Sala Seminari (Dip. Matematica)
Abstract: Complex projective structures are geometric structures locally modelled on the geometry of the Riemann sphere with its group of Möbius transformations PSL(2,C). As this space appears as a natural boundary at infinity for the hyperbolic space, a typical feature of these structures is the interplay between complex analysis and hyperbolic geometry, which gives rise to a rich deformation theory.
Rappresentazioni di Galois triangoline e funtori di Schur
Abstract: In this talk we will talk about the relationship between hyperbolic cone-structures and their holonomy representations. Any hyperbolic structure on a given closed compact and orientable surface S induces in a very natural way a representation of the fundamental group \pi_1(S) in the Lie group PSL(2,R), which encodes geometric data about the structure. The reverse problem to recover a hyperbolic cone-structure from a given representation is more arduous and longer. Even worse it is not always possible.
We consider the linear wave equation outside a compact, strictly convex obstacle in R3with smooth boundary and we show that the linear wave flow satisfies the dispersive estimates as in R3 (which is not necessarily the case in higher dimensions). This is joint work with Gilles Lebeau.
Abstract: Rigidity of lattices in semisimple Lie groups has been widely studied so far. One of the most celebrated theorem is Mostow rigidity, which states the following. Assume n bigger or equal than 3. If two complete hyperbolic n-manifolds of finite volume have isomorphic fundamental groups then they are isometric. Equivalently their fundamental groups are conjugated in the group of isometries of the hyperbolic n-space. There are several ways to prove this theorem.
We study the the moduli space of KSBA stable pairs (X,sS+\sum a_i F_i), consisting of a Weierstrass fibration X, its section S, and some fibers F_i. We find a compactification which is a DM stack, and we prove that there are wall-crossing morphisms when the weights s and a_i change. This recovers the work of Ascher and Bejleri in the case in which s=1 and the one of La Nave when s=1 and a_i=0.
Quasimaps provide a compactification of the space of maps from smooth curves to a GIT quotient; Ciocan-Fontanine and Kim used them to prove beautiful wall-crossing formulae, comparing the resulting invariants with ordinary Gromov-Witten ones. In joint work with Navid Nabijou, we introduce the notion of genus zero relative quasimaps to a toric target X with respect to a smooth (but not necessarily toric) hyperplane section Y, extending work of Gathmann to this setting.