Applications of Algebraic Geometry to Group Theory

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.

Applications of Analysis to Number Theory

The analogy between number fields and fields of meromorphic functions on complex algebraic curves has been known for more than a century.

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

Geometry

One of the principal goals of algebraic geometry is to classify all algebraic varieties up to isomorphism or birational equivalence.

Lattice Polytopes and Toric Varieties

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.

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.

Polynomials with Integer Coefficients

Polynomials with integer and rational coefficients appear in many areas of modern mathematics.

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.