Research interests in mathematics:
Diophantine geometry: Contributions of global geometric approach to solving equations and particularly Diophantine equations. (Or any geometric treatment of arithmetic, algebraic or differential equations.)
Modular forms: Arithmetic structures involved in forms over moduli spaces. (Or in finding arithmetic formulations for every piece of mathematics we are faced to.
Arakelov theory: Mathematical situations where, the role of one piece of a mathematical framework is locally replaced by mathematics coming from other areas, as in Arakelov theory.
Deformation of algebras: Coordinatizing geometric objects and their deformations by translating them to the language of algebras and their deformations. This lead to formulation of Arithmetic Teichmuller Theory.
Knots and three-manifolds: Analogy between prime numbers and knots in three manifolds. I have developed a Chebotarev density theorem for knots in three manifolds. I am trying to develop an analogue of Langlands program for 3-manifolds.
Research interests in mathematics education:
- Modeling of mind, while learning and doing mathematics.
- Modeling of interactions between branches of science and branches of mathematics.
- Modeling of educational systems.
- Psychology of problem solving.
- Understanding the roles which mathematical education can play in cognitive structures of students from elementary school to high-school and university.
- Developing a university curriculum for cognitive approach to mathematics.
Research interests in philosophy of mathematics, science and education:
- Development of a cognitive philosophy of mathematics.
- Development a cognitive philosophy of education.
- Development an Islamic philosophy of science according to Islamic anthropology.
- Categorization of cognitive anthropology in the minds of Islamic philosophers.