Claude LeBrun – Einstein Constants and Differential Topology
A Riemannian metric is said to be Einstein if it has constant Ricci curvature. In dimensions 2 or 3, this is actually equivalent to requiring the metric to have constant sectional curvature. However, in dimensions 4 and higher, the Einstein condition becomes significantly weaker than constant sectional curvature, and this has rather dramatic consequences. In particular, it turns out that there are high-dimensional smooth closed manifolds that admit pairs of Einstein metrics with Ricci curvatures of opposite signs. After explaining how one constructs such examples, I will then discuss some recent results that explore the coexistence of Einstein metrics with zero and positive Ricci curvatures.
Graziano Gentili – Regularity, slice conformality and Riemann-type manifolds on quaternions
The basic features of (slice) regularity of functions over quaternions will be indicated. A few applications of this theory will be mentioned, including that to the study of OCS on domains of R^4. Quaternionic (and octonionic) analogs of the classical Riemann surfaces will be described. The construction of these manifolds has some peculiarities, and the scrutiny of the classical approach to Riemann surfaces, mainly based on conformality, leads to the definition of slice conformal (or slice isothermal) parameterization on quaternions (and octonions). These classes of manifolds include regular quaternionic (and octonionic) curves, graphs of regular functions, the 4 (and 8) dimensional spheres, the helicoidal and catenoidal 4 (and 8) dimensional manifolds. Appropriate Riemann-type manifolds help to give a natural definition of quaternionic (and octonionic) logarithm and n-th root function. A few statements concerning affine quaternionic curves and surfaces will end the presentation.
Caterina Stoppato – Hypercomplex regularity and holomorphy
Several quaternionic function theories have been introduced and studied over the last century: e.g., the theories of Fueter (1934) and of Gentili-Struppa (2006). They have analogs over the real algebra of octonions, due, respectively, to Dentoni (1973) and Gentili-Struppa (2010). They also have analogs over real Clifford algebras: respectively, the celebrated monogenic functions and the slice-monogenic functions introduced by Colombo-Sabadini-Struppa (2009).
The theories of Gentili-Struppa are natural tools for construction and classification of Orthogonal Almost-Complex Structures in the quaternionic and octonionic settings, as proven in collaboration with Gentili and Salamon (2014) and with Ghiloni and Perotti (2022), respectively.
Fueter's theory is also related, in a non-trivial fashion, to holomorphy with respect to appropriate Orthogonal Almost-Complex Structures. This was proven in collaboration with Perotti (2025), in a work inspired by Tarallo.
Recently, a unified theory of regularity has been constructed in collaboration with Ghiloni (2024, 2025). This theory subsumes all of the aforementioned function theories and includes surprising new ones. Holomorphy in this wider setting is an open and fascinating problem.
Diego Conti – Killing spinors and hypersurfaces
Hypersurfaces inside a manifold with a parallel spinor inherit a structure which can be described by a spinor and a symmetric tensor, corresponding to the second fundamental form. This is known as a generalized Killing spinor; in five dimensions (and positive signature), the resulting geometry consists of hypo structures, which were the subject of my thesis written under Simon Salamon; the six-dimensional case corresponds to the half-flat structures introduced by Salamon and Chiossi.
The notion can be generalized if one considers hypersurfaces inside a manifold with a Killing spinor; the resulting structure on the hypersurface can be described in terms of two spinors, a symmetric tensor, and the Killing constant lambda. This construction includes the "torsion analogues" of hypo and half-flat, known as nearly hypo and nearly half-flat, but also applies to indefinite metrics.
All these geometries are fairly flexible, but classifications can be obtained for left-invariant structures on Lie groups in a fixed, small dimension. Traditionally, the approach to classification is via G-structures and differential forms. I will explain how the language of spinors can be used instead, without reference to the structure group G, to obtain a classification of the principal orbits of cohomogeneity one 5-manifolds with an invariant Killing spinor.
This is joint work with Federico A. Rossi and Romeo Segnan Dalmasso.
Robert Bryant – On curvature-homogeneous Riemannian metrics
A Riemannian manifold (M,g) is said to be curvature-homogeneous if, for any two points x and y in M, there is an isometry between T_xM and T_yM that identifies the curvature tensors R_x and R_y. Of course, any locally homogeneous Riemannian manifold is curvature-homogeneous, and the converse is true in dimension 2 (where the condition is equivalent to the constancy of the Gauss curvature), but this is not so in higher dimensions. I will survey the history of this problem, starting with the work of I. Singer and culminating with very recent work of myself and my collaborator Renato Bettiol in which we prove a number of new results, giving sufficient conditions for curvature-homogeneity to imply local homogeneity (for example, we prove that a curvature homogeneous 4-manifold that is either Einstein or conformally flat must be locally homogeneous) and also constructing new examples of curvature-homogeneous Riemannian 4-manifolds that are not locally homogeneous and not products of lower dimensional curvature-homogeneous manifolds.
Vestislav Apostolov – Bi-Hermitian metrics on the complex projective plane: 34 years later
In the paper `Special structures of 4-manifolds’ published in Riv. Mat. Univ. Parma in 1991, Simon Salamon raised the question about the existence of Riemannian metrics in 4 dimensions which are Hermitian with respect to two distinct (up to sign) integrable almost complex structures inducing the same orientation. These metrics are now known as bi-Hermitian structures, and have been in the center of many exciting developments over the years that continue today. In this talk, I will start from the source and give a short survey of some key results in the theory, focussing on what we know about bi-Hermitian structures on the complex projective space CP^2. I will finally present an existence result for bi-Hermitian structures of constant generalized-Kähler scalar curvature on CP^2, as a part of a joint work in progress with Brent Pym and Jeff Streets.
Fiammetta Battaglia – Explicit criteria for the study of LVMB manifolds
We present work in progress with Federico Thiella and Dan Zaffran. We consider LVMB manifolds, a large class of compact, complex, non symplectic manifolds, that have a very rich geometry. A very nice feature of LVMB manifolds is that they are parametrized by certain configurations of points in a complex affine space. We establish some explicit criteria to compare distinct point configurations and draw exact information on the relationship between the corresponding LVMB manifolds. We illustrate constructions and criteria by considering, throughout the exposition, a particular subfamily of the generalized Calabi-Eckmann manifolds. This subfamily is linked with the one-parameter family of complex toric quasifolds, that include all of the Hirzebruch surfaces, constructed jointly with Elisa Prato and Dan Zaffran.
Bobby Acharya – TBA