Algebra, Combinatorics, and Geometry
Algebra, Combinatorics, and Geometry
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. The group also offers a Ph.D. program.
Ph.D. in Mathematics with the the Algebra, Combinatorics, and Geometry Group More>
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 planning and modeling.
Selected recent papers:
- Graphs, networks, and linear unbiased estimates, Discrete Appl. Math., 3, 381-393 (2003).
- Edge-disjoint isomorphic multicolored trees and cycles in complete graphs, SIAM Journal on Discrete Mathematics, 18, 577-580 (2005).
- Colorful isomorphic spanning trees in complete graphs, Annals of Combinatorics, 9, 163-167 (2005).
- In silico design of clinical trials: A method coming of age, Critical Care Medicine, 32, 2061-2070 (2004), with G. Clermont et al.
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 of 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 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, and 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, and categories.
One of the principal goals of algebraic geometry is to classify all algebraic varieties up to isomorphism or birational equivalence. This goal was mostly achieved in dimensions one and two by the classical Italian school in the beginning of the 20th century. In dimension three or more,
however, there was relatively little progress until the 1980s, when the minimal model program was introduced by S. Mori and M. Reid. Since then, this field has exploded into possibly the most active area of algebraic geometry research. Borisov's work on Fano varieties with log-terminal
singularities deals with some of the objects and questions central to this fast-growing field.
Applications of Analysis to Number Theory (Borisov)
The analogy between number fields and fields of meromorphic functions on complex algebraic curves has been known for more than a century. Still, nobody knows exactly how deep it is. The two most successful approaches to understanding this analogy are the Arakelov geometry and Tate's work on harmonic analysis in number fields. Both use the tools of analysis to handle the behavior of number-theoretic objects "at infinity." Borisov's work aims to unite these two approaches. In particular, he used some convolution of measures structures to develop the cohomology theory for Arakelov divisors on a number field. The whole theory is very similar to the corresponding theory for algebraic curves, with Serre duality being the Pontryagin duality of the convolution of measures structures.
Applications of Algebraic Geometry to Group Theory (Borisov)
In a surprising recent development by Borisov and Mark Sapir, periodic orbits of self-maps of affine varieties have been related to the finite quotients of mapping tori of algebraic groups. Study of these orbits, which is also related to the generalizations of Deligne's conjecture and Hrushovski's work in model theory, resulted in a proof of the residual finiteness of all mapping tori of algebraic groups. Borisov is working on the refinements of this technique, which may have applications to some other important problems in group theory.
Lattice Polytopes and Toric Varieties (Borisov)
Toric varieties are a particular class of algebraic varieties that are determined by a combinatorial data based on a collection of convex rational cones in the standard n-dimensional real vector space. These varieties are good examples for the minimal model program, because its basic notions and constructions have combinatorial interpretation there in terms of lattice points in convex cones and polytopes. Borisov has done a lot of work on these objects, including the most general classification result for terminal toric singularities. Recently, Borisov discovered that this sort of problems is related to some families of integer ratios of factorials, hyper geometric functions, and (through the Nyman-Beurling criterion) the zeros of Riemann zeta function. This exciting connection has the potential to produce new results in several areas of mathematics.
Polynomials with Integer Coefficients (Borisov)
Polynomials with integer and rational coefficients appear in many areas of modern mathematics. They have been studied extensively by many people from many different standpoints by a number of different techniques. Although most of these techniques are relatively elementary, they are not easy, and some proofs are highly non-trivial. Some of Borisov's research has been focused on these objects. He proved, in particular, the irreducibility results for some families of integer polynomials and established a simple criterion for a rational polynomial to divide the derivative of a polynomial, which has only rational roots.
Newton-Okounkov Bodies (Kaveh)
The theory of Newton-Okounkov bodies attempts to generalize the correspondence between toric varieties and convex polytopes, to arbitrary varieties (even without presence of a group action). In this generality, one replaces convex polytopes, with convex bodies (i.e. convex compact subsets of Euclidean space). Beside Newton polytopes of toric varieties, many important examples of convex polytopes e.g. moment polytopes (from symplectic geometry), Gelfand-Cetlin polytopes (and their generalization string polytopes) from representation theory fit into this general frame work.
Equivariant Cohomology (Kaveh)
The equivariant cohomology along with the celebrated localization formula provides a strong tool in computing usual cohomology of a geometric object equipped with action of a group. It encompasses several localization theorems in geometry and complex analysis (which have roots in the residue theorem in complex analysis). Surprisingly, in a rich class of examples, known as GKM spaces (named after Goresky, Kottwitz and McPherson), this approach enables one to reduce the description of cohomology, and doing computations in the cohomology, to combinatorics of the so-called GKM graphs. Toric varieties, Grassmannians, flag varieties and many other important examples of varieties are special cases of GKM spaces.
Cryptography and Quantum Computation (Kaveh)
Kaveh has a side interest in applications of algebraic geometry and representation theory in cryptography and quantum computation. Elliptic curves from algebraic geometry are already established as one of the main tools to use for encryption (say of data over internet). A lot of research is going on in regard to security of different encryption schemes as well as finding higher dimensional versions of elliptic curves suitable for cryptography. As for quantum computing, the representation theory (of the unitary group) plays a important role in quantum mechanics and one hopes that applying techniques from representation theory will be fruitful and crucial in development of quantum computing and answering basic questions in this newly emerged computation scheme (in which the future of computing machines may lie).
Symmetries and Dualities in Physics (Sati)
Sati's research is interdisciplinary and lies in the intersection of geometry and mathematical/theoretical physics. Symmetries and various dualities (T-duality, S-duality, U-duality, Pontrjagin duality, Langlands duality) play an important role in his work. Sati has been interested in the appearance of Kac-Moody symmetries, both affine and hyperbolic, in physics. Recent and current projects involve the realization of modular forms in string theory and M-theory, studying Weyl group orbits for discrete forms of exceptional Lie groups, as well as using Bruhat-Tits buildings to describe certain moduli spaces of physical fields.