Recently on arXiv: 'Systole and small eigenvalues of hyperbolic surfaces' / 'Classical KMS Functionals and Phase Transitions in Poisson Geometry'
Montag, 6.12.21, 16:15-17:15, Hörsaal II, Albertstr. 23b
(i) Let S be a closed orientable hyperbolic surface with Euler characteristic \(\bchi\), and let \(\blambda_k(S)\) be the \(k\)-th positive eigenvalue for the Laplacian on \(S\). According to famous result of Otal and Rosas, \(\blambda_−\bchi >0,25\). In this article, we prove that if the systole of S is greater than \(3,46\), then \(\blambda_{−\bchi−1}>0,25\). This inequality is also true for geometrically finite orientable hyperbolic surfaces without cusps with the same assumption on the systole.\n\n(ii)The authors study the convex cone of not necessarily smooth measures satisfying the classical KMS condition within the context of Poisson geometry. They discuss the general properties of KMS measures and its relation with the underlying Poisson geometry in analogy to Weinstein's seminal work in the smooth case. Moreover, by generalizing results from the symplectic case, they focus on the case of \(\bflat\)-Poisson manifolds, where they provide a complete characterization of the convex cone of KMS measures.
The Index theorem on end-periodic manifolds
Montag, 13.12.21, 16:15-17:15, Hörsaal II, Albertstr. 23b
Atiyah and Singer published 1963 a formula for the index of an elliptic operator over a closed oriented Riemannian manifold just containing topological terms, known as the Atiyah-Singer index theorem. Forty-one years later they were awarded with the Abel Prize, among other things, for this deep result connecting topology, geometry and analysis.\nIn this talk the Atiyah-Singer index theorem will be formulated and a proof via the heat equation and it's asymptotic expansion will be sketched. Further, a modification of this proof leads to an index theorem for end-periodic Dirac operators discovered by Mrowka, Ruberman and Saveliev in 2014. This end-periodic index theorem and how it is related to the classical Atiyah-Patodi-Singer index theorem for manifolds with boundary are also treated in the talk.
Weak Dual Pairs in Dirac-Jacobi Geometry
Montag, 20.12.21, 16:15-17:15, BBB (link will be distributed via the Diffgeo mailing list)
Adopting the omni-Lie algebroid approach to Dirac-Jacobi structures, we propose and investigate a notion of weak dual pairs in Dirac-Jacobi geometry. Their main motivating examples arise from the theory of multiplicative precontact structures on Lie groupoids. \n\nIn my talk I will give a short introduction to Dirac-Jacobi geometry, introduce the notion of weak dual pairs, explain some cases where they exist and apply this to prove a normal form theorem, which locally in special cases gives the \nlocal structure theorems by Dazord, Lichnerowicz and Marle for Jacobi structures on the one hand, and the Weinstein splitting theorem on the other hand, which are generalizations of the Darboux theorem for contact (resp. symplectic) manifolds.