## Algebra

Number theory is the research area in which the properties of `numbers' such as I,2,3,... are studied in the beginning, but it is vastly generalized nowadays. Several reaserch subjects in number theory such as algebraic number theory, Galois theory, the theory of automorphic forms and the theory of p-adic numbers are unified and it has been rapidly developed since the latter half of 20th century as arithmetic geometry. One achievement of arithmetic geometry is the proof of Fermat's last theorem in 1995. Our research group has a long tradition in which the theory of class field theory is completed by Teiji Takagi. In keeping with this tradition, we continue to conduct world level research in number theory, in particular in the area of Galois representations, ramification theory, Langlands correspondence, p-adic Hodge theory and p-adic differential equations.

Algebraic geometry is the research area in which shapes defined by polynomials are studied. In 70's, the theory of Kodaira dimension, Hodge theory, toric geometry and the theory of minimal models are developed. Recently, it has been considered to be important also to study algebraic geometry related to mathematical physics. In our research group, we study on higher dimensional algebraic geometry, derived categories, K3 surfaces, Calabi-Yau varieties, Fano varieties, singularities, motives and algebraic geometry in positive characteristic.

Representation theory is the research area in which symmetry of algebraic structure is studied via linear algebraic method. The effectivity of this area has been recognized and it becomes a big area in mathematics nowadays. Recently a representation theory of D-modules, non-commutative algebras consisting of differential operators, provides a strong tool for studying representation theory. As for mathematical physics such as conformal field theory, the role of the representation theory of infinite dimensional Lie algebras becomes increasingly important. In our research group, we study the representations of classical groups via Young diagrams and combinatorial methods, representation theory of real reductive groups and infinite dimensional Lie algebras, categories of representations and their application to mathematical physics and topology.

Algebra is originally the research area based on operations such as addition, multiplication and so on, but it has been widely developed, being tied with geometry, analysis, mathematical physics and computer science.