Charlas 2021
Día y hora: 25 de noviembre de 2021, 14.30 hs (horario Buenos Aires).
Título: Totally geodesic submanifolds in exceptional symmetric spaces.
Resumen: The problem of classifying totally geodesic submanifolds in symmetric spaces has been a relevant topic of research in submanifold geometry in the last decades. This was started in 1963 by Wolf. In this seminal paper, the author classified these objects in symmetric spaces of rank one. For rank two this problem has been addressed by Chen and Nagano, and Klein. Up to now, there are only complete classifications in symmetric spaces of rank less than three.
In this talk I will report on an ongoing joint work with A. Kollross (Universität Stuttgart) where we classify maximal totally geodesic submanifolds in exceptional symmetric spaces.
Día y hora: 11 de noviembre de 2021, 14.30 hs (horario Buenos Aires).
Título: 3-$(\alpha,\delta)$-Sasaki manifolds.
Resumen: This talk is a gentle introduction to the geometry of 3-$(\alpha,\delta)$-Sasaki manifolds, a recent generalisation of 3-Sasaki manifolds. We speak about their geometry, the homogeneous case as well as their remarkable curvature properties.
Joint work with Leander Stecker (now Hamburg) and Giulia Dileo (Bari).
Viviana del Barco (Universidade Estadual de Campinas)
Día y hora: 28 de octubre de 2021, 14.30 hs (horario Buenos Aires).
Título: "Solitary" Lie algebras.
Resumen: An ad-invariant metric on a Lie algebra is a nondegenerate symmetric bilinear form for which inner derivations are skew-symmetric. These are the algebraic counterparts of bi-invariant metrics on Lie groups.
It is known that a positive definite ad-invariant metric can only be defined on compact semisimple Lie algebras, direct sum with an abelian factor. On compact simple Lie algebras, every ad-invariant metric is a multiple of the Killing form which, in addition, is invariant under the Lie algebra automorphisms.
In the pseudo-Riemannian context ad-invariant metrics appear on more general Lie algebras such as semisimple (non-compact), or solvable. For non-semisimple Lie algebras, the orbit space of ad-invariant metrics under the action of the automorphism group has not been systematically described yet.
In this talk, we will discuss characteristics of Lie algebras possessing a unique ad-invariant metric up to automorphisms (and sign). In particular, we will introduce the concept of "solitary" metrics on Lie algebras, which aims to encode the property of being a unique ad-invariant metric. As we will see, this is actually a property of a Lie algebra rather than of the metric itself.
This characterization of uniqueness allowed us to show that Lie algebras admitting a unique ad-invariant metric are necessarily solvable. In addition, we show that many low dimensional Lie algebras carrying ad-invariant metrics are solitary.
Time permitting, generalizations of the solitary conditions will be discussed.
The talk is based on joint works with Diego Conti and Federico A. Rossi (Milano Bicocca).
Día y hora: 12 de octubre de 2021, 14.30 hs (horario Buenos Aires).
Título: Degeneraciones de las álgebras de Lie filiformes complejas de dimensión 9.
Resumen: Los problemas de clasificación para las álgebras de Lie solubles y nilpotentes son muy difíciles de resolver. Dado esto se consideran otros problemas para avanzar en la comprensión de estas álgebras. Un punto de vista muy interesante es el de considerar y estudiar el espacio de todas las álgebras de Lie y las subvariedades de clases especiales, como solubles y nilpotentes. Estos espacios son variedades algebraicas, objetos de estudio de la geometría algebraica. En este contexto el estudio de deformaciones y degeneraciones, dos fenómenos geométricos, que ocurren en estas variedades algebraicas es un tipo de problema natural, la comprensión y solución de estos problemas son de gran interés en el estudio de las variedades algebraicas de las álgebras de Lie.
En esta charla mostraremos que toda álgebra de Lie filiforme, es decir, que toda álgebra de Lie nilpotente de índice de nilpotencia máximo de dimensión 9 es degeneración de otra álgebra de Lie no isomorfa.
La charla está basada en un trabajo conjunto con Felipe Herrera-Granada y Oscar Marquez.
