Applications of Enumerative Geometry to Homogenous Varieties and Moduli Spaces

齐次簇，特别是普通的和各向同性的格拉斯曼簇和旗簇，是代数几何、表示论和组合学的中心研究对象。 研究者提出了积极的算法来计算旗品种和各向同性格拉斯曼的上同调的结构常数。近年来，类似的正演算法已导致许多重要问题的解决，包括Klyachko，Knutson和Tao对Horn猜想的解决和Vakil对Schubert演算实性的解决。 研究人员在以前的工作中，获得了两步旗品种和普通格拉斯曼的量子上同调的正算法。 研究人员将使用退化技术，以获得类似的算法旗品种和各向同性Grassmannians.The一套解决方案的多项式方程，如毕达哥拉斯三元组，已深入研究，因为古代。 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都至关重要。代数几何是研究多项式方程的解的集合的几何性质的学科，称为簇。 变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，理解几何有助于解决多项式系统。 对称交换任意两点的簇称为齐次簇。表示论、组合学和具有大对称群（如旋转或平移对称）的物理学的问题经常会产生齐次变体。 研究者将研究齐次簇的几何不变量，并确定与齐次簇相关联的一组多项式的解的个数，如果有多个解。

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.