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Martedi 14.00-16.00, Sala INdAM Paolo Piazza, Niels Kowalzig, Indrava Roy, Sara Azzali.
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| 22/10/2013 | Niels Kowalzig |
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Introduction to cyclic cohomology (1) - the Hochschild complex - simple definition: cyclic homology of an algebra A via a quotient of the Hochschild complex - simplicial and cyclic objects - Tsygan's bicomplex: cyclic homology as the homology of its total complex - the SBI sequence as resulting from a short exact sequence of complexes and Connes' cyclic operator B |
| 31/10/2013 | (Giovedi, ore 12) Niels Kowalzig |
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Introduction to cyclic cohomology (2) - the double complex lemma - contracting Tsygan's bicomplex to obtain Connes' mixed complex - the Connes-Rinehart operator - the homology of (trivial) mixed complexes associated to cyclic objects - the algebraic HKR-theorem REFERENCES: N. Kowalzig, Hopf Algebroids and Their Cyclic Theory, Ph. D. Thesis, 2009 PDF J.-L. Loday, Cyclic homology, Grundlehren der Mathematischen Wissenschaften, vol 301, Springer-Verlag C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38. |
| 05/11/2013 | Sara Azzali |
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Cyclic cohomology and Banach algebras - cyclic cohomology of locally convex algebras - unbounded derivations and 1-traces on a Banach algera - n-traces |
| 12/11/2013 | Sara Azzali |
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Cyclic cohomology and Banach algebras (2) - proof of the theorem: an n-trace induces a map in K-theory - vanishing of Hochschild cohomology for nuclear C* algebras - example of a 2-cocycle that is not a 2-trace REFERENCES: B. Blackadar, Operator Algebras, Theory of C* Algebras and von Neumann Algebras Encyclopaedia of Math. Sciences Vol.122, Springer 2006 A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, Geometric methods in operator algebras (Kyoto, 1983) 52-144, PDF A. Connes, Noncommutative Geometry, Academic Press 1994 |
| 19/11/2013 | Indrava Roy |
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The transverse fundamental class in the almost isometric case (1) - Crossed product groupoid $X \rtimes \Gamma$ and definition of its reduced C*-algebra - Definition of a Hilbert C*-bimodule on $A:=C_0(X) \rtimes \Gamma$ for non-isometric group actions on equivariant Euclidean vector bundles on X - Construction of a dense holomorphically stable Banach subalgebra B of A |
| 04/12/2013 | (Mercoledi, ore 14.30) Indrava Roy |
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The transverse fundamental class in the almost isometric case (2) - Lemmata to prove that the left action of A is well-defined on the Hilbert bimodule and that it is a closable homomorphism $\lambda$ of C^*-algebras - Definition of almost-isometric action - Example of a geometric situation with an almost-isometric action - Proof that the Banach algebra B (the domain of $\lambda$) is holomorphically closed in A (after unitization). REFERENCES: E. C. Lance. Hilbert C*-modules - a toolkit for operator algebraists. London Math. Soc. Lecture Note Series 210. Cambridge Univ. Press, Cambridge, 1995. A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, Geometric methods in operator algebras (Kyoto, 1983) 52-144, PDF Blackadar, Bruce, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. Encyclopaedia of Mathematical Sciences. Springer-Verlag. (2005) Alain Connes, Noncommutative Geometry, Academic Press, 1994 N.E. Wegge-Olsen, K-theory and C*-algebras, Oxford, 1993. |
| 10/12/2013 | (ore 11.15) Indrava Roy |
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The transverse fundamental class in the almost isometric case (3) - Estimating multilinear functionals on the pre-Hilbert bimodule constructed using the action of \Gamma - Definition of G-structures on a manifold, definition of action of \Gamma preserving the G-structure - Almost-isometric action of $\Gamma$ preserving a G-structure, for a specific subgroup of $SL(n,R)$ |
| 21/01/2014 | Indrava Roy |
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The transverse fundamental class in the almost isometric case (4) - | Sara Azzali |
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Reduction to the almost isometric case (1) - construction of the fibre bundle p: W\to V with W_x= symmetric positive definite quadratic forms on T_xV |
| 28/12/2013 | (ore 10.30) Sara Azzali |