Día y hora: 2 de septiembre de 2021, 14.30 hs (horario Buenos Aires).
Título: Rigidity and stability of Einstein metrics on homogeneous spaces.
Resumen: The question of rigidity of a given Einstein metric, i.e. whether it can be deformed through a curve of Einstein metrics on the same manifold, is closely related to its stability under the Einstein-Hilbert action by the fact that Einstein metrics are critical points of the (normalized) total scalar curvature functional.
The stability problem for irreducible compact symmetric spaces of compact type has been widely investigated by N. Koiso, using the theory of harmonic analysis on homogeneous spaces. However, this problem was not completely settled, leaving open a few cases. I give an overview of the results and the theory involved. In particular, I present my results on the stability of symmetric spaces, completing the investigation of the problem. Furthermore, I present novel results about the rigidity and stability of the non-symmetric homogeneous spaces like, for example, the 6-dimensional homogeneous nearly Kähler manifolds, and give an outlook on how to tackle these questions on general compact homogeneous spaces.
Día y hora: 22 de julio de 2021, 18 hs (horario Buenos Aires).
Título: Non-compact Einstein manifolds with symmetry.
Resumen: In this talk we will discuss recent joint work in collaboration with Christoph Böhm in which we obtain structure results for non-compact Einstein manifolds admitting a cocompact isometric action of a connected Lie group. As an application, we prove the Alekseevskii conjecture (1975): any connected homogeneous Einstein space of negative scalar curvature is diffeomorphic to a Euclidean space.
Día y hora: 8 de julio de 2021, 14.30 hs (horario Buenos Aires).
Título: Spectra of compact quotients of the oscillator group.
Resumen: We consider the four-dimensional oscillator group Osc1, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We classify the lattices of Osc1 up to inner automorphisms of Osc1. For a lattice L in Osc1, we determine the decomposition of the right regular representation of Osc1 on L2(L\Osc1) into irreducible unitary representations. This decomposition allows the explicit computation of the spectrum of the wave operator on the compact locally-symmetric Lorentzian manifold L\Osc1. This is joint work with Mathias Fischer.
Día y hora: 24 de junio de 2021, 14.30 hs (horario Buenos Aires).
Título: Flat solvmanifolds: questions about holonomy and classification in low dimensions.
Resumen: Solvmanifolds are defined as a compact quotient of a simply-connected solvable Lie group by a discrete subgroup. In this talk we will consider flat solvmanifolds, i.e. solvmanifolds endowed with a flat Riemannian metric induced by a flat left invariant metric on the associated Lie group. Milnor gave a characterization of Lie groups which admit a flat left invariant metric and he showed that they are all solvable of a very restricted form, proving that its Lie algebra decomposes orthogonally as an abelian subalgebra and an abelian ideal, where the action of the subalgebra on the ideal is by skew-adjoint endomorphisms.
On the other hand, flat solvmanifolds are contained in the class of compact flat manifolds, which can be studied from the point of view of Bieberbach groups. In particular, the lattice admits a free abelian normal subgroup of finite index, and the holonomy group of the flat manifold is finite. Using these tools we will prove some properties of the holonomy group. Namely, that is abelian and conversely every finite abelian group is the holonomy group of a flat solvmanifold. Moreover we will discuss the minimal dimension of a flat solvmanifold with holonomy group Z_n.
In the last part of the talk we will give the classification of flat solvmanifolds in dimensions 3,4, 5 and 6, which is related to the problem of determining conjugacy classes of subgroups of GL(n,Z).
Ana Carolina Rey (Universidad de Buenos Aires)
Día y hora: 10 de junio de 2021, 14.30 hs (horario Buenos Aires).
Título: Una desigualdad de Sobolev fraccionaria para variedades Riemannianas compactas y sus aplicaciones.
Resumen: En esta charla, consideraremos el problema de Yamabe, que es un problema geométrico que consiste en estudiar la existencia de una métrica en la clase conforme dada, cuya curvatura escalar asociada es constante. Por la invarianza conforme del Laplaciano conforme, el problema se reduce a encontrar soluciones positivas de una ecuación en derivadas parciales cononcida como la ecuación de Yamabe. En nuestro trabajo estudiamos una ecuación no local análoga en una variedad Riemanniana compacta.
