Dual complexes and weight filtrations: Applications to cohomology of moduli spaces and invariants of singularities

Algebraic geometry studies solution sets of systems of polynomial equations. For instance, lines are solution sets of linear polynomial equations, while circles and hyperbolas are solution sets to quadratic polynomial equations, and their study goes back to the ancient Greeks. The solution sets of systems of many polynomial equations in many variables often have beautiful and complicated geometry. The PI will apply new and modern techniques to answer questions of classical interest in the field of algebraic geometry, and to address long standing open problems about the geometry of spaces defined by polynomial equations. He will also continue his energetic engagement with training future generations of mathematicians, including through mentorship of graduate students and postdocs. The PI will pursue three main research directions: cohomology of moduli spaces of stable curves, cohomology of moduli spaces of smooth curves, and the local monodromy conjectures for hypersur- face singularities. He will confirm predictions of the Langlands program and the Hodge conjecture for moduli spaces of stable curves, using new results on the Chow cohomology and cycle class maps for moduli spaces of smooth curves. He will apply new results on the cohomology of moduli spaces of stable curves to study the weight-graded cohomology of moduli spaces of open curves, proving new non-vanishing results for cohomology of mapping class groups and producing new generating functions for weight-graded Euler characteristics. And he will pursue a proof of the motivic, p-adic, and topological local monodromy conjectures for hypersurface singularities, along with related conjectures such as the monodromy and holomorphy conjectures for p-adic local zeta functions twisted by a character. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.