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

> **NSF 01002425DB NSF RESEARCH & RELATED ACTIVIT** · Regents of the University of Michigan - Ann Arbor (MI) · $337,070

## Abstract

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 

## Key facts

- **NSF award ID:** 2542134
- **Awardee organization:** Regents of the University of Michigan - Ann Arbor (MI)
- **SAM.gov UEI:** GNJ7BBP73WE9
- **PI:** Sam Payne
- **Primary program:** 01002425DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $337,070
- **Funds obligated:** $201,798
- **Transaction type:** Continuing Grant
- **Period:** 07/01/2025 → 08/31/2028

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2542134

## Citation

> US National Science Foundation, Award 2542134, Dual complexes and weight filtrations: Applications to cohomology of moduli spaces and invariants of singularities. Retrieved via AI Analytics 2026-06-08 from https://api.ai-analytics.org/grant/nsf/2542134. Licensed CC0.

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