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

代数几何研究多项式方程组的解集。例如，直线是线性多项式方程的解集，而圆和双曲线是二次多项式方程的解集，它们的研究可以追溯到古希腊人。多变量多项式方程组的解集往往具有美丽而复杂的几何形状。PI将应用新的和现代的技术来回答代数几何领域中经典感兴趣的问题，并解决由多项式方程定义的空间几何的长期悬而未决的问题。他还将继续致力于培养未来几代数学家，包括通过指导研究生和博士后。PI将追求三个主要研究方向：稳定曲线的模空间的上同调，光滑曲线的模空间的上同调，以及超曲面奇点的局部单调猜想。他将利用关于光滑曲线模空间的Chow上同调和循环类映射的新结果，证实关于稳定曲线模空间的朗兰兹程序和Hodge猜想的预测。他将应用稳定曲线的模空间的上同调的新结果来研究开曲线的模空间的加权分次上同调，证明映射类群的上同调的新的非零化结果，并产生新的加权分次欧拉特征的母函数。他将对超曲面奇点的动机猜想、p进位猜想和拓扑性局部单调猜想以及相关猜想(如由特征扭曲的p进位局部Zeta函数的单调猜想和全纯猜想)进行证明。这一奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.