Thomas Stieltjes Institute for Mathematics

Analysis 1997/98


Title: Course Operator Theory
Lecturers: M.A. Kaashoek, (); S.M. Verduyn Lunel, ()
Time and place: 3 hours per week, from the beginning of September until the beginning of December; Vrije Universiteit Amsterdam
Contents: This course treats various topics from the theory of non-selfadjoint operators with the aim to develop a qualitative theory for concrete classes of integral- and (functional) differential equations. A strong interaction with complex function theory is typical; operators are studied using analytic functions (determinant functions, characteristic matrix functions, symbols) which appear in a natural way. Often operators will be considered as maps generated by dynamical systems. Much attention will be given to the analysis of particular families of solutions (existence, asymptotic behaviour etc.).
The list of topics for this year includes the following subjects. The resolvent operator and the Riesz-Dunford calculus, in particular for unbounded operators, spectral theory of strongly continuous semi-groups and the corresponding generators, in connection with this the development of a perturbation theory (variation of constants formula), completeness of systems of eigenfunctions and generalised eigenfunctions, aiming at a qualitative theory of dynamical systems, applications to functional differential equations.
Literature: O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel and H.O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995, and I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators I, Birkha"user Verlag, 1990 (to acquire in consultation with the lecturers).
Prerequisites: introduction to functional analysis and complex function theory
Examination: via exercises and/or presentations
Remark : the first six weeks of this course also serve as a preparation for the workshops on "Dynamics of Differential Equations with Delays" and "Operators and Dynamical Systems" which will be held at Leiden University from 13-15 and 23-26 October, respectively.

Title: Seminar Analysis and Linear Operators
Organiser: M.A. Kaashoek, ()
Time and place: Each Thursday morning from 9.15 - 11.30 a.m.,
Vrije Universiteit Amsterdam, room R2.40 of the building De Boelelaan 1081
Contents: Various research topics from Analysis, Operator Theory, and related fields. See the announcements in ET-NA.

Title: Course computer algebra algorithms for special functions
Lecturer: T.H. Koornwinder (), (http://turing.wins.uva.nl/~thk/).
Time and place: Trimester I, 1997, University of Amsterdam
Aim: Zeilberger and Gosper algorithms and applications
Contents: There exists a large number of sums of products of binomial coefficients equal to some elementary expression. Such combinatorial identities can be rewritten in terms of so-called hypergeometric functions. The classical analytic proofs of such identities have recently been supplemented with algorithmic proofs that can be implemented in computer algebra systems such as Maple or Mathematica. These algorithmes have been developed by Gosper and Zeilberger and they are the starting point for much new research. In this course various aspects are discussed: from the underlying mathematical theory to the computer implementation. The related special functions are being discussed whenever they come along. Depending on their interests and previously followed courses, students may put the emphasis either on the various computer algebra aspects or on the theoretical background and special functions.
Structure: lectures
Literature: M. Petkovsek, H.S. Wilf and D. Zeilberger, ``A=B'', published by A.K. Peters, Wellesley, Massachusetts, 1996.
Prerequisite: Calculus, Linear Algebra, some familiarity with Maple or Mathematica Suited for 4th year math students, graduate students, also for other students interested in computer algebra.
Examination: hand-in exercises (both theoretical and computer) and/or take home exam and/or oral exam

Title: Course advanced Theory of Functions
Lecturer: Dr. J.J.O.O. Wiegerinck ( )
Time and place: trimester 1, 4 hours each week (probably Tuesday 9-11 and Wednesday 13-15; will be announced later). location:UvA
Contents: The aim of this course is the derivation of a theory of functions on open Riemann surfaces. Specific attention will be paid to the Riemann mapping theorem, multiply connected domains, uniformising Riemann surfaces, harmonic and subharmonic functions, Hardy spaces on the disk and on open Riemann surfaces.
Literature: lecture notes (by J. Korevaar) for the first part, furthermore Ahlfors L. en Sario: Riemann Surfaces Princeton, 1960. Fisher, S. D.: Function theory on planar domains, Wiley, 1983. [additional material will be indicated later]

