# 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. 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 Abstracts

We intend to build a large library of mathematical theorems which are stated in both a human-readable and machine-readable way. This library, called Formal Abstracts, will compass almost all areas of mathematics and all definitions appearing in a theorem statement will be implemented in and checked by a proof assistant. This allows for a semantic search of the library: you can search for theorems which are equivalent to the theorem you searched for, even if it is formulated differently. Other applications include analysis via machine learning, translation of theorems, and exploration of mathematical concepts. We do not include formally verified proofs of the theorems, because giving a formal proof of a theorem takes substantially more effort than writing out a proof on paper.

### 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.

### Homotopy Type Theory

Type theory is a collection of formal languages that can be used as a foundation to mathematics which can also model computation. In the late 2000s, Vladimir Voevodsky and others found a homotopical interpretation of type theory by constructing a model in simplicial sets. This gave rise to a new variant of type theory, called homotopy type theory, which combines the geometry of homotopy theory with the logic of type theory. In homotopy type theory you can do homotopy theory synthetically, without interpreting the basic concepts like "space" and "path." Furthermore, computer proof assistants are able to verify these synthetic proofs. Van Doorn has made advances in synthetic homotopy theory by constructing the Serre spectral sequence and Atiyah-Hirzebruch spectral sequence synthetically in homotopy type theory, and these constructions were verified in the proof assistant Lean.

### 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.

### 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.

### 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.