En la primera parte de esta charla hablaremos sobre una desigualdad de Sobolev clásica en variedades Riemannianas y cómo ésta ayudó a resolver el problema de Yamabe. En la segunda parte, probaremos una desigualdad óptima de Sobolev fraccionaria, y la aplicaremos a la resolución de la ecuación no local mencionada.
Este es un trabajo en conjunto con Nicolás Saintier (Universidad de Buenos Aires).
Día y hora: 27 de mayo de 2021, 14.30 hs (horario Buenos Aires).
Título: On cohomogeneity-one Hermitian non-Kähler metrics.
Resumen: In this talk, we will consider Hermitian manifolds acted by a (real) compact Lie group by holomorphic isometries with principal orbit of codimension one. In particular, we will focus on a special class of these manifolds constructed by following Bérard-Bergery, which includes, among the others, the holomorphic line bundles over the complex projective spaces, the linear Hopf manifolds and the Hirzebruch surfaces. On such spaces, we characterize the invariant special Hermitian non-Kähler metrics, such as balanced, pluriclosed, locally conformally Kähler, Vaisman. Furthermore, we construct new examples of cohomogeneity one Hermitian metrics solving the second-Chern-Einstein equation and the constant Chern-scalar curvature equation. This is a joint work with Daniele Angella.
Día y hora: 13 de mayo de 2021, 14.30 hs (horario Buenos Aires).
Título: The geodesic flow on Lie groups.
Resumen: We study the geodesic flow on nilmanifolds equipped with a left-invariant metric. We write the underlying definitions and find general formulas for the Poisson involution. We find formulas for invariant functions in the cases of 2- and 3-step nilpotent Lie groups. Complete integrability is proved in low dimensions. With time, we can discuss some generalizations.
Día y hora: 29 de abril de 2021, 14.30 hs (horario Buenos Aires).
Título: Symmetric and skew-symmetric complex structures.
Resumen: We study the geometry of a complex manifold (M,J) endowed with a closed and non-degenerate 2-form with respect to which J is symmetric or skew-symmetric. This leads, respectively, to complex symplectic (a.k.a. holomorphic symplectic) and pseudo-Kähler structures. A complex symplectic structure is related to many remarkable geometric structures, such as hyperkähler and hypercomplex, while a pseudo-Kähler structure is the generalisation of a Kähler structure to the non-definite case.
The goal of this talk is to describe the interaction of these structures on a fixed complex manifold and to construct explicit examples.
Joint work with M. Freibert, A. Gil García, A. Latorre, B. Meinke.
Día y hora: 15 de abril de 2021, 14.30 hs (horario Buenos Aires).
Título: Rigidez de álgebras de Lie de grafos.
Resumen: En 2018 María Alejandra Álvarez presentó una condición necesaria y suficiente para rigidez de álgebras de Lie 2-pasos nilpotentes. Como consecuencia de este resultado y un argumento combinatorio, logramos probar que, además de los grafos completos K_n, hay solamente 5 grafos rígidos en su correspondiente variedad de álgebras de Lie 2-pasos nilpotentes. Este resultado forma parte de un trabajo realizado en conjunto con Paulo Tirao.
Día y hora: 18 de marzo de 2021, 14.30 hs (horario Buenos Aires).
Título: Infinitesimal Maximal Symmetry of Ricci solitons on solvable Lie groups.
Resumen: A left-invariant Riemannian metric on a Lie group G is said to be maximally symmetric if its isometry group contains a copy of the isometry group of every other left-invariant Riemannian metric on G. Left-invariant Einstein metrics on simply-connected solvable Lie groups are always maximally symmetric. We introduce a weaker notion of infinitesimal maximal symmetry and show that left-invariant Ricci soliton metrics on simply-connected solvable Lie groups are always infinitesimally maximally symmetric but not always maximally symmetric.
This is joint work with Michael Jablonski.