Arithmetic of Shimura Varieties

The theory of Shimura varieties lies at the intersection of number theory, algebraic geometry, and the theory of automorphic representations. It plays a key role in the Langlands program as a source of Galois representations attached to automorphic f

Bergman kernel and deformation quantization

The asymptotic expansions for the heat kernel and Bergman kernel have many applications.

Combinatorial and Statistical Designs, Set and Graph Partitions

Combinatorial and Statistical Designs, Set and Graph Partitions

Cryptography and Quantum Computation

Kaveh has a side interest in applications of algebraic geometry and representation theory in cryptography and quantum computation.

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.

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 e

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.

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 r

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.

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).

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.

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.

Sphere Packings and Discrete Geometry

The Kepler conjecture asks what is the densest packing of congruent balls in three-dimensional Euclidean space.

Symmetries and Dualities in Physics

Sati's research is interdisciplinary and lies in the intersection of geometry and mathematical/theoretical physics.