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.

多项式方程系统的代数几何研究解决方案集。 例如，线是线性多项式方程的解决方案集，而圆圈和双曲线是对二次多项式方程的溶液集，它们的研究可以追溯到古希腊人。许多变量中许多多项式方程的系统的解决方案集通常具有美丽而复杂的几何形状。 PI将采用新的现代技术来回答代数几何学领域的经典兴趣问题，并解决有关多项式方程定义的空间几何形状的长期开放问题。他还将继续与培训子孙后代的数学家进行充满活力的参与，包括通过研究生和博士后的指导。 PI将追求三个主要的研究方向：稳定曲线的模量空间的共同体学，光滑曲线的模量空间的同一个同谋以及局部的单型猜想，以实现超曲面奇异性。他将使用关于平滑曲线模量的模量空间的新结果来确认langlands计划的预测和稳定曲线模量空间的Hodge猜想。他将对稳定曲线的模量空间的共同体进行新的结果，以研究开放曲线模量空间的重量级别的共同体，证明了对映射类群体的共同研究的新的非变化结果，并为重量分级的Euler特征产生新的生成功能。他将寻求有关超表面奇点的动机，p-呼吸和拓扑当地单构的猜想，以及相关的猜想，例如p-adadic本地Zeta的构造和全体形状的猜想，以扭转局限性，这是由NSF奖励的智力及其授予的智力依据。 标准。