Abstracts

Ugo Bruzzo

Title: Crepant resolutions of quotient singularities and Ricci-flat metrics

Abstract: The quotients C^3/G, where G is a finite subgroup of SL(3,C), admit crepant resolutions, i.e., they are resolved by noncompact Calabi-Yau manifolds. In view of theorems by Joyce and others, these admit Ricci-flat metrics, which are either generalized ALE or quasi-ALE metrics, depending on whether the locus in C^3 with nontrivial isotropy under the G-action is compact or not. While the existence of such Ricci-flat metric is ensured by these theorems, their explicit construction is far from being trivial. In my talk I will review some more or less recent known results and will relate on work in progress about the problem, which uses the Kähler quotient construction of the resolutions.

Pedro Ontaneda

Title: 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 show that these singularities can be removed (provided the hyperolization piece is large). Hence the strict hyperbolization process can be done in the Riemannian setting.

Anna Fino

Title: Balanced Hermitian metrics and the Hull-Strominger system

Abstract: A Hermitian metric on a complex manifold is balanced if its fundamental form is co-closed. An important tool for the study of balanced manifolds is the Hull-Strominger system.

In the talk I will review some general results about balanced Hermitian metrics and present new smooth solutions to the Hull-Strominger system, showing that the Fu-Yau solution

on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. The talk is based on a joint work with G. Grantcharov and L. Vezzoni.

Jorge Lauret

Title: Prescribing Ricci curvature on homogeneous spaces

Abstract: Given a symmetric 2-tensor T on a manifold M, it is a classical problem in Riemannian geometry to ask about the existence (and uniqueness) of a metric g on M such that Ric(g) = T (see e.g. [Besse,Chap.5]). Assuming that M is a homogeneous manifold, we will consider in the talk the prescribed Ricci curvature problem in a G-invariant setting, where G is a Lie group acting transitively on M.

After an overview of results and questions, we will give a formula for the derivative dRic of the map Ric at a G-invariant metric g, in terms of the moment map for the variety of algebras. As an application, we study metrics which are Ricci locally invertible, in the sense that the map Ric is, locally, as injective and surjective as it can be (e.g., when the kernel of dRic at g consists only of the subspace generated by g). Our main result is that such property is generic in the compact case.

This is joint work with Cynthia Will.

Luis Hernández Lamoneda

Title: Banach's isometric problem

Abstract: In his 1932 book, Banach posed the following question: Fix a positive integer n and let V be a real (finite or infinite dimensional) Banach space. Assume that all n-dimensional subspaces of V are isometric to each other. Does it follow that V is Hilbert?

In 1967, Gromov discovered a connection between the isometric problem and G-structures on S^n; from this, he gave a positive answer for all even n.

Recently, also using Gromov’s key observation, we have managed to settle half of the remaining open cases: namely, for every n=4k+1 (but different from 133).

This is joint work with G. Bor, V. Jiménez and L. Montejano.

Paolo Piccinni

Title: Some remarks on Spin(9) and related geometries.

Abstract: It will be a survey on Spin(9) structures in Riemannian geometry, and on the related notions of Clifford system and of even Clifford structure. I will discuss examples, mostly exceptional symmetric spaces of compact type, their cohomological properties, and the construction of a twistor space and of a diamond diagram over parallel even Clifford manifolds. This latter construction is a collaboration I had with G. Arizmendi and R. Herrera.

Alexander Quintero

Title: Scattering amplitudes and L∞-algebras

Abstract: In this talk I will explain how the information contained in a classical field theory (symmetries, field content, field equations and Noether currents) can be conveniently encoded in an algebraic structure known as L∞-algebra. The latter are generalisations of the notion of a graded Lie algebra in which the Jacobi identity is satisfied only up to homotopy. I will also show how the scattering amplitudes (at tree level) of certain quantum field theories, as well as the recursion relations that they satisfy, can be directly extracted from the minimal model of the L∞-algebra that governs the associated classical field theory. This is based on joint work with Humberto Gomez, Renann Lipinski Jusinskas and Cristhiam Lopez Arcos.

John Ratcliffe

Harmonic spinors on the Davis hyperbolic 4-manifold.

Abstract: This talk is on joint work with Daniel Ruberman and Steven Tschantz. In this talk, I will discuss how we used the G-index theorem for the Dirac operator to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic 4-manifold that admits harmonic spinors.

Andrew Swann

Title: A shear construction, solvable Lie algebras and SKT geometry

Abstract: In joint work with Marco Freibert, we introduced a shear construction, that gives a foliated version of T-duality. Applied to approriate data on Lie groups, by iteration starting with an Abelian group, this produces all solvable Lie groups. We will describe the construction and give applications to construction and classification of SKT structures.