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163
ON GLOBAL DEFORMATION QUANTIZATION IN THE ALGEBRAIC CASE
, 2006
"... We give a proof of Yekutieli’s global algebraic deformation quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems. Instead we use an argument inspired by algebraic De Rham cohomology. ..."
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We give a proof of Yekutieli’s global algebraic deformation quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems. Instead we use an argument inspired by algebraic De Rham cohomology.
A Survey of Characteristic Classes of Singular Spaces
"... 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 va ..."
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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 Lclass. 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.
Double affine Hecke algebras and 2dimensional local fields
 JAMS
"... The concept of an ndimensional local field was introduced by A.N. Parshin [Pa 1] with the aim of generalizing the classical adelic formalism to (absolutely) ndimensional schemes. By definition, a 0dimensional local field is just a finite field and an ndimensional local field, n> 0, is a compl ..."
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The concept of an ndimensional local field was introduced by A.N. Parshin [Pa 1] with the aim of generalizing the classical adelic formalism to (absolutely) ndimensional schemes. By definition, a 0dimensional local field is just a finite field and an ndimensional 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 1dimensional 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 biinvariant with respect to an appropriate subgroup (i.e., as the algebra of double cosets). Harmonic analysis on groups over 2dimensional local fields has not been developed, the main difficulty being the infinite dimensionality (absense of local compactness) of such fields. However,
Rational computations of the topological Ktheory of classifying spaces of discrete groups
 J. Reine Angew. Math
"... We compute rationally the topological (complex) Ktheory of the classifying space BG of a discrete group provided that G has a cocompact GCWmodel for its classifying space for proper Gactions. For instance wordhyperbolic groups and cocompact discrete subgroups of connected Lie groups satisfy thi ..."
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We compute rationally the topological (complex) Ktheory of the classifying space BG of a discrete group provided that G has a cocompact GCWmodel for its classifying space for proper Gactions. For instance wordhyperbolic groups and cocompact discrete subgroups of connected Lie groups satisfy this assumption. The answer is given in terms of the group cohomology of G and of the centralizers of finite cyclic subgroups of prime power order. We also analyze the multiplicative structure. Key words: topological Ktheory, classifying spaces of groups.
Topological Ktheory of Algebraic Ktheory Spectra
 J. Algebraic KTheory
, 1999
"... Introduction One of the central problems of algebraic Ktheory is to compute the Kgroups K n X of a scheme X. Since these groups are, by denition, the homotopy groups of a spectrum KX, it makes sense to analyze the homotopytype of the spectrum, rather than just the disembodied homotopy groups. ..."
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Introduction One of the central problems of algebraic Ktheory is to compute the Kgroups K n X of a scheme X. Since these groups are, by denition, the homotopy groups of a spectrum KX, it makes sense to analyze the homotopytype of the spectrum, rather than just the disembodied homotopy groups. In addition to facilitating the computation of the Kgroups themselves, knowledge of the spectrum KX can be applied to the study of other topological invariants. For example, if X = Spec R, then the homology groups of the zeroth space 1 KX are of interest since they are the homology groups of the innite general linear group GLR; but they are not determined by the homotopy groups of KX alone. Topological complex Ktheory is another important invariant. Let K denote the periodic complex Ktheory spectrum, and let ^ K denote its Bouseld `adic completion
Regulators and Characteristic Classes of Flat Bundles
"... In this paper, we prove that on any nonsingular algebraic variety, the characteristic classes of CheegerSimons and Beilinson agree whenever they can be interpreted as elements of the same group (e.g. for at bundles). In the universal case, where the base is BGL(C)^δ, we show that the uni ..."
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In this paper, we prove that on any nonsingular algebraic variety, the characteristic classes of CheegerSimons and Beilinson agree whenever they can be interpreted as elements of the same group (e.g. for at bundles). In the universal case, where the base is BGL(C)^&delta;, we show that the universal CheegerSimons class is half the Borel regulator element. We were unable to prove that the universal Beilinson class and the universal CheegerSimons classes agree in this universal case, but conjecture they do agree.
The construction problem in Kähler geometry
, 2004
"... 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 theoretic ..."
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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 differentialgeometric 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 overarching problem in algebraic geometry is to understand the classification of algebrogeometric objects. The topology of the usual complexvalued points of a variety plays
Periodic Floer ProSpectra from the SeibergWitten equations
"... Abstract. We use finite dimensional approximation to construct from the SeibergWitten equations invariants of threemanifolds with b1> 0 in the form of periodic prospectra. Their homology is the SeibergWitten Floer homology. Then we proceed to construct relative stable homotopy SeibergWitten ..."
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Abstract. We use finite dimensional approximation to construct from the SeibergWitten equations invariants of threemanifolds with b1> 0 in the form of periodic prospectra. Their homology is the SeibergWitten Floer homology. Then we proceed to construct relative stable homotopy SeibergWitten invariants of fourmanifolds with boundary. 1.
Completions of prospaces
 Math. Zeit
"... Abstract. For every ring R, we present a pair of model structures on the category of prospaces. 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 Rmodules (or equivalently ..."
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Abstract. For every ring R, we present a pair of model structures on the category of prospaces. 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 Rmodules (or equivalently by prohomology with coefficients in R). In the second model structure, fibrant replacement is essentially just the BousfieldKan Rtower. When R = Z/p, the first homotopy category is equivalent to a homotopy theory defined by Morel but has some convenient categorical advantages. 1.