# Algebra, Combinatorics, and Geometry

### Bergman kernel and deformation quantization

The asymptotic expansions for the heat kernel and Bergman kernel have many applications. We are interested in the algebraic structures of these asymptotic expansions and related subjects of deformation quantization, Feynman graphs and invariant differential operators on symmetric spaces.

### 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.

### Cryptography and Quantum Computation

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).

### 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, called Flyspeck, gives a complete formal proof of the Kepler conjecture in sphere packings.

### Intersection theory of moduli space of curves

The intersection theory on moduli spaces of curves is connected to KdV hierarchy through the celebrated Witten-Kontsevich theorem. There is much recent progress on Faber's remarkable conjectures about the structure of tautological rings of moduli spaces of curves, especially their Gorensteiness. The subject is closely related to Gromov-Witten invariants, Weil-Petersson volumes, integrable systems and matrix integrals (Eynard-Orantin theory).

### 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.

### Motivic Integration and Representation Theory

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.

### 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.

### Spectral graph theory and random walk

Spectral graph theory is a subfield of graph theory that mainly concerns properties of a graph pertinent to eigenvalues and eigenvectors of its adjacency or Laplacian matrix. In 1990s, Chung and Yau revolutionized spectral graph theory by introducing spectral geometric methods. On the other hand, a random walk on a graph is a special case of a Markov chain.

We are interested in topics like curvature properties of graphs, electric networks and various applications of random walks.

### 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.