Schubert calculus is the study of the (singular) cohomology ring of a homogeneous space X, with an emphasis on multiplication of Schubert classes. Such products can be interpreted as the number of points in an intersection in general Schubert varieties, and therefore provide an essential tool in enumerative algebraic geometry. While this ring contains information about the components of an intersection of Schubert varieties, the K-theory ring also provides information about how these components intersect. The equivariant cohomology ring refines the ring structure by taking a group action into account. And the quantum cohomology ring encodes the Gromov-Witten invariants, which give the number of curves of fixed degree that meet general Schubert varieties. All these rings can be combined to one single equivariant quantum K-theory ring (to rule them all). Until recently, not much has been known about the structure of this ring, since the Gromov-Witten invariants used to define it have been hard to compute. The investigator will attempt to overcome this problem and uncover the ring structure in as many cases as possible. He will also seek answers to some important open questions concerning the orbit closures of quiver representations. These objects from representation theory generalize many interesting degeneracy loci, including Schubert varieties and determinantal varieties, and their classes generalize important polynomials from geometry and combinatorics, including Schur, Schubert, and Grothendieck polynomials. Studies of orbit closures for relatively simple quivers have in the past resulted in many beautiful formulas, constructions, and positivity results. The proposer hopes to generalize these results to more general quivers.
The investigator will attack problems in the broad area of enumerative geometry, especially problems concerning quantum rings and quiver cycles. Enumerative geometry aims to provide the tools required to count the number of geometric objects of given type that satisfy specified conditions. A typical example is the fact that exactly two lines (possibly with complex coordinates) meet 4 randomly chosen fixed lines in 3-space. Enumerative problems are often approached by constructing a moduli space, which has one point for each geometric object of the given type, and then translating the specified conditions into polynomial equations on the moduli space. This transforms the problem into one of counting the solutions to such equations. Quiver cycles can be understood as a way to organize equations that arise in many important situations, and knowledge about geometric invariants of quiver cycles adds to their utility for counting solutions. Ideas from string theory in physics led to the definition of the quantum cohomology ring of a homogeneous space, which provides an efficient tool for counting the number of curves of given degree that meet fixed subvarieties in the space, at least when this number is finite. When the number of solution curves is infinite, the set of these curves form a space called a Gromov-Witten variety, and geometric invariants of this space can tell us much about the problem. For example, if the Gromov-Witten variety has non-zero Euler characteristic, then solution curves do exist. Quantum K-theory is a generalization of quantum cohomology that encodes the Euler characteristic of Gromov-Witten varieties. The investigator plans to study these topics with a combination of geometric and combinatorial methods.
