Tropical and nonarchimedean analytic methods in algebraic geometry

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曲线几何的长期悬而未决的问题。这个项目为研究生提供了研究培训的机会。在一个非阿基米德领域，人们可以将理解这样的解集的问题分成两部分。解决方案的可能估值是什么？在给定估值的情况下，解决方案是什么？解的赋值集具有丰富的组合多面体结构，是热带几何的主要研究对象；具有给定赋值的解可以通过非阿基米德解析几何来研究。这些领域的最新发展使得使用新的和创新的工具和技术来解决关于实际解集的几何的微妙问题成为可能。这个项目将提炼、抽象和推广这些新方法，并探索更深层次的应用于代数几何中的公开问题。PI将使用热带和非阿基米德方法继续他在精细曲线计数和曲线模空间的不稳定上同调方面的工作，超越权过滤的顶级分次段来检查完全权分次上同调环。他还将利用热带交换簇的模空间的拓扑和由么模拟阵和完全二次型构成的复形的组合代数，开始对交换簇的模空间的加权分次上同调进行类似的研究。其他项目将利用相对局部h-多项式的组合学和施泰普隆的单调公式来攻击牛顿非退化奇点的动机猜想、p-进猜想和拓扑单列猜想，并试图解决Kontsevich和Morita在交换图复合体的同调中的突出(和相互矛盾的)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。

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 curves defined by polynomial equations. This project provides research training opportunities for graduate students.Over a nonarchimedean field, one can split the problem of understanding such solution sets into two parts. What are the possible valuations of solutions? And what are the solutions with a given valuation? The set of valuations of solutions has a rich combinatorial and polyhedral structure, and is the primary object of study in tropical geometry; the solutions with a given valuation can be studied via nonarchimedean analytic geometry. Recent developments in these fields make it possible to resolve subtle questions about the geometry of the actual solution set using new and innovative tools and techniques. This project will refine, abstract, and generalize these new methods and explore deeper applications to open problems in algebraic geometry.The PI will use tropical and nonarchimedean methods to continue his work on refined curve counting, and on unstable cohomology of moduli spaces of curves, proceeding beyond the top graded piece of the weight filtration to examine the full weight-graded cohomology ring. He will also initiate an analogous study of the weight graded cohomology of the moduli space of abelian varieties, using the topology of moduli spaces of tropical abelian varieties and the combinatorial algebra of complexes built out of unimodular matroids and perfect quadratic forms. Additional projects will attack the motivic, p-adic, and topological monodromy conjectures for Newton nondegenerate singularities, using the combinatorics of relative local h-polynomials and Stapledon’s formulas for monodromy, and attempt to resolve outstanding (and mutually contradictory) conjectures of Kontsevich and Morita in the homology of commutative graph complexes.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.