CAREER: The cohomology and birational geometry of moduli spaces

齐次簇，特别是普通的和各向同性的格拉斯曼簇，是代数几何、表示论和组合学的中心研究对象。研究者提出了积极的算法计算各向同性格拉斯曼和各向同性旗品种的上同调的结构常数。近年来，类似的正演算法已经解决了许多关于A型格拉斯曼的重要问题，包括饱和猜想和舒伯特演算的实性。研究人员还将研究曲线和稳定映射的模空间的有效锥。曲线的模空间是数学中研究最多的对象之一。他们的锥有效因子是重要的不变量，密切相关的问题，如肖特基问题，科代拉维的模空间的曲线，并存在模块化的形式。最近，理解模空间的有效锥在Hacon，McKernan及其合作者在最小模型程序（MMP）上的开创性工作之后获得了新的动力。研究者建议在模空间上运行MMP，例如亏格零稳定映射的Kontsevich模空间或射影平面上点的Hilbert方案。多项式方程组出现在生活的许多方面，从进化生物学到物理学，从密码学到计算机科学。代数几何学研究多项式系统解的几何性质。 具有许多对称性的解是特别有趣和重要的。例如，球面是一个完全对称的空间，因为任何点都可以旋转到任何其他点。调查研究的解决方案的数量多项式系统涉及这种完美对称空间称为齐次品种。调查员还计算更微妙的几何不变量系统的多项式方程，如如何行为的空间的解决方案的变化下扰动的系统。

Homogeneous varieties, in particular, ordinary and isotropic Grassmannians, are central objects of study in algebraic geometry, representation theory, and combinatorics. The investigator proposes to develop positive algorithms for computing the structure constants of the cohomology of isotropic Grassmannians and isotropic flag varieties. In recent years, similar positive algorithms have led to the solutions of many important problems for Type A Grassmannians, including the saturation conjecture and the reality of Schubert calculus. The investigator will also study the effective cones of the moduli space of curves and stable maps. The moduli spaces of curves are among the most studied objects in mathematics. Their cones of effective divisors are important invariants, intimately tied to problems such as the Schottky problem, the Kodaira dimension of the moduli space of curves, and the existence of modular forms. Recently, understanding the effective cone of moduli spaces has received new impetus following the seminal work of Hacon, McKernan, and their collaborators on the Minimal Model Program (MMP). The investigator proposes to run MMP on moduli spaces such as the Kontsevich moduli spaces of genus zero stable maps or Hilbert scheme of points on the projective plane. Systems of polynomial equations occur in many facets of life ranging from evolutionary biology to physics and from cryptography to computer science. Algebraic geometry studies geometric properties of solutions of polynomial systems. The solutions that have many symmetries are especially interesting and important. For example, a sphere is a perfectly symmetric space in the sense that any point can be rotated to any other point. The investigator studies the number of solutions to polynomial systems involving such perfectly symmetric spaces called homogeneous varieties. The investigator also calculates more subtle geometric invariants of systems of polynomial equations such as how the behavior of the space of solutions changes under perturbations of the system.