Algebraic Cobordism and \'Etale Cohomology

Thomason's \'{e}tale descent theorem for Bott periodic algebraic $K$-theory \cite{aktec} is generalized to any $MGL$ module over a regular Noetherian scheme of finite dimension. Over arbitrary Noetherian schemes of finite dimension, this generalizes the analog of Thomason's theorem for Weibel's homotopy $K$-theory. This is achieved by amplifying the effects from the case of motivic cohomology, using the slice spectral sequence in the case of the universal example of algebraic cobordism. We also obtain integral versions of these statements: Bousfield localization at \'etale motivic cohomology is the universal way to impose \'etale descent for these theories. As applications, we describe the \'etale local objects in modules over these spectra and show that they satisfy the full six functor formalism, construct an \'etale descent spectral sequence converging to Bott-inverted motivic Landweber exact theories, and prove cellularity and effectivity of the \'{e}tale versions of these motivic spectra.Comment: 68 pages, results generalized and arguments clarified, comments still welcome

Geometry and categorification

We describe a number of geometric contexts where categorification appears naturally: coherent sheaves, constructible sheaves and sheaves of modules over quantizations. In each case, we discuss how "index formulas" allow us to easily perform categorical calculations, and readily relate classical constructions of geometric representation theory to categorical ones.Comment: 23 pages. an expository article to appear in "Perspectives on Categorification.

Relations between slices and quotients of the algebraic cobordism spectrum

We prove a relative statement about the slices of the algebraic cobordism spectrum. If the map from MGL to a certain quotient of MGL introduced by Hopkins and Morel is the map to the zero-slice then a relative version of Voevodsky's conjecture on the slices of MGL holds true. We outline the picture for K-theory and rational slices.Comment: 15 pages; misprints correcte