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。多元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的经典问题，并解决长期存在的关于多项式方程定义的空间几何的开放问题。他还将继续积极参与培训未来几代数学家，包括指导研究生和博士后。PI将追求三个主要的研究方向：稳定曲线的模空间的上同调，光滑曲线的模空间的上同调，以及超曲面奇异性的局部单值性。他将确认预测的朗兰兹计划和霍奇猜想的模空间的稳定曲线，使用新的成果周上同调和循环类地图的模空间的光滑曲线。他将应用新的成果上的模空间的稳定曲线的上同调研究的权重分级上同调的模空间的开放曲线，证明新的非零结果上同调的映射类组和生产新的生成函数的权重分级欧拉特征。他还将致力于证明超曲面奇点的motivic、p-adic和拓扑局部单值性，沿着相关的证明，如被字符扭曲的p-adic局部zeta函数的单值性和全纯性。该奖项反映了NSF的法定使命，并被认为值得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来支持。