Gradient Flows

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Description

Tuesdays 11.00am-1.00pm, starting week 3 / 22nd October

 

Brief description

The course will provide a survey of the theory of gradient flows.

Probable topics:

  • Review of semiflows with a Lyapunov function and convergence to rest points.
  • Gradient flows in finite dimensions, elements of Morse theory, mountain pass theorem, Lojasiewicz inequality.
  • Gradient flows in Hilbert Space. Brezis-Komura theorem.
  • Gradient flows in metric spaces.
  • Wasserstein gradient flows and other applications to PDE.

 

Course material:

Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second Edition, Birkhauser, 2008.

Luigi Ambrosio, Elia Brué, Daniele Semola, Lectures on Optimal Transport, Springer, 2021.

Haim Brézis,  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies, vol. 5. North-Holland, Amsterdam (1973).

Yukio Matsumoto. An Introduction to Morse Theory, AMS, 2002.

 

Assessment:  by short presentations on topics expanding or complementing course content.