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Grassmannian簇和旗簇，是代数几何、表示论和组合学的中心研究对象。研究人员提出了计算旗簇和各向同性Grassmannians上同调的结构常数的正算法。近年来，类似的正演算法已经解决了许多重要问题，包括Klyachko，Knutson和Tao对Horn猜想的解以及Vakil对Schubert微积分实在性的解。在以前的工作中，研究者得到了两步旗簇和普通Grassmannians的量子上同调的正算法。研究人员将使用退化技术为旗帜变种和各向同性Grassmannians获得类似的算法。自古以来，多项式方程的解集合，如勾股三元组，一直被广泛研究。对于自然科学、计算机科学、密码学和数学的许多分支来说，能够求解多项式系统是至关重要的。代数几何是研究被称为簇的多项式方程的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，了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。具有大对称群(如旋转对称或平移对称)的表示论、组合学和物理学问题往往会产生齐次变种。研究者将研究齐次簇的几何不变量，并在有有限多个解的情况下确定与齐次簇相关的一组多项式的解的数目。

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.