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.
