Two weeks on Global Analysis

course: Topological invariants in bifurcation theory of Fredholm maps

speaker: Jacobo Pejsachowicz (Politecnico di Torino)

abstract: I will analyze bifurcation of zeroes of parameterized families of Fredholm maps from a topological point of view. Given a family of maps f: P × X ! Y between Banach spaces X, Y parameterized by a manifold P, assuming that the equation f(p, x) = 0 has a known (trivial) branch of solutions e.g. f(p, 0) = 0, a bifurcation point for solutions of the equation f(p, x) = 0 from the trivial branch is a point p in P such that every neighborhood of (p, 0) in P ×X contains nontrivial solutions of this equation. By the implicit function theorem, bifurcation can occur only at points p of the trivial branch where the linearization Lp = Dxf(p, 0) is singular. However the existence of such points is only a necessary condition for bifurcation but in general it is not sucient. It turns out that, if each Lp is a Fredholm operators of index 0, then the presence of bifurcation can be determined from the homotopy class of the map L: P ! 0(X, Y ). The space 0(X, Y ) of all Fredholm operators of index 0 has a rich topology. The purpose of linearized-bifurcation theory is the study and computation of homotopy invariants of the map L whose nontriviality entails bifurcation. If the family f is given by gradients of a family of C2 functionals, then each Lp is self-adjoint and stronger invariants can be found. When the parameter space is one dimensional then bifurcation arise if either the parity of a path L or ( in the variational case) the spectral flow of L is non trivial. For general parameter space, it s is natural to look at the K-theory of P and derived groups because the homotopy classes of maps from P to 0(X, Y ) are in one to one correspondence with stable equivalence classes of vector bundles over P. I will briefly discuss this, together with some applications to bifurcation of geodesics, periodic orbits and solutions of nonlinear elliptic boundary values problems.

timetable:
Mon 14 Feb, 10:05 - 11:00, Sala Conferenze Centro De Giorgi
Tue 15 Feb, 10:05 - 11:00, Sala Conferenze Centro De Giorgi
Wed 16 Feb, 10:05 - 11:00, Sala Conferenze Centro De Giorgi
Thu 17 Feb, 10:05 - 11:00, Sala Conferenze Centro De Giorgi
Fri 18 Feb, 10:05 - 11:00, Sala Conferenze Centro De Giorgi
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