most recent 30 from http://mathoverflow.net 2013-06-20T09:04:54Z http://mathoverflow.net/feeds/question/25282 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25282/are-there-applications-of-algebraic-geometry-into-algebraic-topology Akela 2010-05-19T21:24:56Z 2010-05-19T22:32:37Z

<p>It is described in many sources that algebraic topology had been a major source of innovation for algebraic geometry. It is said that the uses of cohomology, sheaves, spectral sequences etc. in algebraic geometry were motivated by algebraic topology. Moreover it is said that Weil conjectures arose out of inspiration from algebraic topology.</p> <p>So it seems a very clear thing that algebraic topology tremendously influenced algebraic geometry, at least historically.</p> <p>But are there influences in the other way? Did it ever happen that the modern developments in algebraic geometry were ever taken back to algebraic topology and led to developments over there?</p> <p>Edit: Topological K-theory is one application. Are there more?</p>

http://mathoverflow.net/questions/25282/are-there-applications-of-algebraic-geometry-into-algebraic-topology/25283#25283 Steve Huntsman 2010-05-19T21:32:25Z 2010-05-19T21:32:25Z

<p>The Riemann-Roch (<em>et al.</em>) theorems were generalized by the Atiyah-Singer theorem.</p>

http://mathoverflow.net/questions/25282/are-there-applications-of-algebraic-geometry-into-algebraic-topology/25286#25286 Tilman 2010-05-19T21:57:21Z 2010-05-19T21:57:21Z

<p>elliptic cohomology, topological modular forms, stacks, formal groups, genera, to name but a few.</p>

http://mathoverflow.net/questions/25282/are-there-applications-of-algebraic-geometry-into-algebraic-topology/25291#25291 Urs Schreiber 2010-05-19T22:32:37Z 2010-05-19T22:32:37Z

<p>A nice conceptual link between derived algebraic geometry and algebraic topology is established in</p> <p>Bertrand Toen, <em>Champs affines</em> , <a href="http://arxiv.org/abs/math/0012219" rel="nofollow">http://arxiv.org/abs/math/0012219</a></p> <p>An impression of the central conceptual picture developed there is given at</p> <p><a href="http://ncatlab.org/nlab/show/rational+homotopy+theory+in+an+(infinity%2C1)-topos" rel="nofollow">nLab: rational homotopy theory in an (oo,1)-topos</a>.</p> <p>The main point is, in slight paraphrase of what Toen writes, that he shows that the relation between dg-algebras and rational homotopy theory factors through derived algebraic stacks. This allows to closely connect algebraic topology with derived algebraic geometry and improve on ordinary rational homotopy theory by admitting algebraic models for spaces with arbitrary fundamental groups.</p> <p>This is supposed to go back to and provide a solution to Grothendiecks "schematization problem" in terms of "schematic homotopy types".</p>