Professor Shimozono is an expert in the Schubert calculus of the infinite-dimensional flag varieties associated with affine Lie algebras and with large rank limits of finite-type simple Lie algebras.
He has studied the Quantum Equals Affine phenomenon which asserts that the Gromov-Witten invariants for the equivariant cohomology and K-theory of finite-dimensional flag varieties, coincide with equivariant structure constants for the Schubert basis in the homology and K-homology of the affine Grassmannian.
He has expertise in the computation of explicit combinatorial formulas for equivariant cohomology and K-theory classes of degeneracy loci, in particular solving a version of the Buch-Fulton Factor Sequence Conjecture with Allen Knutson and Ezra Miller. He (joint with Thomas Lam and Seungjin Lee) has initiated back-stable Schubert calculus, the study of infinite-dimensional flag varieties based on a type A Dynkin diagram that is infinite in both the forward and backward direction.
Professor Shimozono has studied the combinatorics of quantum affine Lie algebras, particularly of the crystal graph theory for highest weight and level zero modules and their connections with nonsymmetric Macdonald polynomials and modules supported on the nullcone.
Currently he is working with Professor Dan Orr on the combinatorics of representations of quantum toroidal algebras and double affine Hecke algebras and their connection with the geometry of quiver varieties and the Hilbert scheme of points in the plane.