SPP 1786:  Homotopy Theory and Algebraic Geometry

• Subject Area: Mathematics
• Term: from 2015 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 255373834

The Priority Programme 1786 in Homotopy Theory and Algebraic Geometry builds upon recent developments in two central pillars of modern mathematics, algebraic geometry and homotopy theory, bringing together mathematicians working in these areas to encourage interactions among researchers and break down the classical boundaries between homotopy theory and algebraic geometry. Algebraic geometry studies the solution sets of algebraic equations, applying methods from algebra, analysis and topology, and has wide-ranging applications to other branches of mathematics as well as to physics and engineering. Homotopy theory is an important branch of topology, dealing with properties of spaces that are preserved in continuous families and studying them through algebraic invariants. This Priority Programme is based on several areas of interaction between these two fields:Motivic homotopy theory. Derived algebraic geometry. Differential homotopy theory and Arakelov Theory. Equivariant homotopy theory and its links to motivic homotopy theory.

• DFG Programme: Priority Programmes
• International Connection: France, Ireland, Norway, Switzerland, United Kingdom, USA

Projects

• Applying motivic filtrations (Applicants Levine, Marc ;  Röndigs, Oliver ;  Spitzweck, Markus )
• Beyond Spin bordism (Applicant Noel, Justin )
• C_2-equivariant Schubert Calculus for homogeneous spaces (Applicants Hudson, Thomas ;  Tilson, Ph.D., Sean )
• Characterizations and Uniqueness of the stable motivic homotopy theory (Applicant Hornbostel, Jens )
• Chern classes of automorphic bundles (Applicant Esnault, Hélène )
• Chromatic Derived Algebraic Geometry and Equivariant Homotopy Theory (Applicant Meier, Lennart )
• Classification of vector bundles over affine varieties (Applicant Rosenschon, Ph.D., Andreas )
• Computations of Chow-Witt groups for split quadrics and other smooth varieties (Applicants Hornbostel, Jens ;  Zibrowius, Ph.D., Marcus )
• Coordination Funds (Applicant Levine, Marc )
• Enumerative geometry with quadratic forms (Applicant Levine, Marc )
• Expansion principles of topological automorphic forms (Applicant Laures, Gerd )
• Exponential motivic homotopy theory, foliations and applications (Applicant Pepin Lehalleur, Simon )
• Global equivariant homotopy theory II (Applicant Schwede, Stefan )
• Mixed Hodge structures in homotopy theory (Applicants Cirici, Joana ;  Levine, Marc )
• Motives with modulus (Applicants Esnault, Hélène ;  Kerz, Moritz ;  Levine, Marc )
• Motivic and homotopical methods in classical and p-adic Hodge theory (Applicant Huber-Klawitter, Annette )
• Motivic homotopy theory for local quotient stacks (Applicant Heinloth, Jochen )
• Motivic invariants of moduli spaces (Applicants Hoskins, Victoria ;  Schmitt, Alexander )
• Motivic iterated integrals and integral points (Applicant Dan-Cohen, Ishai )
• Motivic Stack Inertia of Moduli Spaces of Curves, Variation of Periods and Multiple Zeta Values in Genus 0 and 1 (Applicant Collas, Ph.D., Benjamin )
• New examples for logarithmic ring spectra (Applicant Richter, Birgit )
• Odd primary equivariant rigidity and equivariant derivators (Applicant Schwede, Stefan )
• Operads in algebraic geometry and their realizations (Applicants Hornbostel, Jens ;  Röndigs, Oliver ;  Spitzweck, Markus ;  Stelzer, Manfred )
• Oriented cohomology theories and equivariant motives (Applicant Geldhauser, Nikita )
• Quadratic forms, quadrics, sums of squares and Kato's cohomology in positive characteristic (Applicant Hoffmann, Detlev )
• Ramified extensions of commutative ring spectra (Applicant Richter, Birgit )
• The summand BtmfP of MU<6> (Applicant Laures, Gerd )
• Topological dihedral homology of group rings (Applicant Reich, Holger )
• Trace maps for real algebraic K-theory (Applicant Dotto, Ph.D., Emanuele )

• Spokesperson: Professor Dr. Marc Levine