Mondays, 16.00-17.30, room WI 401, begin: 29 January
The notion of homotopy principle or h-principle is one of the key concepts in an elegant language developed by Gromov to deal with a host of questions in geometry and topology. Roughly speaking, for a certain differential geometric problem to satisfy the h-principle is equivalent to saying that a solution to the problem exists whenever certain obvious topological obstructions vanish.
The foundational examples for applications of Gromov's ideas include (i) Hirsch-Smale immersion theory, (ii) Nash-Kuiper C 1-isometric immersion theory, (iii) existence of symplectic and contact structures on open manifolds.
Gromov has developed several powerful methods to prove h-principles. In this seminar we want to discuss two of these: the covering homotopy method and convex integration theory.
Students receive credit (7 stp.) for giving one of the 90 min. presentations. The first four lectures (after the introductory survey) require as background knowledge only a first course in manifold theory. Please contact if you are interested in participating actively.
Main source for this seminar will be the book Embeddings and Immersions by M.Adachi (on order).
N.B.The previously announced seminar on geometries of surfaces and 3-manifolds will not take place.
|Introduction to Gromov's language of partial differential relations and h-principles
|Regular closed curves in the plane
|K. van Winden
|Morse functions and handlebody decompositions
|Spaces of maps and their topologies
|The h-principle for open, invariant relations I
|The h-principle for open, invariant relations II
|The h-principle for open, invariant relations III
|Convex integration theory
|O. van Koert
|Complex structures on open manifolds I
|O. van Koert / M. Lübke
|Complex structures on open manifolds II
F. Pasquotto, January 8, 2001.