeren Bartels
Harmonic maps are stationary points of the Dirichlet energy among vector fields with values in the unit sphere. Owing to the nonconvex constraint, harmonic maps are non-unique and fail to admit higher regularity properties. Moreover, the constraint prohibits the use of standard tools for the numerical approximation. In this talk we discuss stability and weak convergence of three numerical schemes. The first scheme consists in the minimization of the Dirichlet energy over suitable tangent spaces and a renormalization of the update in each iteration. The second approach penalizes the constraint and leads to a time-dependent Ginzburg-Landau equation. Implicit and explicit methods for the discretization of the harmonic map heat flow are the basis for the third approach. Besides stating sufficient conditions for stability and weak convergence to an exact solution, we indicate generalizations to the approximation of p-harmonic maps. Applications include liquid crystal theory, image processing, and micromagnetics. Part of this talk is based on joint work with A. Prohl.
| Uhrzeit: | 17:15 |
| Ort: | M3 |
| Gruppe: | Kolloquium |
| Einladender: | Prohl |
Freitag, 16.02.2007: Class Numbers of Cyclotomic Fields
Rene Schoof
Even since Kummer's 19th century attempts to prove Fermat's Last Theorem, cyclotomic fields have been at the center of attention. Indeed, in the second half of the 20th century the subject enjoyed a strong revival in the hands of Iwasawa. Despite tremendous progress, many questions remain. Vandiver's conjecture is perhaps the most outstanding one. In this talk we discuss the problem of calculating class numbers of cyclotomic fields. This is a computational problems that, even using the fastest computers, has only been solved in a handful of cases. We present an experimental approach to determining these class numbers.
| Uhrzeit: | 17:15 |
| Ort: | M3 |
| Gruppe: | Kolloquium |
| Einladender: | Kaup |