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Titles:

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Graduate students talks:
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I   Hellwig, Sieg	Conley index for flows
II  Noeske, Robertz	Attractor repeller pairs 
III Korneffel, Petera	Morse decomposition  
IV  Soemers, Pankratz   Algebra and homology of cubical sets
V   Pasyuga		Relative and reduced homology, exact sequences
VI  Neunhöffer, Dymkou	Connecting homomorphism for attractor repeller pairs
VII Gawron, Eberlein	Connection matrices 


Expert talks:
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Mischaikow I	 Preview/ Recap Lecture 
Mischaikow II	 Computational Homology I 
Mischaikow III	 Computational Chain Recurrence 
Junge I		 Global Numerics I: Invariant Sets, GAIO 
Wanner I	 Structure Theorems from Conley Index I 

Mischaikow IV	 Structure Theorems from Conley Index II 
Mischaikow V	 Computational Homology II 

Junge II	 Global Numerics II: Isolating Neighborhoods and Statistics 
Wanner II	 Infinite- Dimensional Dynamical Systems 
Junge III	 Local Numerics 
Wanner III	 Applications: The Cahn- Hilliard Model I 
Roberts, Barakat Computation of Homology Groups in Maple 

Wanner IV	 Applications: The Cahn- Hilliard Model II 
Junge IV	 Applications: Kuramoto- Sivashinsky and Kot- Schaffer


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Abstracts Oliver Junge
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Global Numerics I: Invariant Sets, GAIO
     The lecture covers the basics of set-oriented numerics
     for the computation of invariant sets and invariant
     manifolds: the subdivision algorithm, the continuation
     algorithm, and statements about their convergence.
     We will discuss implementational issues and give an
     overview of the associated software package GAIO.

Global Numerics II: Isolating Neighborhoods and Statistics
     Building on the first lecture on global numerics, algorithms
     for the computation of isolating neighborhoods and index
     pairs will be presented.  In a second part, it will be shown
     how to derive global statistical information on the dynamics.
     In particular, the concept of invariant measures and almost
     invariant sets will be presented, together with algorithms
     for their computation.

Local Numerics
     This lecture covers "classical" numerical methods for the
     local analysis of dynamical systems.  We will focus on
     pseudo-arclength continuation and bifurcation analysis
     for equilibria and on the computation of homo- and heteroclinic
     solutions.

Applications: Kuramoto-Sivashinsky and Kot-Schaffer
     On the basis of these two example systems, it will be shown
     how to derive rigorous results about the dynamics of infinite
     dimensional systems by combining the theoretical and numerical
     methods presented in the previous lectures.  This includes
     in particular the reduction to a finite dimensional multivalued
     system, the associated numerical computation, as well as the
     lifting of the computed information to the original system
     via Conley index arguments.

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Abstracts Thomas Wanner
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Structure Theorems from Conley Index I
     This lecture introduces the fundamental concepts of connection
     and transition matrices. In addition to their basic properties,
     I will discuss how symmetries of the underlying dynamical
     system are reflected in the structure of these matrices.

Infinite-Dimensional Dynamical Systems
     To address the varied background of the audience, this lecture
     provides background material on infinite-dimensional dynamical
     systems, which is needed for the subsequent applications of the
     theory. Specifically, I will discuss a dynamical systems setting
     to study deterministic and stochastic partial differential equations
     and describe basic numerical techniques for approximating the solutions
     of such equations. Moreover, basic results from bifurcation theory
     will be discussed.

Applications: The Cahn-Hilliard Model I
     This lecture is the first of two lectures on a model equation
     from materials science due to Cahn and Hilliard, as well as its
     stochastic extension. I will discuss the physical background of
     the models, as well as aspects of their derivation. The major part
     of the lecture is concerned with a phase separation phenomenon
     called spinodal decomposition. It will be demonstrated how
     computational homology can be used to test the validity of these
     models against experiments, as well as to uncover significant
     differences between the deterministic and the stochastic version
     of the model.

Applications: The Cahn-Hilliard Model II
     The second lecture on the Cahn-Hilliard model is concerned with
     rigorous results on the dynamics of these partial differential
     equations. I will present results on the structure of the equilibrium
     set, as well as on the long-term dynamics as described by a global
     attractor. In addition, it will be indicated how these results relate
     to a second phase separation phenomenon called nucleation, which can
     be observed in the stochastic model.

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