SEMINARI DI GEOMETRIA
Kodaira fibrations are surfaces of general type with a non-isotrivial fibration,
which are differentiable fibre bundles. First examples were constructed by
Kodaira and Atiyah to show that the signature need not be multiplicative in
We review approaches to construct such surfaces with focus on controlling the
signature. From this we obtain new examples with signature 16. (Joint work
with Ju A Lee and Michael Lönne)
In his PhD thesis, Paul Seidel found examples of symplectomorphisms which are smoothly isotopic to the identity, but not isotopic to the identity within the symplectomorphism group.
The problems of conjugacy rigidity and of entropy rigidity have a long history, with major breakthroughs in the last twenty years, for negatively curved compact manifolds and symmetric spaces. On the other hand, little is known for finite volume manifolds. I will survey the state of the art of these problems, and explain what can be generalized to finite volume manifolds, pointing out the main difficulties that arise in the non-compact case.
Kazhdan's property (T) is a strong negation of amenability, relevant in the study of algebraic properties of groups, in the construction of expander graphs, in dynamics and in connection to the Baum-Connes conjectures. Various strengthened versions of property (T) have been formulated in recent years.
The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal R-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal R-matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal R-matrix. On the other hand, R.
A classical theorem of Gromov states that the Betti numbers, i.e. the size of the free part of the homology groups, of negatively curved manifolds are bounded by the volume. We extend this theorem to the torsion part of the homology in all dimensions d > 3. From Gromov’s work it is known that in dimension 3 the size of torsion homology cannot be bounded in terms of the volume. In dimension 4 we give a somewhat precise estimate for the number of negatively curved manifolds of finite volume, up to homotopy, and in dimension d > 4 up to homeomorphism.
Consider a group G action on a metric space X by isometries, and subgroups M_1,..., M_l. One interesting and important problem is to study conditions under which non-emptyness of each M_i-fixed points ensures that of global (G-)fixed points. One extreme case is where G is a simple Lie group, X is an origin-excluded Hilbert space, and the action is given by a (strongly continuous) unitary representation. Then, the Howe--Moore property, based on the Mautner phenomenon, implies that for each(!) non-compact closed subgroup M, the existence of M-fixed points suffices that of G-fixed points.
Le rappresentazioni massimali sono le famiglie di sottogruppi di