**Abstracts**

**Nelia Charalambous**

*"The spectrum of the Laplacian on noncompact manifolds"*

**Abstract: **The essential spectrum of the Laplacian on functions has been extensively studied. It is known that on hyperbolic space a spectral gap appears, whereas is has been conjectured that on manifolds with uniformly subexponential volume growth and Ricci curvature bounded below the essential spectrum is the nonnegative real line. In our work with Zhiqin Lu we prove a generalization of Weyl's criterion for the essential spectrum. We then apply this generalized criterion to expand the set of manifolds on which the essential spectrum is the nonnegative real line. We also use our criterion to compute the essential spectrum of complete shrinking Ricci solitons and weighted manifolds, as well as to study the essential spectrum of the Laplacian on forms.

**Noé Bárcenas**

"**Segal's spectral sequence for twisted equivariant K Theory"**

**Abstract:** In joint work with Uribe, Velasquez, Espinoza and Joachim, we construct a spectral sequence for computing twisted equivariant $K$-Theory for proper actions. We Give an application to the Baum-Connes conjecture.

**Pedro Ontaneda**

*"Riemannian hyperbolization"*

**Abstract:** The strict hyperbolization process of R. Charney and M. Davis produces a large and rich class of negatively curved spaces (in the geodesic sense). This process is based on an earlier version introduced by M. Gromov and later studied by M. Davis and T. Januszkiewicz. If M is a manifold its Charney-Davis strict hyperbolization is also a manifold, but the negatively curved metric obtained is far from being Riemannian because it has a large and complicated set of singularities. We will discuss whether these singularities can be removed, so that this process can be done smoothly.

**Andrés Pedroza**

*"Gromov-Witten invariants of blow ups"*

**Abstract:** We will describe how the moduli space of holomorphic curves of the one point blow up \tilde{M} fits inside the moduli of holomorphic curves of M. Moreover it contains enough information so that the Gromov-Witten invariants of M can be computed in terms of the Gromov-Witten invariants of \tilde{M}.

**Gabriel Ruíz**

*"Generalization of the Laplacian formula of the angle function"*

**Abstract: **James Simons proved a called "Extrinsic rigidity theorem" for minimal hypersurfaces in S^n: A compact minimal hypersurface in S^n, whose normal vector makes a positive inner product with a fixed vector in $R^{n+1}$

is totally geodesic. In his proof, he calculated the Laplacian of the angle function. In this talk I will give a formula for the Laplacian with respect to a closed and conformal vector field in the ambient. It will be true for a hypersurface with constant mean curvature and invariant under the curvature tensor of the ambient. This is part of a work with A. J. Di Scala and C. Barrera-Cadena.

**Pablo Suárez Serrato**

*"On the topology and geometry of higher graph manifolds"*

**Abstract:**Our understanding of $3$--manifolds has illuminated two distinct classes of importance; hyperbolic manifolds and graph manifolds. These are by now considered the basic blocks featured in the geometrisation programme of Thurston, famously consolidated by Perelman. From one perspective graph manifolds are exactly the manifolds that {\it collapse}, in the sense that they admit a family of smooth metrics whose volumes tend to zero while their sectional curvatures remain bounded. In fact the work of Shioya-Yamaguchi was instrumental in the last steps of Perelman's approach; they pointed out that manifolds that collapse are precisely graph manifolds.

Historically the term {\it graph manifold} was introduced by Waldhausen in the 1960's, it really highlited the fact that the fundamental group can be described as a graph of groups and that the manifolds were built up from fundamental pieces that are (heuristically) described in terms of singular circle bundles over 2-orbifolds.

In a recent (and fundamental) paper Frigerio, Lafont and Sisto proposed a family of {\it generalised graph manifolds}; products of $k$--tori with hyperbolic $(n-k)$--manifolds with truncated cusps are glued along their common $n$--toral boundaries. They explored multiple topological aspects of this family and raised some questions. For example, in their definition $k$ is allowed to equal zero, so that one subfamily is made up of hyperbolic manifolds glued along truncated cusps. They asked if the minimal volume of such manifolds is achieved by the sum of the hyperbolic volumes of the pieces.

Together with Chris Connell we answered this question positively. In so doing we realised that a natural family we termed {\it higher graph manifolds} could be defined; bundles of infranilpotent manifolds over negatively curved bases are glued along boundaries (when possible). This family further extends the one proposed by Frigerio, Lafont and Sisto. (Although in contrast to Waldhausen's definition both families are to be though of as {\it nonsingular}).

We first characterise the higher graph manifolds that admit volume collapse, by explicitly constructing sequences of metrics with volume collapse (this builds on earlier work by Fukaya). Various results about the simplicial volume and volume entropy of this family are calculated. Then we exploit the graph structure of the fundamental group to show that these manifolds obey the Baum-Connes conjecture, have finite asymptotic dimension and do not admit metrics of positive scalar curvature. Finally we use several of the produced results to prove that when the infranilmanifold fibre has positive dimension the Yamabe invariant vanishes.

**Ferrán Valdez **

*"Polynomial invariants of hyperbolic 3-manifolds"*

**Abstract:** A hyperbolic 3-manifold that fibers over the circle gives rise to a family of pseudo-Anosov maps on surfaces of different genus. In this talk we will explain how Thurston's theory of fibered faces gives rise to a coherent picture of this situation and how the Teichmuller polynomial (introduced by McMullen) is defines and used to calculate the dilations of such pseudo-Anosov maps.