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

对偶复形和权重过滤：模空间上同调和奇点不变量的应用

• 批准号: 2302475
• 负责人: Sam Payne
• 金额: $ 33.71万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2028-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302475&HistoricalAwards=false
• 关键词: Dual; complexes; weight; filtrations; Applications

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.

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优美而复杂的几何形状。 PI 将应用新的和现代的技术来回答代数几何领域中经典的问题，并解决有关由多项式方程定义的空间几何的长期悬而未决的问题。他还将继续积极参与培养未来几代数学家，包括通过指导研究生和博士后。 PI将致力于三个主要研究方向：稳定曲线模空间上同调、平滑曲线模空间上同调以及超曲面奇点的局部单向猜想。他将利用 Chow 上同调和平滑曲线模空间的循环类图的新结果，确认朗兰兹纲领和霍奇猜想对稳定曲线模空间的预测。他将应用稳定曲线模空间上同调的新结果来研究开曲线模空间的权重分级上同调，证明映射类群上同调的新的非消失结果，并为权重分级欧拉产生新的生成函数特征。他将追求超曲面奇点的动机、p-adic 和拓扑局部单向猜想的证明，以及相关猜想，例如由字符扭曲的 p-adic 局部 zeta 函数的单向和全纯猜想。该奖项反映了 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。