This proposal continues the investigator's research on elliptic analogues of special functions (hypergeometric functions, Painleve transcendents, and Macdonald polynomials), with specific attention to the study of degenerations. One major theme of the research is the classification of difference and differential equations (viewed as difference equations on a possibly very singular elliptic curve) using certain moduli spaces of sheaves on (possibly noncommutative) surfaces, generalizations of the spaces of initial conditions of the Painleve equations. For instance, this reduces the problem of classifying degenerations of the elliptic hypergeometric equation to one of classifying -2 curves on certain rational surfaces. Other problems involve classifying and studying degenerations of the investigator's elliptic analogues of Macdonald/Koornwinder polynomials (with a long-term goal of understanding how the Hecke algebra approach might be generalized); proving special cases for Hall-Littlewood-type polynomials of various multivariate quadratic transformations conjectured by the investigator; and studying limiting cases of a random tiling model with elliptic probabilities, generalizing the uniform distribution on lozenge tilings of a hexagon.
Historically, the study of "special functions" originated in the fact that quite a few functions of interest in applications turned out to be members of a single family, the hypergeometric functions; more recently, the "Painleve transcendents" have also begun to play a significant role in applications. Under previous grants, the investigator constructed and studied so-called "elliptic" analogues of these functions (a significant generalization that includes most of the generalized hypergeometric functions and generalied Painleve transcendents in the literature); the current research program continues this investigation, with particular attention to the implications for more classical special functions.
The Painlevé transcendents are a family of special functions (satisfying certain nonlinear differential equations known as the Painlevé equations) which have gained special prominence in the past decade and a half (through relations to random matrix theory and combinatorics in particular). A number of generalizations have been studied, and the main purpose of this grant was to understand the structure of these generalizations, both enabling further generalization to be done, and clarifying the nature of the Painlevé equations themselves. Although some of the PI's work in this context is still ongoing, the PI made significant progress, by showing that a certain limiting case of the generalized (elliptic) Painlevé equations was itself a generalization of a family of equations studied by Hitchin. Moreover, joint work of the PI with Okounkov gave an explanation of this fact by showing that one could recover one of the more important generalized Painlevé equations by replacing the geometric object appearing in Hitchin's work by an object from "noncommutative" geometry; in the relevant limit, the geometric object becomes commutative, aand one recovers an integrable system à la Hitchin. One of the more important interpretations of the Painlevé transcendents is the so-called "isomonodromy" interpretation, which uses the transcendents to parametrize a particularly nice family of linear differential equations (with mild singularities and fixed "monodromy"). A similar interpretation exists for various generalizations of the Painlevé transcendents, leading to the natural question of how this might relate to the noncommutative geometric interpretation. The PI performed an important initial step in understanding this by understanding how the relevant Hitchin-type integrable system was related to discrete analogues of differential equations (difference equations); this connection turned out to be explicit enough to enable the PI to make a solid conjecture about the structure of new generalizations of the Painlevé equations. Future work of the PI will use this connection between geometry and difference equations not only to further explore the relation between Painlevé theory and noncommutative geometry, but also to make new contributions entirely within noncommutative geometry itself. Beyond the above work, the PI also mentored five graduate students (three graduated, two current) and a postdoctoral scholar (current).
