Operads in algebraic geometry and their realizations

• Applicants: Professor Dr. Jens Hornbostel; Professor Dr. Oliver Röndigs; Professor Dr. Markus Spitzweck; Privatdozent Dr. Manfred Stelzer
• Subject Area: Mathematics
• Term: from 2015 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 269680815

A major objective in algebraic topology is to investigate and classify topological spaces via algebraic invariants. Topological spaces which are equivalent up to deformation (homotopy) usually have the same algebraic invariants. The fundamental group, consisting of homotopy classes of based loops in a pointed topological space, is a basic such invariant. By definition, it coincides with the path components of the associated loop space, the topological space of based maps from the circle to the given space. Special properties of the circle provide these loops with a multiplication which is associative up to homotopy. In the loop space of a loop space, the multiplication is even commutative up to homotopy. Further iterations improve this multiplication, and infinite loop spaces are equivalent to connective spectra, highly structured and somewhat manageable invariants.So-called recognition principles characterize the additional structure a topological space has to possess in order to be an iterated loop space. These principles can be phrased via actions of special collections of topological spaces dubbed operads. Despite their topological origin, operads abound in algebra, geometry, and mathematical physics. Spectacular work that Morel and Voevodsky accomplished in the 90s transferred homotopical methods to the realm of algebraic geometry, whose objects of interest are rather rigid, being defined via polynomials. Connections with Grothendieck‘s vision of universal invariants (Motifs) for these algebraic varieties coined the term „motivic homotopy theory“.The major objective of this project is the investigation, construction and modification of explicit operads, closely connected to moduli spaces of algebraic curves of genus zero, in algebraic geometry. Topological realizations of various flavors will be our preferred investigation device. Only those algebraic operads whose topological realizations act on usual loop spaces will stand a chance of acting on loop spaces in motivic homotopy. These loop spaces inherit amazing complexity from an additional "circle". This circle produces certain transfer maps. Incorporating suitable transfer maps in the aforementioned algebraic operads will be one modification, as well as completion of partial operads.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1786: Homotopy Theory and Algebraic Geometry