Applications of Enumerative Geometry to Homogenous Varieties and Moduli Spaces

齐次变体，特别是普通的和各向同性的格拉斯曼变体和旗变体，是代数几何、表示理论和组合学研究的中心对象。研究人员提出了一种计算旗类和各向同性格拉斯曼类上同调结构常数的正算法。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猜想的解和Vakil对Schubert微积分的实解。研究者在之前的工作中，获得了普通格拉斯曼子的两步标志变异和量子上同调的正算法。研究者将使用退化技术来获得旗子品种和各向同性格拉斯曼种的类似算法。多项式方程的解集，如毕达哥拉斯三元组，自古以来就被广泛研究。能够求解多项式系统对自然科学、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一组解的几何性质的学科。品种通常具有丰富而美丽的对称性，这有助于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变种称为齐次变种。具有大对称群（如旋转或平移对称）的表示理论、组合学和物理问题经常产生齐次变体。研究者将研究齐次变量的几何不变量，并在有有限多个解的情况下确定与齐次变量相关的一组多项式的解的个数。

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.