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Hyperbolic 3-manifolds with uniform spectral gap for coclosed 1-forms

Expositor: Ben Lowe - University of Chicago
Ter 05 mar 2024, 15:30 - SALA 236Seminário de Geometria Diferencial

Resumo: We study for hyperbolic 3-manifolds two quantifications of the property of having vanishing first Betti number, one geometric and one topological: the spectral gap for the Laplacian on coclosed 1-forms and the size of the first torsion homology group. First we construct a sequence of closed hyperbolic integer homology spheres with volume tending to infinity and a uniform coclosed 1-form spectral gap, which answers a question asked by Lin-Lipnowski. We also find sequences of hyperbolic rational homology spheres with the same property that geometrically converge to a tame limit manifold, but on the other hand we show that any such sequence must have unbounded torsion homology growth. Finally we show that a sequence of closed hyperbolic 3-manifolds with uniformly bounded rank and a uniform coclosed 1-form spectral gap must have exponential in volume torsion homology growth. Based on joint work with Amina Abdurrahman, Anshul Adve, Vikram Giri, and Jonathan Zung.


Magnetic systems on closed surfaces: Zoll systems, periodic orbits, and systolic inequalities

Expositor: Luca Asselle - Bochum University
Qua 06 mar 2024, 15:30 - SALA 236Seminário de Geometria Simplética

Resumo: The goal of these two talks is to cover several recent results about magnetic systems on closed surfaces. After a short introduction, we will show that, for all but very few exceptional „fully resonant“ systems, every sufficiently low energy level carries infinitely many closed magnetic geodesics, and discuss how such a result fits into previous literature. In the second talk we will instead focus on Zoll systems, that is, systems for which the magnetic flow restricted to a (given) energy level induces a free $S^1$-action: after recalling their central role in systolic geometry, we will show how to construct examples of such systems by means of the Nash-Moser implicit function theorem.

(This is Part I of a joint seminar with the Seminário de Geometria Diferencial).


Magnetic systems on closed surfaces: Zoll systems, periodic orbits, and systolic inequalities

Expositor: Luca Asselle - University of Bochum
Qui 07 mar 2024, 15:30 - SALA 236Seminário de Geometria Diferencial

Resumo: Abstract: The goal of these two talks is to cover several recent results about magnetic systems on closed surfaces. After a short introduction, we will show that, for all but very few exceptional „fully resonant“ systems, every sufficiently low energy level carries infinitely many closed magnetic geodesics, and discuss how such a result fits into previous literature. In the second talk we will instead focus on Zoll systems, that is, systems for which the magnetic flow restricted to a (given) energy level induces a free $S^1$-action: after recalling their central role in systolic geometry, we will show how to construct examples of such systems by means of the Nash-Moser implicit function theorem.

(This is Part II of a joint seminar with the Seminário de Geometria Simplética)