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Giovedì 13.00-15.00, Sala INdAM Paolo Piazza, Niels Kowalzig, Indrava Roy.
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| 30/10/2014 | Niels Kowalzig |
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Gel'fand-Fuchs cohomology and the Godbillon-Vey cocycle - the Lie algebra of formal vector fields and its associated cochain complex - the Weil algebra W and its quotient W_n - Theorem of Gel'fand-Fuchs: a quasi-isomorphism between W_n and the cochain spaces of the Lie algebra of formal vector fields. - finite-dimensionality of the Gel'fand-Fuchs cohomology groups - the Godbillon-Vey cocycle - étale groupoid cohomology and Haefliger's differentiable cohomology - Haefliger's theorem: H^*(\Gamma_M, \mathbb{R}) \simeq H^*(\mathfrak{a}_n, O) REFERENCES: [1] C. Godbillon, Cohomologies d'algèbres de Lie de champs de vecteurs formels PDF [2] H. Cartan, Cohomologie réelle d'un espace fibré principal différentiable. I: notions d'algèbre différentielle, algèbre de Weil d'un groupe de Lie PDF [3] I. Gel'fand and D. Fuchs, Cohomology of the Lie algebra of formal vector fields PDF [4] A. Haefliger Differential Cohomology PDF |
| 06/11/2014 | Niels Kowalzig |
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Étale groupoid cohomology and differentiable equivariant cohomology
- nerve and classifying space of an étale groupoid - Dupont's simplicial version of the de Rham complex and integration along the fibres - (differentiable) equivariant cohomology and Haefliger-van Est theorem - Haefliger's conjecture: H^*(G,A) \simeq H^*(BG, \tilde{A}) for an abelian G-sheaf A. REFERENCES: [1] H. Moscovici, Geometric construction of Hopf cyclic characteristic classes PDF [2] I. Moerdijk, Proof of a conjecture of A. Haefliger PDF [3] M. Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes PDF |
| 11/11/2014 | Indrava Roy |
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Action groupoids and equivariant cohomology
- Action groupoid and its thick realization - Relation with equivariant cohomology REFERENCES: [1] J. Dupont, Simplicial de Rham cohomology and characteristic classes of flat bundles PDF [2] Peter May, The geometry of iterated loop spaces PDF |
| 27/11/2014 | Indrava Roy |
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Geometric van Est-Haefliger isomorphism theorem
- jet bundles and the Lie algebra of formal vector fields - explicit form of the quasi-isomorphism between the Gelfand-Fuchs complex and the differential Haefliger-de Rham complex REFERENCES: [1] R. Bott, On characteristic classes in the framework of Gelfand-Fuks cohomology, Astérisque, No. 32-33, 1976, p. 113-139 [2] I. Kolář, P. Michor, I. and Slovák, Natural operators in differential geometry PDF [3] V. Yumaguzhin, Introduction to Differential Invariants PDF |
| 04/12/2014 | Indrava Roy |
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Chern-Weil theory on simplicial de Rham complex of frame bundles
- Review of classical Chern-Weil theory - rôle of Weil algebra and the extended Chern-Simons morphism - the truncated Weil algebra and Vey complex - Vey basis of the Vey complex REFERENCES: [1] S. Morita, Geometry of Differential forms, Translations of Mathematical Monographs 201, AMS, 2001 [2] S. Morita, Geometry of Characteristic classes, Translations of Mathematical Monographs 199, AMS, 2001 |
| 18/12/2014 | Indrava Roy |
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The Connes-Moscovici Hopf algebra I
- prolongation of action of local diffeomorphisms on frame bundles - description of vertical and horizontal vector fields for a choice of connection on the frame bundle - generalised Leibniz rules for the action of vector fields on the crossed product algebra of smooth compactly supported functions by the pseudogroup action of local diffeomorphisms, denoted A = C_c^\infty(FM) \rtimes Gamma_M REFERENCES: [1] R. Wulkenhaar, On the Connes-Moscovici algebra associated to the diffeomorphisms of a manifold PDF [2] A. Connes, H. Moscovici, Hopf algebras, cyclic cohomology and the transverse index theorem PDF [3] A. Connes, H. Moscovici, Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry PDF [4] S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, 1995 |
| 22/01/2015 | Indrava Roy |
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The Connes-Moscovici Hopf algebra II
- Lie algebra structure of multipliers on A generated by vector fields and "holonomy" of horizontal vector fields under push-forward and pull-back by local diffeomorphisms - Universal enveloping algebra of the above infinite-dimensional Lie algebra with a coproduct structure compatible with the generalized Leibniz rules - description of counit and antipode, completing the Hopf algebra description |
| 29/01/2015 | Niels Kowalzig |
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Hopf-cyclic cohomology
- the Connes-Moscovici characteristic map via an invariant trace - twisted antipodes and modular pairs in involution - the quotient of coinvariants and "partial integration" - (co)cyclic modules and cyclic cohomology for Hopf algebras REFERENCES: [1] M. Crainic, Cyclic cohomology of Hopf algebras PDF |
| 06/02/2015 | Niels Kowalzig |
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The bicrossed product construction
- the canonical splitting of the group of diffeomorphisms - matched pairs of Hopf algebras - bicrossed product Hopf algebras - the isomorphism between the Connes-Moscovici algebra H_n and a bicrossed product of the function algebra F and U(h) REFERENCES: [1] H. Moscovici and B. Rangipour, Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology PDF |