A Morse function on a smooth manifold M is a smooth function f: M -> R such that the Hessian of f is invertible at all critical points. It turns out that such a function always exists on a quite general class of smooth manifolds (including all closed ones) and reflects the topology of M quite closely: For example, if the only critical points of a Morse function on M are one global maximum and one global minimum, then M is a sphere.

Among the more spectacular applications of Morse Theory are:
• Computation of the singular homology of closed manifolds; for example, the proof of Poincare? duality in this context is especially illuminating
• The Bott periodicity theorem on the homotopy groups of unitary and orthogonal groups
• The h-cobordism theorem; this led to a proof of the generalized Poincaré conjecture in dimensions greater than four (which earned Stephen Smale a Fields medal)
• Floer homology, an application of Morse theory for (infnite dimensional) Hilbert manifolds. This is an active area of research

The aim of the seminar will be to give a mostly self-contained account of the technical details of Morse theory and some of the above applications. It will also contain a crash-course in Riemannian geometry.

Necessary qualifications for participants include a good acquaintance with analysis on manifolds and linear Algebra including the basic theory of quadratic forms. Knowledge of Algebraic Topology or Differential Geometry is not necessary/can be acquired during the seminar but will be helpful especially for the more advanced talks.

• Registration for course work/examination/ECTS: FlexNow

• Giving a seminar talk and writing a detailed written report