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Title: Course finite groups of Lie type
Lecturers: H.T. Koelink (), (http://turing.wins.uva.nl/~koelink/) and E.M. Opdam ()
Time and place: Tuesday 11.15-13.00 & 14.15-16.00
Spring semester 1998
First meeting January 20, 1998.
Mathematical Institute, Leiden University
Aim: Understand character theory of finite groups of Lie type
Contents: Apart from a few exceptions (cyclic and alternating groups and a finite list of sporadic groups) the finite groups arise from simple Lie groups defined over a finite field. These finite groups of Lie type have a rich structure, and this enables us to understand the characters of these groups. In this course we discuss the relation between the character theory and geometric structures (orbits). We also study Harish-Chandra induction, which gives all characters, except to so-called cuspidal characters, if we assume that the characters of certain subgroups are known. If possible the Delign-Lusztig induction functor will be discussed. Other keywords for this course are Weyl groups, BN-pairs and Hecke algebras
Structure: lectures
Literature: R.W. Carter, ``Finite Groups of Lie Type'', Wiley, 1985
Prerequisite: Some acquaitance with Lie groups and Lie algebras. Suited for 4th year math students and graduate students.
Examination: take home exam and/or oral exam

Title: International Workshop 1997 Twente Conference on Lie Theory
Organisors: UT, KUN, UU, RUG, RUL, UvA.
Coordination: G.F. Helminck (UT) ()
Time and place: December 15-18, 1997, Twente University.
More information: http://www.math.utwente.nl/~lie/

Title: International Summer School European School of Group Theory
Contact: G. van Dijk, ()
Time and place: June 21 - July 4, 1998, Lorentz Center, Leiden.
Contents: The summer school is organised on yearly basis by European mathematians working in Group Theory. In 1997 the summerschool was held in Luminy, France, and in 1996 it was held in Schloss Hirschberg, Germany.
More information: http://www.wi.leidenuniv.nl/~esgt98/esgt98.html
For information about the Lorentz Center: http://www.lc.leidenuniv.nl.
Title: Seminar Analysis on Lie Groups
Organisors: E.P. van den Ban, G.J. Heckman, E.M. Opdam, (), T.H. Koornwinder (),
Time and place: Two-weekly seminar on Friday at Utrecht University starting January or Februaryr 1998
Contents: The subject will be vertex algebras and some related papers by Borcherds.
Literature: V.G. Kac, ``Vertex Algebras for Beginners'', 141 pp, AMS, 1997. R.E. Borcherds, ``Automorphic Forms and Lie Algebras'', Current Developments in Mathematics, 1996, pp. 1-27.
More information: http://turing.wins.uva.nl/~thk

Title: Course Functional Analytic Methods for Partial Differential Equations.
Lecturers: Ph. Clement, () ; B. de Pagter, () .
Time and place: 4 hours weekly, 2x7 weeks.
(the precise schedule and location will be announced later)
Technische Universiteit Delft.
Aim: To introduce the students in certain functional analytic methods in the study of partial differential equations.
Contents: This course will consist of two parts: (1) Bifurcation theory; (2) Sobolev spaces and regularity theory. The following topics will be covered: (1)- Differential calculus in Banach spaces; - Implicit function theorem - Bifurcation theorem of Crandall- Rabinowitz; - Brouwer and Leray-Schauder degree; - Global bifurcation theorem of Rabinowitz; (2)- Definition and elementary properties of Sobolev spaces; - Embedding theorems; - The trace operator; - regularity theory for elliptic problems.
Prerequisites: It is assumed that the students are familiar with the basic principles of functional analysis and operator theory.
Examination: oral examination.

Title: Course Modern Computational Fluid Dynamics
Lecturer: B. Koren ()
Time and place: Second half 1997, on Thursday, 13.45-15.30 h. (starting from October 2, 1997) Mathematical Institute, Niels Bohrweg 1, Leiden
Aim: Getting insight into theoretical and practical aspects of modern computational fluid dynamics
Contents:
  1. Equations of fluid dynamics
  2. History of computational fluid dynamics
  3. Upwind discretization methods (1-D, multi-D, nonlinear systems, limiters)
  4. Multigrid solution methods
  5. Robust coarse-grid corrections (guest lecturer: dr. P.M. de Zeeuw, CWI)
  6. Local grid refinement
  7. Sparse-grid solution methods
  8. Parallel computing software (guest lecturer: drs. C.T.H. Everaars, CWI)
  9. Conditioning of stiff systems of equations
  10. Industrial applications
Course material: Reprints of papers and reports (to be distributed during the course)
Literature: 1. Ch. Hirsch: Numerical Computation of Internal and External Flows, Volumes 1 and 2, Wiley, Chichester (1988 and 1990). 2. R.J. LeVeque: Numerical Methods for Conservation Laws, Birkhauser, Basel (1990). 3. P. Wesseling: An Introduction to Multigrid Methods, Wiley, Chichester (1992).
Prerequisites: Basic theory of partial differential equations
Examination: oral or take-home exam

