Applications of Enumerative Geometry to Homogenous Varieties and Moduli Spaces

齐次簇，特别是普通和各向同性格拉斯曼簇和旗簇，是代数几何、表示论和组合学的中心研究对象。 研究人员建议开发正算法来计算标志簇和各向同性格拉斯曼函数的上同调的结构常数。近年来，类似的正算法导致了许多重要问题的解决，包括Klyachko、Knutson 和Tao 对Horn 猜想的解决以及Vakil 对舒伯特微积分现实的解决。 研究人员在之前的工作中获得了两步旗簇的正算法和普通格拉斯曼量的量子上同调。 研究人员将使用退化技术来获得旗簇和各向同性格拉斯曼方程的类似算法。多项式方程组的解，例如毕达哥拉斯三元组，自古以来就已被深入研究。 能够求解多项式系统对于自然科学、计算机科学、密码学和数学的许多分支至关重要。代数几何是对称为簇的多项式方程组解的几何性质的研究。 品种通常具有丰富而美丽的对称性，可以帮助理解它们的几何形状。反过来，理解几何有助于求解多项式系统。 对称性交换任意两点的簇称为同质簇。具有大对称群（例如旋转或平移对称）的表示论、组合学和物理学问题常常会产生同质簇。 研究者将研究同质簇的几何不变量，并在解数有限的情况下确定与同质簇相关的一组多项式的解数。

Homogeneous varieties, in particular ordinary and isotropic Grassmannians and flag varieties, 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 flag varieties and isotropic Grassmannians. In recent years, similar positive algorithms have led to the solution of many important problems, including Klyachko, Knutson and Tao's solution of Horn's conjecture and Vakil's solution of the reality of Schubert calculus. The investigator, in previous work, obtained positive algorithms for two-step flag varieties and the quantum cohomology of ordinary Grassmannians. The investigator will use degeneration techniques to obtain similar algorithms for flag varieties and isotropic Grassmannians.The set of solutions of polynomial equations, such as Pythagorean triplets, have been studied intensively since Antiquity. Being able to solve polynomial systems is crucial to many branches of natural sciences, computer science, cryptography and mathematics. Algebraic geometry is the study of the geometric properties of the set of solutions of polynomial equations called varieties. Varieties often have rich and beautiful symmetries that can help understand their geometry. In turn, understanding the geometry helps solve the polynomial systems. The varieties where the symmetries exchange any two points are called homogeneous varieties. The problems of representation theory, combinatorics and physics with large symmetry groups (such as rotational or translational symmetry) often give rise to homogeneous varieties. The investigator will study the geometric invariants of homogeneous varieties and determine the number of solutions of a set of polynomials associated to a homogeneous variety in case there are finitely many solutions.