Schubert Calculus, Quiver Varieties, and Kazhdan-Lusztig Coefficients

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Many questions in mathematics can be phrased as follows: if one imposes a certain list of independent conditions on the points in a manifold (for example, a vector space or sphere), are there any points that satisfy all the conditions? Such questions have linear approximations more prone to systematic attack, but yet still quite difficult. This field of linear intersection questions goes by the 19th-century name "Schubert calculus". Two of the PI's three projects are in the realm of Schubert calculus. One of the unusual features of this pursuit is that (long, slow) formulae are generally available to count the number of solutions exactly, but they are ill-suited to easily check whether this number is positive. The project provides research training opportunities for graduate students.The first project replaces (intersection theory on) flag manifolds with their cotangent bundles, a small change, but then realizes the latter as special cases of "Nakajima quiver varieties". The PI's "Schubert calculus puzzles" are best interpreted on these larger quiver varieties, an intermediate ground between the (cotangent bundles of) flag manifolds of actual interest. The second concerns a recent formula of Goldin and the PI computing this intersection theory succinctly (although not manifestly positively, a long-term goal in the field). This involved the creation of some operators with intriguing algebraic properties, but no clear geometric origin; part of the project is a search for this geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.