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Differential cohomology in a cohesive ∞topos
"... We formulate differential cohomology and ChernWeil theory the theory of connections on fiber bundles and of gauge fields abstractly in the context of a certain class of higher toposes that we call cohesive. Cocycles in this differential cohomology classify higher principal bundles equipped with c ..."
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We formulate differential cohomology and ChernWeil theory the theory of connections on fiber bundles and of gauge fields abstractly in the context of a certain class of higher toposes that we call cohesive. Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive structure (topological, smooth, synthetic differential, supergeometric, etc.) and equipped with connections. We discuss various models of the axioms and wealth of applications revolving around fundamental notions and constructions in prequantum field theory and string theory. In particular we show that the cohesive and differential refinement of universal characteristic cocycles constitutes a higher ChernWeil homomorphism refined from secondary caracteristic classes to morphisms of higher moduli stacks of higher gauge fields, and at the same time constitutes extended geometric prequantization – in the sense of extended/multitiered quantum field theory – of higher dimensional ChernSimonstype field theories and WessZuminoWittentype field theories. This document, and accompanying material, is kept online at ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos 1 We formulate differential cohomology and ChernWeil theory the theory of connections on fiber bundles
ON THEORIES OF SUPERALGEBRAS OF DIFFERENTIABLE FUNCTIONS
"... This is the first in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we study theories of supercommutative algebras for which infinitely differentiable functions can be evaluat ..."
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This is the first in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we study theories of supercommutative algebras for which infinitely differentiable functions can be evaluated on elements. Such a theory is called a super Fermat theory. Any category of superspaces and smooth functions has an associated such theory. This includes both real and complex supermanifolds, as well as algebraic superschemes. In particular, there is a super Fermat theory of C ∞superalgebras. C ∞superalgebras are the appropriate notion of supercommutative algebras in the world of C ∞rings, the latter being of central importance both to synthetic differential geometry and to all existing models of derived smooth manifolds. A super Fermat theory is a natural generalization of the concept of a Fermat theory introduced by E. Dubuc and A. Kock. We show that any Fermat theory admits a canonical superization, however not every super Fermat theory arises in this way. For a fixed super Fermat theory, we go on to study a special subcategory of algebras called nearpoint determined algebras,
RESEARCH STATEMENT
"... In mathematics, the most elegant results emerge only once one has found the right setting for them to do so. For example, the famous Bezout’s theorem in algebraic geometry, which says that any two plane curves of degree d and e intersect in exactly de points, only holds if one uses the complex numbe ..."
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In mathematics, the most elegant results emerge only once one has found the right setting for them to do so. For example, the famous Bezout’s theorem in algebraic geometry, which says that any two plane curves of degree d and e intersect in exactly de points, only holds if one uses the complex numbers instead of the reals, uses the projective plane instead of the affine plane, counts with multiplicity rather than naively, and works in the derived setting rather than the discrete one (in order to deal with cases of selfintersection). The appropriate structure is necessary to produce the most perfect result. My research in both pure and applied mathematics has tended to revolve around this principle. On one end of the spectrum, my Ph.D. thesis is a generalization of the above Bezout’s theorem, except that it takes place in the category of manifolds rather than schemes, and hence it takes place in a similarlystructured category. On the other end of the spectrum, my work in applied mathematics has been focused on finding categories which best express the dynamics of a given subject in computer science, such as realizing databases as sheaves on simplicial sets. In the following sections, I will discuss my current work and future goals in both the pure and the applied side of my research.
HIGHER ORBIFOLDS AND DELIGNEMUMFORD STACKS AS STRUCTURED INFINITY TOPOI
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