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Reduction to the almost isometric case (2) - the total space W has a G_q- structure (here q= n(n+1)/2) - the action of \Gamma on W preserves the G_q-structure - construction of an element in the equivariant KK-theory: \beta_0\in KK_\Gamma^{i}(C_0(V), C_0(W)), i=n(n+1)/2 mod 2 - using \beta_0, construct a 'Thom map' \beta: K_{j}(C_0(V)\rtimes \Gamma)\to K_{j+q}(C_0(W)\rtimes \Gamma) - the algebraic HKR-theorem REFERENCES: A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, Geometric methods in operator algebras (Kyoto, 1983) 52-144, PDF G. Kasparov, K-theory, group C* algebras and higher signatures (Conspectus), Novikov conjectures, index theorems and rigidity Vol. 1, Oberwolfach 1993 - London Mathematical Society Lecture Notes 226 T. Bröcker, T. tom Dieck, Representations of Compact Lie Groups, Springer. |
| 13/03/2014 | Paolo Piazza |
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Detailed summary. Further remarks on the reduction to the almost isometric case - In preparation of the main theorem next time, I have stated Corollary 4.7 and I have given a sketch of the proof. - I have stated Theorem 5.6 and Theorem 5.8 and I have given a sketch of their proof. REFERENCES: A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, Geometric methods in operator algebras (Kyoto, 1983) 52-144, PDF |
| 18/03/2014 | Paolo Piazza |
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Main theorem - K-homology and the Baum-Coones map - Old and new definitions - Main theorem (THeorem 6.9) and main steps of its proof REFERENCES: A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, Geometric methods in operator algebras (Kyoto, 1983) 52-144, PDF P. Baum and A. Coones, Geometric K-Theory for Lie Groups and Foliations, L'Enseignement Mathematique, Tome 46, 2000 P. Baum, A. Connes and N. Higson, Classifying space for proper actions and K-Theory of group C^*-algebras. Contemporary Math. Vol 167. J. Raven, An equivariant bivariant chern character. Ph. Thesis 2004, Pennsylvania State University. Available at The Pennsylvania Digital Library. |
| 04/04/2014 | Niels Kowalzig |
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Gel'fand-Fuchs Cohomology I - 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 formalk vector fields. - finite-dimensionality of the Gel'fand-Fuchs cohomology groups - existence of a special cocycle in each cohomology group REFERENCES: 1] C. Godbillon, Cohomologies d'algebres de Lie de champs de vecteurs formels PDF 2] H. Cartan, Cohomologie reelle d'un espace fibre principal differentiable. I: notions d'algebre differentielle, algebre 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 |
| 18/04/2014 | Niels Kowalzig |
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Gel'fand-Fuchs Cohomology II - the Godbillon-Vey cocycle - the relative cohomolgoy groups H^*(WSO_n) and H^*(WO_n) - etale groupoid cohomology and Haefliger's differentiable cohomology - Haefliger's theorem: H^*(\Gamma_M, \mathbb{R}) \simeq H^*(\mathfrak{a}_n, O) ADDITIONAL REFERENCES: [1] Andre HAEFLIGER Differential Cohomology PDF |
| 12/05/2014 | Indrava Roy |
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Bott-Thurston cocycle and the transverse fundamental class - Definition of Bott-Thurston 2-cocycle arising from a Radon-Nikodym derivative associated with pullback of orientation-preserving diffeomorphisms - Connes' 2-trace on $A:=C(S^1)\rtimes \Gamma$ associated with the Bott-Thurston cocycle - Modular automorphism groups and left Hilbert-algebras - Modular derivative of the transverse fundamental class and its invariance under the modular automorphism group - General construction of contraction of a cyclic cocycle invariant under an automorphism group with its generator - Application of above construction relating the Bott-Thurston-Connes 2-trace on $A$ and the modular derivative of the transverse fundamental class |
| 29/05/2014 | Indrava Roy |
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The Godbillon-Vey index theorem - Reinterpretation of the Godbillon Vey class as a differential form on the jet bundles of positive frames J^+_k. - Construction of 1-trace on the reduced C*-algebra over J^+_j, j=1,2, for with an action of a discrete subgroup of Diff^+ (S^1). - Connes analogue of Thom isomorphism and dual weights on crossed-product C*-algebras - Pairing with the 1-trace coming from the Bott-Thurston cocycle and its relationship with the pairing with the Godbillon-Vey class - Deducing the following result of Hurder for foliations: if the Godbillon-Vey class of a foliation is non-zero then its von Neumann algebra must have a type III component. |