Part I: Computability and Complexity in Topology and Differential Geometry
(A. Nabutovsky)

Computability (Turing machines, recursive functions, degrees of unsolvability), algorithmic information theory, unsolvable problems in group theory, group homology, non-computability in topology, applications of non-computability and algorithmic information theory in geometric calculus of variations (``thick" knots, local minima of Riemannian functionals, geometry of moduli spaces arising in differential geometry).

Literature: S. Weinberger, ``Computers, Rigidity and Moduli", Princeton
University Press, 2005 and papers of the instructor.

Part II: Computability and Complexity in Dynamics (M. Yampolsky)

Basic notions of computable analysis: computable real functions, computable sets in R^n. Dynamics of rational maps of the Riemann sphere: Fatou and Julia sets. Computability and complexity of Julia sets. Algorithms used to draw Julia sets versus theoretical complexity bounds. Constructing non-computable examples of Julia sets. Computability of filled Julia sets of polynomials. Computable properties and the topology of Julia sets. Open problems.

Literature: M. Braverman, M. Yampolsky, "Computability of Julia sets", Springer, 2008