Arithmetic and Quantum Intersection Theory on Homogeneous Spaces

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This research is in different aspects of intersection theory in algebraic geometry: higher dimensional Arakelov theory, a part of arithmetic algebraic geometry, and quantum cohomology, a theory on the border between mathematics and quantum physics. The investigator studies the arithmetic and quantum intersection rings of homogeneous spaces, such as flag manifolds, as part of a general program of extending results of classical algebraic geometry to the new settings. The following are some of the problems considered: a) Understanding the arithmetic and quantum Schubert calculus for homogeneous spaces of classical Lie groups; b) finding determinantal formulas for Lagrangian and orthogonal degeneracy loci and developing further the investigator's theory of double Schubert polynomials; c) obtaining arithmetic analogues of Fulton's results on degeneracy loci and numerically positive polynomials; d) computing arithmetic intersections and heights on Shimura varieties.During the 19th century, mathematicians and physicists were fascinated by the symmetries observed in geometric objects. The many experiments and computations made at that time eventually led to the cohomology theories of the 20th century, which were applied to solve long-standing open problems in both fields. Today, we are in a similar situation in modern number theory and quantum field theory, and it is important to have calculations of specific examples to support and guide our intuition. The investigator examines two new theories, one motivated by number theory and the other by quantum physics, in many specific examples which are prototypes for this purpose. The potential applications are a better understanding of Diophantine equations and approximation, used in coding theory and theoretical computer science, and enumerative geometry of curves, related to string theory and symmetry in physics.