The notion of quandles is originated in knot theory and can be regarded as a generalization of (discrete) symmetric spaces. In this talk, we will introduce this concept and describe some simple examples. We will also mention that some ideas from symmetric spaces can be transferred to quandles. Although quandles are discrete and do not possess a topology, one can apply the notions of flatness and Euler characteristics to them.
Geometry day
Santiago de Compostela - 18 February 2025
Venue
Facultade de Matemáticas (Aula 8), Universidade de Santiago de CompostelaProgram
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In the study of geometric structures on nilmanifolds, two-step nilpotent Lie algebras obtained by graphs play an important role. Recently, we have constructed nilpotent Lie algebras from quivers. In this talk, we focus on nilpotent Lie algebras obtained by quivers that are two-step nilpotent, and we prove that they can also be obtained by graphs. Using this relationship, we demonstrate that every two-step nilpotent Lie algebra obtained by a quiver admits a pseudo-Riemannian Ricci-flat metric. Additionally, we also classify these nilpotent Lie algebras that admit symplectic structures.
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We are interested in the classification or finding conditions for the existence of left-invariant symplectic structures on Lie groups. In this paper, we study this problem for the case of almost abelian Lie algebras, which are Lie algebras that contain a codimension 1 abelian subalgebra. We show that, for this case, the problem is reduced to studying a certain matrix congruence relation. We present a complete classification result for all almost abelian Lie algebras and state necessary and sufficient conditions for the existence of such structures.
- 18:15 - 18:45 Break
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A Riemannian manifold can be endowed with a density function, modifying the Riemannian volume element and giving rise to a smooth metric measure space. The study of such spaces relies on weighted geometric objects that incorporate information on the density. In the context of smooth metric measure spaces, weighted Einstein manifolds are the natural Einstein-type structure, appearing in the study of the weighted Yamabe problem and discussions around Gagliardo-Nirenberg-Sobolev inequalities. From the point of view of conformal geometry, weighted conformal classes can be defined, and the question of finding which of these classes admit more than one weighted Einstein structure arises naturally. In this talk, we will give local and global classification results for weighted Einstein manifolds in these classes, and touch on their relationship with weighted Einstein manifolds satisfying a certain harmonicity condition on the weighted Weyl tensor.
Symmetric spaces are one of the best-understood classes of Riemannian manifolds and can also be seen as an immediate generalization of space forms. The index of symmetry is a geometric invariant that, in some sense, measures how far a Riemannian manifold is from being a symmetric space. The index (and co-index) of symmetry has been mainly studied in the context of compact homogeneous Riemannian manifolds, yielding general structural results in these cases.
Within compact homogeneous spaces, a natural class for which seems natural to compute the index of symmetry is that of the homogeneous spaces that are diffeomorphic to compact-type symmetric spaces. Specifically, in this talk, we focus on studying the index of symmetry for homogeneous spaces diffeomorphic to CROSSes (compact rank-one symmetric spaces).
Organizers
- Miguel Domínguez Vázquez, CITMAga - Universidade de Santiago de Compostela.
- Víctor Sanmartín López, CITMAga - Universidade de Santiago de Compostela.
Sponsors
- Grant PID2022-138988NB-I00 funded by MICIU/AEI/ 10.13039/501100011033 and by ERDF, EU
- Centro de Investigación e Tecnoloxía Matemática de Galicia (CITMAga), Spain
- Red Española de Análisis Geométrico
