Bücher der Reihe Oberwolfach Seminars

Higher Dimensional Algebraic Geometry presents recent advances in the classification of complex projective varieties. Recent results in the minimal model program are discussed, and an introduction to the theory of moduli spaces is presented.

These notes form an extended version of talks given at the DMV-seminar in Giinzburg, September 1990. The seminar consisted of two parts: 1) "The isomorphism problem for integral group rings", with the main talks given by K. W. Roggenkamp and shorter contributions by W.Kimmerle, J.llitter and A. Zimmermann (Part 1). 2) "Galois-Module structure", with the main talks given by M.Taylor and shorter contributions by N.Byott (Part 2). We greatly appreciate the opportunity, given us by the DMV to hold this seminar. DMV-Seminar Part 1 Group Rings: Units and the Isomorphism Problem K. W. Roggenkamp with contributions by W. Kimmerle and A. Zimmermann Contents 3 Table of Contents I Some general facts ... 7 § 1 Ring reduction to PID . 7 § 2 Modules ........ . 7 II Some notes on representation theory 9 §1 Blocks .................. . 9 §2 Normalizers of p-Sylow subgroups of group bases 11 §3 Cohomology rings 12 Profinite groups .. . . . . . . . . 13 §4 III The leading coefficient of units 15 IV Class sum correspondence ... . . . . . . . . . . . .. . . 21 .

They are related to representations of finite groups and group actions on spheres, and are considered as a generalisation of the spherical space form problem. The second part reviews the general setting of surgery theory and reports on the computation of the surgery abstraction groups.

One of the key concerns in modern control theory is the design of steering strategies. The implementation of such strategies is done by a regulator. Presented here is a self-contained introduction to the mathematical background of this type of regulator design. The topics selected address the matter of greatest interest to the control community, at present, namely, when the design objective is the reduction of the influence of exogeneous disturbances upon the output of the system. In a first scenario the disturbance signal is regarded as a deterministic time series with known dynamics but unknown parameters. The design objective is then the asymptotic disturbance compensation. In a second scenario, no information about the disturbance signal is available apart from some bounds. Here, in an H-approach, control strategies are worked out which will prove efficient for all such disturbances. The intention of this book is to present ideas and methods on such a level that the beginning graduate student will be able to follow current research. New results are included, especially for nonlinear control systems, and as a service to the reader, an extensive appendix presents topics from linear algebra, invariant manifolds and calculus of variations, information which is hardly to be found in standard textbooks. Contents: Introduction ¿ The problem of output regulation ¿ Introduction ¿ Problem statement ¿ Output regulation via full information ¿ Output regulation via full error feedback ¿ A particular case ¿ Well-posedness and robustness ¿ The construction of a robust regulator ¿ Disturbance attenuation via H-methods ¿ Introduction ¿ Problem statement ¿ A characterization of the L2-gain of a linear system ¿ Disturbance attenuation via full information ¿ Disturbance attenuation via measured feedback ¿ Full information regulators ¿ Problem statement ¿ Time-dependent control strategies ¿ Examples ¿ Time-independent control strategies¿ The local case ¿ Nonlinear observers ¿ Problem statement ¿ Time-dependent observers ¿ Error feedback regulators ¿ Examples ¿ Nonlinear H-techniques ¿ Introduction ¿ Construction of the saddle-point ¿ The local scenario ¿ Disturbance attenuation via linearization ¿ Matrix equations ¿ Linear matrix equations ¿ Algebraic Riccati equations ¿ Invariant manifolds ¿ Existence theorem ¿ Outflowing manifolds ¿ Asymptotic phase ¿ Convergence for T ¹ ¿ A special case ¿ Dichotomies and Lyapunov functions ¿ Hamilton-Jacobi-Bellman-Isaacs equation ¿ Introduction ¿ Method of characteristics ¿ The equation of Isaacs ¿ The Hamiltonian version of Isaacs' equation ¿ Bibliography

thus the connectedness theorem can be used to prove the inequality-part of Hartshorne's conjecture on linear normality, whereas Deligne's generalisation of the connectedness theorem leads to a refinement of Barth's results on the topology of varieties with small codimension in a projective space.

The first part of the book provides an introduction to key tools and techniques in dispersive equations: Strichartz estimates, bilinear estimates, modulation and adapted function spaces, with an application to the generalized Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation.

Serre's GAGA-theorems [56J and base change for field extensions the algebraic analogue was obtained for projective manifolds over a field k of characteristic p = 0, for a long time no algebraic proof was known and no generalization to p > 0, except for certain lower dimensional manifolds.

Finite element methods and the closely related boundary element methods nowadays belong to the standard routines for the computation of solutions to boundary and initial boundary value problems of partial differential equations with many applications as e.g.