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. 

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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. 

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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.