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Higher topos theory
, 2006
"... Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain com ..."
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Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain complex of Gvalued singular cochains on X. An alternative is to regard H n (•, G) as a representable functor on the homotopy category
On the moduli stack of commutative, 1parameter formal Lie groups
, 2007
"... Abstract. We attempt to develop a general algebrogeometric study of the moduli stack of commutative, 1parameter formal Lie groups, in full comportment with the modern foundations of algebraic geometry. We emphasize the proalgebraic structure of this stack: it is the inverse limit, over varying n, ..."
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Abstract. We attempt to develop a general algebrogeometric study of the moduli stack of commutative, 1parameter formal Lie groups, in full comportment with the modern foundations of algebraic geometry. We emphasize the proalgebraic structure of this stack: it is the inverse limit, over varying n, of moduli stacks of nbuds, and these latter stacks are algebraic. Our main theorems pertain to the height stratification relative to fixed prime p on the stacks of formal Lie groups and of nbuds. Notably, we show that the stack of nbuds of height ≥ h is smooth and universally closed over Fp of dimension −h; we characterize the stratum of nbuds of (exact) height h and the stratum of formal Lie groups of (exact) height h as classifying stacks of certain groups, smooth algebraic in the bud case; and we obtain some structure results on these groups. We also obtain a second characterization of the stratum of formal Lie groups of height h as an inverse limit of classifying stacks of certain finite étale algebraic groups.
Representation and character theory
 in 2categories, Advances in Mathematics, Volume 213, Issue
, 2007
"... The goal of this paper is to develop a (2)categorical generalization of the theory of group representations and characters. It is classical that a representation ̺ of a group G is often determined by its character χ(g) = tr(̺(g)), ..."
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The goal of this paper is to develop a (2)categorical generalization of the theory of group representations and characters. It is classical that a representation ̺ of a group G is often determined by its character χ(g) = tr(̺(g)),
CIRCLEEQUIVARIANT CLASSIFYING SPACES AND THE RATIONAL EQUIVARIANT SIGMA GENUS
"... Abstract. We analyze the circleequivariant spectrum MStringC which is the equivariant analogue of the cobordism spectrum MU〈6 〉 of stably almost complex manifolds with c1 = c2 = 0. In [Gre05], the second author showed how to construct the ring Tspectrum EC representing the Tequivariant elliptic c ..."
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Abstract. We analyze the circleequivariant spectrum MStringC which is the equivariant analogue of the cobordism spectrum MU〈6 〉 of stably almost complex manifolds with c1 = c2 = 0. In [Gre05], the second author showed how to construct the ring Tspectrum EC representing the Tequivariant elliptic cohomology associated to a rational elliptic curve C. In the case that C is a complex elliptic curve, we construct a map of ring Tspectra MStringC → EC which is the rational equivariant analogue of the sigma orientation of [AHS01]. Our method gives a proof of a conjecture of the first author in [And03b]. Contents
Algebraic
"... Application for a five year (60 months) ERC starting grant Project Summary: Some mathematical notions can be described in both algebraic and geometric ways. The geometric descriptions are sometimes richer and can reveal hidden symmetries. Ktheory is an example of a theory that has both and algebrai ..."
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Application for a five year (60 months) ERC starting grant Project Summary: Some mathematical notions can be described in both algebraic and geometric ways. The geometric descriptions are sometimes richer and can reveal hidden symmetries. Ktheory is an example of a theory that has both and algebraic and geometric description. It links algebraic topology with geometry on curved spaces, to the mutual benefit of both disciplines. We believe that the notion of topological modular forms (TMF) can play a similar role by connecting algebraic topology with geometry on loop spaces. The latter is a subject with many open questions, and it is strongly influenced by string theory.
K3 spectra
"... We introduce the notion of a K3 spectrum in analogy with that of an elliptic spectrum and show that there are “enough ” K3 spectra in the sense that for all K3 surfaces X in a suitable moduli stack of K3 surfaces there is a K3 spectrum whose underlying ring is isomorphic to the local ring of the mod ..."
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We introduce the notion of a K3 spectrum in analogy with that of an elliptic spectrum and show that there are “enough ” K3 spectra in the sense that for all K3 surfaces X in a suitable moduli stack of K3 surfaces there is a K3 spectrum whose underlying ring is isomorphic to the local ring of the moduli stack in X with respect to the etale topology, and similarly for the ring of formal functions on the formal deformation space. 1