# Algebra, Combinatorics, and Geometry

### Combinatorial and Statistical Designs, Set and Graph Partitions

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.

### Congruences Between Modular Forms

Modular forms have a natural action by Hecke operators, and the eigenvalues of these operators are known to be algebraic integers. This makes it possible to discuss congruences modulo a prime between the Hecke eigenvalues of modular forms. Such congruences between modular forms have been used for a wide variety of arithmetic applications, with Wiles's proof of Fermat's last theorem being an especially notable advance. Wang-Erickson is especially interested in the case where one of these modular forms is an Eisenstein series. This case has special connections to arithmetic, namely, to ideal class groups and Iwasawa theory. This is a long-established idea in algebraic number theory, and it has also seen a resurgence of new developments.

### Cryptography and Quantum Computation

Kaveh has a side interest in applications of algebraic geometry and representation theory in cryptography and quantum computation but he has not done any research in this direction in a long time.

### Equivariant Cohomology

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.

### Formal Theorem Proving

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.

In collaboration with a large international research group, Hales has completed one of the largest formal proof projects ever attempted. The project gives a complete formal proof of the Kepler conjecture in sphere packings.

### Lie theory, Representation theory

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.

### Moduli Spaces of Galois Representations

Representations of Galois groups of number fields encode symmetries among algebraic numbers. Wang-Erickson is interested in moduli spaces where each point corresponds to a p-adic Galois representation, and families in the space correspond to p-adically interpolated p-adic Galois representations. His projects have included efforts to refine the technologies used to understand and measure these spaces, which especially involve deformation theory and homotopical algebra. A particular goal is to identifying close relationships between arithmetic phenomena and deformations of Galois representations.

### Newton-Okounkov Bodies

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.

### Non-Commutative Algebra and Geometry

Ion maintains an active interest in several topics in non-commutative algebra/geometry: deformation quantization, (finite dimensional) Hopf algebras, graded rings, and categories.

### Principal bundles and the Langlands Program

The Langlands Program is a series of far-reaching conjectures, which first emerged in number theory but then extended to many areas such as algebraic geometry, representation theory, and mathematical physics. The geometric Langlands program is a statement about equivalence of certain categories of moduli spaces of principal bundles on algebraic curves. The research of Fedorov is about the Langlands duality for Hitchin systems, the Langlands program with ramifications, and motivic classes of moduli spaces occurring in Langlands program. Fedorov is also interested in applying the philosophy of Langlands program to classical questions of algebraic geometry such as studying principal bundles over local rings.

### Sphere Packings and Discrete Geometry

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.