The theory of characteristic classes of vector bundles and smooth manifolds plays an important role in the theory of smooth manifolds. An investigation of reasonable notions of characteristic classes of singular spaces started with a systematic study of singular spaces such as singular algebraic varieties. We give a quick survey of characteristic classes of singular varieties, mainly focusing on the functorial aspects of some important ones such as the singular versions of the Chern class, the Todd class and Thom–Hirzebruch’s L-class. Further we explain our recent “motivic” characteristic classes, which in a sense unify these three different theories of characteristic classes. We also discuss bivariant versions of them and characteristic classes of proalgebraic varieties, which are related to the motivic measures/integrations. Finally we explain some recent work on “stringy” versions of these theories, together with some references for “equivariant” counterparts.

Fix a prime number ℓ. In this paper we prove ℓ-adic versions of two related conjectures of Deligne, [4, 8.2, p. 163] and [4, 8.9.5, p. 168], concerning mixed Tate motives over the punctured spectrum of the ring of integers of a number field. We also prove a conjecture [11, p. 300], which Ihara attributes to Deligne, about the

The concept of an n-dimensional local field was introduced by A.N. Parshin [Pa 1] with the aim of generalizing the classical adelic formalism to (absolutely) n-dimensional schemes. By definition, a 0-dimensional local field is just a finite field and an n-dimensional local field, n> 0, is a complete discrete valued field whose residue field is (n − 1)-dimensional local. Thus for n = 1 we get the locally compact fields such as Qp, Fq((t)) and for n = 2 we get fields such as Qp((t)), Fq((t1))((t2)) etc. In representation theory, harmonic analysis on reductive groups over 0- and 1-dimensional local fields leads, in particular, to consideration of the finite and affine Hecke algebras Hq, H • q associated to any finite root system R and any q ∈ C ∗. These algebras can be defined in several ways, one being by generators and relations, another as the convolution algebra, with respect to the Haar measure, of functions on the group bi-invariant with respect to an appropriate subgroup (i.e., as the algebra of double cosets). Harmonic analysis on groups over 2-dimensional local fields has not been developed, the main difficulty being the infinite dimensionality (absense of local compactness) of such fields. However,

Abstract. We use finite dimensional approximation to construct from the Seiberg-Witten equations invariants of three-manifolds with b1> 0 in the form of periodic pro-spectra. Their homology is the Seiberg-Witten Floer homology. Then we proceed to construct relative stable homotopy Seiberg-Witten invariants of four-manifolds with boundary. 1.

One of the most surprising things in algebraic geometry is the fact that algebraic varieties over the complex numbers benefit from a collection of metric properties which strongly influence their topological and geometric shapes. The existence of a Kähler metric leads to all sorts of Hodge theoretical restrictions on the homotopy types of algebraic varieties. On the other hand, a sparse collection of examples shows that the remaining liberty is nontrivially large. Paradoxically, with all of this information, the research field remains as wide open as it was many decades ago, because the gap between the known restrictions, and the known examples of what can occur, only seems to grow wider and wider the more closely we look at it. In spite of the differential-geometric nature of the questions and methods, the origins of the situation are very algebraic. We look at subvarieties of projective space over the complex numbers. The main over-arching problem in algebraic geometry is to understand the classification of algebro-geometric objects. The topology of the usual complex-valued points of a variety plays

Abstract. For every ring R, we present a pair of model structures on the category of pro-spaces. In the first, the weak equivalences are detected by cohomology with coefficients in R. In the second, the weak equivalences are detected by cohomology with coefficients in all R-modules (or equivalently by pro-homology with coefficients in R). In the second model structure, fibrant replacement is essentially just the Bousfield-Kan R-tower. When R = Z/p, the first homotopy category is equivalent to a homotopy theory defined by Morel but has some convenient categorical advantages. 1.

Abstract. We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen’s small object argument). The necessity of such a generalization arose with appearance of several important examples of model categories which were proven to be non-cofibrantly generated [2, 6, 8, 20]. Our current approach allows for construction of functorial factorizations and localizations in the equivariant model structures on diagrams of spaces [10] and diagrams of chain complexes. We also formulate a non-functorial version of the argument, which applies in two different model structures on the category of pro-spaces [11, 20]. The examples above suggest a natural extension of the framework of cofibrantly generated model categories. We introduce the concept of a classcofibrantly

Abstract. We compare Friedlander’s definition of étale homotopy for simplicial schemes to another definition involving homotopy colimits of pro-simplicial sets. This can be expressed as a notion of hypercover descent for étale homotopy. We use this result to construct a homotopy invariant functor from the category of simplicial presheaves on the étale site of schemes over S to the category of pro-spaces. After completing away from the characteristics of the

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Michael Gromov has recently initiated what he calls “symbolic algebraic geometry”, in which objects are proalgebraic varieties: a proalgebraic variety is by definition the projective limit of a projective system of algebraic varieties. In this paper we introduce characteristic classes of proalgebraic varieties, using Grothendieck transformations of Fulton–MacPherson’s Bivariant Theory, modeled on the construction of MacPherson’s Chern class transformation of proalgebraic varieties. We show that a proalgebraic version of the Euler–Poincaré characteristic with values in the Grothendieck ring is a generalization of the so-called motivic measure.