CAREER: The cohomology and birational geometry of moduli spaces

齐次变体，特别是普通的和各向同性的格拉斯曼变体，是代数几何、表示理论和组合学研究的中心对象。研究者建议发展计算各向同性格拉斯曼和各向同性标志变体的上同调结构常数的正算法。近年来，类似的正算法已经解决了许多A型格拉斯曼人的重要问题，包括饱和猜想和舒伯特微积分的现实。研究者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。它们的有效因子锥是重要的不变量，与Schottky问题、曲线模空间的Kodaira维以及模形式的存在性等问题密切相关。最近，随着Hacon， McKernan和他们的合作者在最小模型计划（MMP）上的开创性工作，对模空间有效锥的理解得到了新的推动。研究者建议在模空间上运行MMP，如零属稳定映射的Kontsevich模空间或投影平面上点的Hilbert格式。多项式方程系统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。代数几何研究多项式系统解的几何性质。具有许多对称性的解特别有趣和重要。例如，球体是一个完全对称的空间，从某种意义上说，任何一点都可以旋转到任何其他点。研究者研究了多项式系统的解的数目，这些解涉及到称为齐次变分的完全对称空间。研究者还计算了多项式方程系统的更微妙的几何不变量，例如在系统扰动下解空间的行为如何变化。

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.