Hodge Theory and Classifying Spaces

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Mathematics has become central to the modern economy, based on information and computation. In particular, algebraic geometry studies the geometric spaces defined by algebraic formulas. The grand challenges of algebraic geometry, such as the Hodge conjecture, seek to describe all the subspaces of a given space in computable terms. This project will engineer new connections between geometric spaces and the corresponding formulas. Looking at familiar spaces through the eyes of algebraic geometry yields new results in topology and group representation theory. The project will bring undergraduates, graduate students, and postdocs into mathematical research.In more detail, the theory of algebraic cycles aims to describe all the subvarieties of a given algebraic variety. This subject has recently led to dramatic progress on the old problem of which varieties have a rational parametrization. One main theme of the project is to study varieties in positive characteristic, where there are direct connections between algebraic cycles and more computable theories such as Hodge cohomology. The PI will use these connections to make new calculations in representation theory and the theory of algebraic groups. Finally, the PI will prove new cases of the integral Hodge conjecture for complex varieties.