Combinatorics and Algebraic Geometry -- Macdonald Polynomials, Hilbert Schemes, and Related Topics

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Haiman 9701218 Haiman has been working on a project of several years duration to prove conjectures of Garsia and Haiman giving a combinatorial model for Macdonald polynomials, along with a series of conjectures of Haiman on diagonal harmonics, using sheaf-cohomological methods applied to the Hilbert scheme of points in the plane and related algebraic varieties. Once that is done, it will be possible to apply the resulting tools to seek unified geometric explanations of the Macdonald polynomials' many remarkable properties. This constitutes the first part mf the planned research. Haiman and his student W. Brockman, seeking to apply recent work of Broer to certain aspects of the above study, have discovered new connections linking other algebraic varieties to the one- and two-parameter Kostka polynomials. These discoveries have led to a geometric explanation of Lascoux's atoms and to tantalizing conjectures whose further study will form the second part of the planned research. As a third and final part of the planned research, Haiman will resume his earlier work on Hecke algebras and Kazhdan-Lusztig polynomials, motivated by his still unsolved conjectures on Hecke algebra characters, and the related problem of finding a satisfactory combinatorial interpretation of Kazhdan-Lusztig polynomials. This research concerns the interplay between combinatorics and algebraic geometry. One of the goals of combinatorics is to find efficient methods of studying how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis, topology, and combinatorics, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.