Invited Talks

This is a joint work with P.Bressler, R. Nest and B. Tsygan. We will discuss problems in which formal deformations of etale groupoids and gerbes arise and give an explicit description of the differential graded Lie algebra which controls this deformation theory. • Maxim Kontsevich (IHES) Title: On the degeneration of the Hodge to de Rham spectral sequence. Abstract: Several years ago I proposed a conjecture saying that for a smooth proper dg algebra over a field of characteristic zero, the spectral sequence from Hochschild homology to the periodic cyclic homology collapses. In the case of the dg algebra describing perfect complexes over a smooth proper scheme (Bondal van der Bergh theorem) this is just the usual classical degeneration of the Hodge to de Rham spectral sequence. Recently D.Kaledin claimed a proof of my conjecture in the special case of a sheaf of algebras on a site, generalizing Deligne-Illusie method. In my talk I wille describe 3 conjectures related the degeneration conjecture, weaking some assumptions on dg algebras. Several years ago I proposed a conjecture saying that for a smooth proper dg algebra over a field of characteristic zero, the spectral sequence from Hochschild homology to the periodic cyclic homology collapses. In the case of the dg algebra describing perfect complexes over a smooth proper scheme (Bondal van der Bergh theorem) this is just the usual classical degeneration of the Hodge to de Rham spectral sequence. Recently D.Kaledin claimed a proof of my conjecture in the special case of a sheaf of algebras on a site, generalizing Deligne-Illusie method. In my talk I wille describe 3 conjectures related the degeneration conjecture, weaking some assumptions on dg algebras. • Marc Levine (Northeastern) Title: Algebraic cobordism. Abstract: We will give a report on some recent work on algebraic cobordism, joint with R. Pandharipande. Algebraic cobordism was defined by myself and Fabien Morel as a geometric construction of the “classical” part of MGL-theory. We have also shown that algebraic cobordism is the universal oriented cohomology theory on smooth varieties, and have related algebraic cobordism to the Lazard ring via its formal group law. This theory has been useful in proving the degree formulas of Markus Rost, as well as giving a systematic understanding of Riemann-Roch theorems. We will give a report on some recent work on algebraic cobordism, joint with R. Pandharipande. Algebraic cobordism was defined by myself and Fabien Morel as a geometric construction of the “classical” part of MGL-theory. We have also shown that algebraic cobordism is the universal oriented cohomology theory on smooth varieties, and have related algebraic cobordism to the Lazard ring via its formal group law. This theory has been useful in proving the degree formulas of Markus Rost, as well as giving a systematic understanding of Riemann-Roch theorems. Together with Pandharipande, we have given a simple presentation of the algebraic cobordism groups of a variety via “double point cobordisms”. We use this to verify some conjectures of Pandharipande et al. on generating functions arising in Donaldson-Thomas theory. • Wolfgang Lück (Münster) Title: The Farrell-Jones Conjecture for algebraic K-theory holds for word-hyperbolic groups and arbitrary coefficients. Abstract: This is joint work with Arthurs Bartels and Holger Reich from Muenster. The main result is that the Farrell-Jones Conjecture for algebraic K-theory is true for wordhyperbolic groups and arbitrary cefficient rings. This result has many application, e.g., it implies the Bass Conjecture for fields of characteristic zero and Moody’s induction theorem for word hyperbolic groups. We show that the Farrell-Jones Conjecture for algebraic Ktheory is true for all coefficient rings for those groups which were defined by Gromov and for which the Baum-Connes Conjecture with coefficients fails as shown by Higson, Lafforgue and Skandalis. The methods of proof are based on K-theory, geometric group theory and controlled topology. This is joint work with Arthurs Bartels and Holger Reich from Muenster. The main result is that the Farrell-Jones Conjecture for algebraic K-theory is true for wordhyperbolic groups and arbitrary cefficient rings. This result has many application, e.g., it implies the Bass Conjecture for fields of characteristic zero and Moody’s induction theorem for word hyperbolic groups. We show that the Farrell-Jones Conjecture for algebraic Ktheory is true for all coefficient rings for those groups which were defined by Gromov and for which the Baum-Connes Conjecture with coefficients fails as shown by Higson, Lafforgue and Skandalis. The methods of proof are based on K-theory, geometric group theory and controlled topology.