The central focus of the course is the interplay of the topology and dynamical systems, in particular, the following type of questions will be addressed: What constraints does the topology of phase space impose on the dynamics? Given a region of the phase space, what can be said about the orbits inside it? What algebraic invariants can tell us about a dynamical system and its bifurcations?
The course is organized in five modules:
The first two lay the geometric foundations: flows and limit sets, hyperbolic fixed points and their stable and unstable manifolds, connecting orbits, attractors, repellers, and Morse decompositions. The third module, the theoretical heart of the course, develops Conley index theory: isolating neighbourhoods, index pairs, the continuation theorem, and the connection matrix — an algebraic object that detects the existence of connecting orbits from purely homological data.. The fourth module turns to parametric dynamics: structural stability, local bifurcations (saddle-node, Hopf), global bifurcations (homoclinic and heteroclinic orbits), symbolic dynamics, and chaos. The fifth and final module develops Forman's discrete Morse theory from scratch — critical cells, discrete gradient vector fields, the discrete Morse theorem and Morse inequalities. Moreover, it establishes the discrete Conley index, closing the loop with Module III and connecting the theory to persistent homology and computational dynamics.
Goals:
- Understand the qualitative structure of flows using limit sets, Morse decompositions, and Lyapunov functions, and apply Conley's decomposition theorem.
- Compute the Conley index of isolated invariant sets and use continuation invariance and the connection matrix to establish the existence of connecting orbits.
- Classify local and global bifurcations and relate bifurcation values to changes in the Conley index.
- Establish chaotic behaviour rigorously using symbolic dynamics, the Smale horseshoe, and topological entropy.
- Construct discrete Morse functions on simplicial complexes and apply the discrete Morse theorem and discrete Conley index.
Target group: The course is designed for first- and second-year PhD students with a general mathematical background. The student should have completed graduate-level courses in real analysis and linear algebra and has some familiarity with ordinary differential equations, but need not have a specialisation in topology or dynamical systems.
Prerequisites: Graduate real analysis, point-set topology, linear algebra, basic algebraic topology (singular homology, homotopy), some familiarity with ordinary differential equations. No need to have a specialization in topology or dynamical systems.
Evaluation: The final grade will be determined based on student participation in lectures and by the quality of solutions provided to the problems discussed during recitations.
Teaching format: None
ECTS: 3 Year: 2026
Track segment(s):
Elective
Teacher(s):
Michal Lipinski
Teaching assistant(s):
- Trainer/in: Michal Lipinski