The Morse theory of critical points, originally developed by Morse with his work on closed geodesics and later extended by Palais and Smale to a certain class of functionals on Hilbert manifolds, is inapplicable for problems defined in a Banach setting, as the usual notion of nondegeneracy for critical points cannot be satisfied, and, even in a Hilbert setting, it is of no use in case the critical points have infinite Morse index and coindex, as the critical groups always vanish. One way to overcome these issues is to use the Morse complex approach, which relies on a different notion of nondegeneracy as well as on the theory of stable and unstable manifolds, and uses their intersections to build a chain complex generated by critical points. In this talk I will discuss advantages and drawbacks of the Morse complex approach with respect to the classical approach, both from a general point of view and for concrete examples, such as the Hamiltonian action functional on cotangent bundles, functionals involving the pLaplacian, and the functionals of Diracharmonic maps. This is joint work with Maciej Starostka.
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Morse theory in infinite dimension old and new
Research Group:
Speaker:
Luca Asselle
Institution:
Bochum University
Schedule:
Thursday, November 9, 2023  14:00
Abstract:
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