Division of algebra and foundations of mathematics was founded in 1978.
The main topics of research are the following.
Recently, there was a very important advancement in the Langlands program, namely, the proof of the existence of the endoscopic transfer of local and global discrete series representations from the split classical groups to GL(n), mainly due to Arthur. The research in our group is coordinated with that development. For example, we calculate Jacquet modules of the discrete series representations, which will enable the full understanding of the parabolically induced representations from the discrete series (generalized principal series). In the theory of automorphic forms, we develop explicit constructions based on the work of Arthur and Moeglin. In that way, we explicitly construct new series not only of the discrete series representations, but also of the isolated unitary representations.
The main topics of the research in the number theory group are the elliptic curves, modular forms, Diophantine equations, Diophantine approximations, and the application of the number theory in cryptography. We study the structure of the groups attached to the elliptic curves over rational numbers and over algebraic number fields. We examine the relations between arithmetic properties of the Fourier coefficients of the modular forms and arithmetic geometry. We research Diophantine m-tuples and their various generalizations, especially in the ring of integers of the fields of small degree. In the area of Diophantine approximations, we examine the problem of separation of the roots of polynomials and connections with the classifications of transcendental numbers. We also research into applications of the elliptic curves and Diophantine approximations in cryptography.
In this division there is a group which studies vertex-algebra theory and related infinite-dimensional Lie algebras. We study C_2 finite vertex-algebras which are closely related with mathematical physics and quantum group theory. Special emphasis is on the construction of new vertex-algebras, their representations and intertwining operators. The vertex operator theory is also used in constructions of the new combinatorial basis of representations of affine Kac-Moody Lie algebras, and in proving combinatorial identities. We also examine embeddings of finite dimensional Lie algebras and related conformal embeddings of affine vertex algebras.
The main topic of our research is interpretability logics. We consider the following problems on interpretability logics: semantics, completeness, filtration, finite model property, decidability and complexity. We use generalized Veltman model and bisimulations of these models.
The members of Division of algebra and foundations of mathematics are offering the following courses: Algebra, Algebraic number theory, Algebraic curves, Algebraic structures, Number theory, Elementary number theory, Lie algebras, Elliptic curves in cryptography, Cryptography and network security, Set theory, and Mathematical logic.