Project description

To classify and study geometric objects that appear in continuous families in which some members admit a large set of symmetries, algebraic stacks have proven to be a powerful geometric tool. These objects are spaces in which the building blocks admit Symmetries. In this research project we construct and study algebraic invariants of these objects that can be obtained using methods from homotopy theory.The basis for our construction is the motivic homotopy theory introduced by Morel and Voevodsky for the study of algebraic and arithmetic varieties. We want to extend their thoery in a way that allows to obtain information on the geometry of a large class of algebraic stacks and remove previous technical restrictions.In many important examples the known geometric invariants of these objects turned out to have unexpectedly favorable properties and the origin of this behavior has not yet found a satisfactory explanation. We want to provide a class of fine invariants for these space and use them to obtain a better understanding of the observed unexpected behavior.

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