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Algebra, Combinatorics, and Geometry Professors Chao, Constantine, Dickinson, Fulman, Ion, Hales, Lukic, and Whitehead.
[Standing left to right: Jason Fulman, Denis Lukic,
Jyotsna Diwadkar, Dana Mihai, Ken-Hsien Chuang, Bogdan Ion,
Mark Dickinson, Greg Constantine. Seated left to right:
Tom Hales, Glen Whitehead, Chris Jones]
Algebra, Combinatorics, and Geometry are areas of
very active research at the University of Pittsburgh.
A number of the ongoing research projects are described
below. These include
Combinatorial and Statistical Designs, Set and Graph Partitions (Constantine) Constantine's research interests include combinatorial and statistical designs, set and graph partitions, combinatorics on finite groups, and mathematical and statistical planing and modeling. Selected recent papers:
Modularity of Galois representations (Dickinson) Some famous conjectures due to Fontaine and Mazur link geometric two-dimensional p-adic representations of the absolute Galois group of the rational numbers with certain classical modular forms. Substantial parts of these conjectures have been proved, especially as part of Andrew Wiles' proof of Fermat's Last Theorem and the subsequent proof of the elliptic modularity conjecture. Perhaps surprisingly, there are difficulties in establishing this link for a particularly simple class of representations: those whose mod-p reduction is trivial when restricted to a decomposition group at p. Dickinson is working on ways of extending Wiles' original methods to understand these representations.Explorations in Stein's Method (Fulman) Stein's method is a remarkable technique for proving limit and approximation theorems in probability. The advantage to Stein's method is that one can prove results with different (and often less) information than by the traditional methods, and that explicit error terms emerge. The purpose of this project is to explore applications of Stein's method to combinatorics, and to expand the scope of the method as much as possible.Combinatorics of the Symmetric Group (Fulman) The symmetric group is a fundamental object, and the purpose of this project is to explore its combinatorial properties. Combinatorial questions about random walk and Markov chains on the symmetric group are emphasized. There are interesting connections with representation theory, which are still only partly understood.Motivic integration and representation theory (Hales) Several years ago, M. Kontsevich created a new type of integration, called motivic integration, where the values of integrals are not numbers but geometric objects. Hales's research explores connections between representation theory and motivic integration.Formal theorem proving (Hales) In a formal proof, all the intermediate logical steps of a proof are supplied. No appeal is made to intuition, even if the translation from intuition to logic is routine. Thus, a formal proof is less intuitive, and yet less susceptible to logical errors than a traditional proof. Recently, Hales has produced a formal proof of the classical Jordan curve theorem. This is part of a larger project, called flyspeck.Sphere Packings and Discrete Geometry (Hales) The Kepler conjecture asks what is the densest packing of congruent balls in three dimensional Euclidean space. Hales and a graduate student Sam Ferguson solved this conjecture in 1998. The proof requires a number of long computer calculations. These include linear programming, computer classification of certain planar graphs, and interval arithmetic calculations.Another problem in discrete geometry that Hales solved is the honeycomb conjecture, which asserts that the most efficient partition of the plane into equal area cells is the hexagonal honeycomb. Representations, Macdonald theory, and Hecke algebras (Ion) Ion's main research area is Lie theory/Representation theory. Most recently, he has been interested in Macdonald theory which provides an uniform framework for the study of several questions regarding the spherical harmonic analysis of real/p-adic reductive groups. His work in this area makes use of various connections with affine Kac-Moody groups, Hecke algebras, the geometry of the affine Grassmannians and the affine flag manifolds, combinatorics of Coxeter groups and root systems, symmetric functions, hypergeometric functions.Another subject Ion works on, still deeply intertwined with the above topics but of considerable independent interest, is the representation theory of double affine Hecke algebras. Non-commutative algebra and geometry (Ion) Ion maintains an active interest in several topics in non-commutative algebra/geometry: deformation quantization, (finite dimensional) Hopf algebras, graded rings, categories.Algebro-geometric methods of representation theory (Lukic) Lukic's research interest lies in the field of representation theory of groups and algebras, especially the algebro-geometric methods of their study. One of the basic problems in representation theory is to understand representations of semisimple Lie groups. The algebraization of this problem leads to the study of modules over complex semismple Lie algebras. His research focuses on a geometric approach to the study of the various categories of modules over complex semisimple Lie algebras using D-modules techniques and Beilinson-Bernstein localization theory. Particularly he is interested in irreducibility and multiplicity problems in these categories.Graph Polynomials (Whitehead) Prof. Whitehead is an internationally known expert on graph polynomials. He has studied the chromatic polynomial, the flow polynomial, the matching polynomial, and the Tutte polynomial. A list of his published research papers is available from his webpage. Graph polynomials count various numbers related to properties of graphs. For example, the chromatic polynomial counts the number of proper colorings of a graph. Whitehead organized a Special Session on Graph Polynomials as part of the meeting of the American Mathematical Society held at the University of Pittsburgh, November 6-7, 2004. |
