Integrability of Lie algebroids and closed geodesics in positive curvature

This combined lecture series brings together two significant advances in differential and Riemannian geometry, focusing on the interplay between algebraic structures and global geometric phenomena.

Integrability of Lie Algebroids — Rui Loja Fernandes

In a comprehensive 1 hour 32 minute lecture, Rui Loja Fernandes reviews the foundational and recent developments in the integrability of Lie algebroids. Lie algebroids, which generalize Lie algebras and tangent bundles, serve as a crucial framework connecting differential geometry with algebraic structures. Fernandes systematically covers:

• The main criteria and obstructions for integrability, including the role of monodromy groups and differentiation of Lie groupoids.
• Geometric and topological conditions under which a Lie algebroid can be integrated to a Lie groupoid.
• Applications highlighting how these integrability conditions impact the study of foliations, Poisson structures, and geometric quantization.

This lecture deepens understanding of how algebraic and analytic methods converge in the study of smooth manifolds, providing tools that bridge local infinitesimal data with global topological properties.

Lengths of Closed Geodesics in Manifolds of Positive Scalar Curvature — Yevgeny Liokumovich

In a complementary 58 minute talk, Yevgeny Liokumovich investigates the lengths of closed geodesics in Riemannian manifolds with positive scalar curvature. His analysis addresses fundamental questions about how curvature constraints influence the global geometry of closed geodesics:

• Techniques to estimate minimal and maximal lengths of closed geodesics on manifolds exhibiting positive scalar curvature.
• Interactions between scalar curvature bounds and the existence, multiplicity, and length spectrum of closed geodesics.
• Implications for rigidity phenomena and geometric inequalities, linking scalar curvature conditions to global topological and dynamical properties.

Liokumovich’s talk contributes to understanding how curvature conditions impose constraints on geodesic behavior, a central theme in global Riemannian geometry.

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Significance and Interconnections

Together, these lectures highlight a profound relationship linking algebraic integrability conditions of Lie algebroids with global geometric features such as closed geodesics under curvature constraints:

• The integrability of Lie algebroids provides a conceptual and technical framework that can influence the study of foliations and geodesic flows on manifolds.
• Scalar curvature conditions, by controlling geodesic lengths, relate to topological invariants and can be studied via techniques that are informed by integrability properties of underlying algebraic structures.
• These results illustrate the deep synergy between algebraic and analytic methods in geometry, opening pathways for future research bridging differential geometry, topology, and global analysis.

This pairing of talks advances the frontier of geometric understanding by connecting algebraic integrability with global metric properties, enriching both the theory and applications within modern geometry.