Title: Defect Correction and Multigrid Methods
Lecturer: Prof.dr P.W.Hemker (P.W.Hemker@cwi.nl)
Time and place: (probably)
Trimester I, 1997 - 1998 (sept-nov,1997)
Thursday 15h15 - 17h00
Gebouw Euclides, Plantage Muidergracht 24 1018 TV Amsterdam
Aim: To acquire a working knowledge of the use of multigrid techniques for the solution of elliptic partial differential equations.
Contents: Starting from the defect correction principle, in this course its use for the construction of accurate discretisation methods for PDEs and the construction of fast iterative processes will be treated. The course consists of the following chapters:
  1. Defect correction processes
  2. Discretisation of continuous equations
  3. Defect correction and discretisation
  4. Example of a two-grid algorithm
  5. Multigrid algorithms
  6. Multigrid convergence
  7. Convergence proofs for multigrid methods
  8. Local mode analysis
Literature: Special lecture notes for the course will be available. P. Wesseling: Introduction to Multigrid Methods, Wiley, 1992.
W. Hackbusch: Multigrid Methods and Applicatios, Springer, 1985.
Prerequisites: Elementatry knowledge of functional analysis.
Principles of numerical mathematics.
Some experience in computer programming in a procedural language
Examination: To conclude the course, the student is asked to write a paper on an exercise problem.

Title: Course Numerical analysis and dynamical systems
Lecturer: K.J. in 't Hout, ()
Time and place: Thursday 11.15 -- 13.00,
Fall semester,
Leiden University,
starting October 2, 1997
Contents: A fundamental question in the numerical solution of initial value problems for ordinary differential equations is whether the long-time dynamics of ordinary differential equations are preserved under numerical discretization. For example one can think of the convergence of solutions to an equilibrium point or a periodic orbit, or of a particular (physical) quantity that remains constant through time. In this course, the above question is addressed. In particular the following topics are covered:
  1. attractors of differential equations under numerical discretization
  2. numerical methods for Hamiltonian systems
Literature: A.M. Stuart & A.R. Humphries: Dynamical systems and numerical analysis. Cambridge University Press, 1996.
Prerequisites: Some knowledge about differential equations and their numerical solution is assumed

Title: Conference of the Dutch Community of Numerical Mathematicians
Time and place: September 24, 25 and 26 of 1997,
the Woudschoten Conference Centre, Zeist, the Netherlands.
Topics of this year's conference are:
  1. nonlinear boundary-value problems (with special attention to continuation methods and bifurcation)
  2. generalized eigenvalue problems and singular-value decomposition
  3. numerical treatment of financial models

Invited speakers are:

Contributed, 25-minute presentations:
The programme allows incorporation of about four contributed presentations by participants, relevant to either of the conference topics.

Conference fees, due upon registration, are

Registration and optional submission of a contributed presentation can be done by mailing the registration form ultimately on 22 August 1997 (the form is in Dutch, it is obvious that the information needed includes name, address, optional title of contributed presentation, fee category, dietary requirements and signature; when using the URL given below, after completion of the form, please press [click hier] and print the appearing page, the print can then be signed and sent in.

Title and abstract of a contributed presentation in a LaTeX file must be e-mailed to the secretary no later than 18 August.

For further information please apply to the secretary of the organizing committee:

Jan Kok
CWI - Centrum voor Wiskunde en Informatica
Organizing committee Woudschoten Conference
P.O. Box 94079
NL-1090 GB Amsterdam
Telephone: +31 20 592 4107

This information and a (clickable) registration form can be reached using:
http://WWW.cwi.nl/~jankok/woudschoten.html


This page is constucted by , date: 14/10/1997