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Homotopy Theory and Algebraic Geometry (SPP 1786)
Termin:
23.07.2014
Fördergeber:
Deutsche Forschungsgemeinschaft (DFG)
In March 2014 the Senate of Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) established the Priority Programme "Homotopy Theory and Algebraic Geometry" (SPP 1786). Applications are now invited for the first three-year funding period of the programme.
Ideas from algebraic geometry have influenced modern homotopy theory, for example, the use of the moduli stack of elliptic curves in the construction of the topological modular forms spectrum. In the other direction, the introduction of motivic homotopy theory has enabled the application of methods and constructions from homotopy theory to problems in algebraic geometry. The slice spectral sequence was invented in motivic homotopy theory, but its counterpart in equivariant stable homotopy theory was a key ingredient in the solution of the Kervaire invariant one problem. The central purpose of this programme is to advance research at the nexus between homotopy theory and algebraic geometry, with the goal of furthering the cross-fertilisation between these areas.
We expect the individual research projects to contribute to at least one of the following research areas, to which an application should explicitly refer.
Motivic homotopy theory:
- chromatic homotopy theory in the motivic setting
- slice towers and related spectral sequences
- the introduction of aspects of classical homotopy theory in the motivic setting
- construction and study of motivic cohomology operations and their application to problems in algebraic geometry and arithmetic
- extensions to non-A1-invariant theories such as higher Chow groups with modulus
- the development and application of a motivic homotopy theory of rigid analytic spaces and other adic spaces
- computations of motivic homotopy groups of special varieties and applications of these to problems in algebraic geometry and K-theory
- the use of homotopical invariants in arithmetical settings, such as existence of rational points and related questions
Derived algebraic geometry in relation to homotopy theory:
- K-theory of ring spectra, logarithmic structures on ring spectra, logarithmic topological Hochschild homology
- extensions of the construction of the topological modular forms spectrum to other formal groups and to the motivic setting
- characteristic classes for String bundles, especially the use of the topological modular forms spectrum and motivic liftings of connective covers of MU and MO
Differential homotopy theory and motivic aspects of classical homotopy theory:
- homotopical and motivic invariants arising from differential homotopy theory and motivic versions of Deligne cohomology
- the development and application of differential aspects of motivic cohomology theories
- equivariant aspects of differential homotopy theory
- differential cobordism invariants, Deligne and Arakelov cobordism
- cobordism categories and motivic analogs
- motivic aspects of rational homotopy theory
Equivariant stable homotopy theory:
- foundations of equivariant stable homotopy theory, global equivariant stable homotopy theory, equivariant formal group laws
- equivariant motivic stable homotopy theory
- real motivic homotopy theory, Hermitian K-theory and Chow-Witt groups
- motivic aspects of real and tropical enumerative geometry
Contact:
Faculty of Mathematics
University of Duisburg-Essen
45127 Essen
Prof. Dr. Marc Levine
marc.levine@uni-due.de
Further Information:
http://www.dfg.de/foerderung/info_wissenschaft/info_wissenschaft_14_23/index.html
Ideas from algebraic geometry have influenced modern homotopy theory, for example, the use of the moduli stack of elliptic curves in the construction of the topological modular forms spectrum. In the other direction, the introduction of motivic homotopy theory has enabled the application of methods and constructions from homotopy theory to problems in algebraic geometry. The slice spectral sequence was invented in motivic homotopy theory, but its counterpart in equivariant stable homotopy theory was a key ingredient in the solution of the Kervaire invariant one problem. The central purpose of this programme is to advance research at the nexus between homotopy theory and algebraic geometry, with the goal of furthering the cross-fertilisation between these areas.
We expect the individual research projects to contribute to at least one of the following research areas, to which an application should explicitly refer.
Motivic homotopy theory:
- chromatic homotopy theory in the motivic setting
- slice towers and related spectral sequences
- the introduction of aspects of classical homotopy theory in the motivic setting
- construction and study of motivic cohomology operations and their application to problems in algebraic geometry and arithmetic
- extensions to non-A1-invariant theories such as higher Chow groups with modulus
- the development and application of a motivic homotopy theory of rigid analytic spaces and other adic spaces
- computations of motivic homotopy groups of special varieties and applications of these to problems in algebraic geometry and K-theory
- the use of homotopical invariants in arithmetical settings, such as existence of rational points and related questions
Derived algebraic geometry in relation to homotopy theory:
- K-theory of ring spectra, logarithmic structures on ring spectra, logarithmic topological Hochschild homology
- extensions of the construction of the topological modular forms spectrum to other formal groups and to the motivic setting
- characteristic classes for String bundles, especially the use of the topological modular forms spectrum and motivic liftings of connective covers of MU and MO
Differential homotopy theory and motivic aspects of classical homotopy theory:
- homotopical and motivic invariants arising from differential homotopy theory and motivic versions of Deligne cohomology
- the development and application of differential aspects of motivic cohomology theories
- equivariant aspects of differential homotopy theory
- differential cobordism invariants, Deligne and Arakelov cobordism
- cobordism categories and motivic analogs
- motivic aspects of rational homotopy theory
Equivariant stable homotopy theory:
- foundations of equivariant stable homotopy theory, global equivariant stable homotopy theory, equivariant formal group laws
- equivariant motivic stable homotopy theory
- real motivic homotopy theory, Hermitian K-theory and Chow-Witt groups
- motivic aspects of real and tropical enumerative geometry
Contact:
Faculty of Mathematics
University of Duisburg-Essen
45127 Essen
Prof. Dr. Marc Levine
marc.levine@uni-due.de
Further Information:
http://www.dfg.de/foerderung/info_wissenschaft/info_wissenschaft_14_23/index.html