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Algebra
Number theory is the research area which begins with the study of simple `numbers' such as 1,2,3,..., but has since been vastly generalized. Several major areas of number theory--such as algebraic number theory and Galois theory, which study symmetries arising from extensions of number systems; the theory of automorphic forms, rooted in the moduli of elliptic curves; and the theory of p-adic numbers, which involves completing number systems with respect to a prime number p--have been unified and developed rapidly since the latter half of the 20th century under the framework of arithmetic geometry. One achievement of arithmetic geometry is the proof of Fermat's last theorem in 1995. In our research group, we carry on a tradition that dates back to Teiji Takagi's class field theory. 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. The field has grown by adopting methods from complex analysis, differential geometry, and topology. 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 the symmetry of algebraic structures is studied via linear algebraic methods. The effectiveness of this area has been recognized, and it has become a major field in mathematics today. The representation theory of real and p-adic reductive groups continues to develop, guided in part by the Langlands program in automorphic representation theory. In addition to algebraic methods, geometric representation theory, which employs geometric techniques, has also been rapidly advancing. Furthermore, in mathematical physics, infinite-dimensional Lie algebras and their representations play an important role. In our research group, we study the representation theory of real and p-adic reductive groups, categories of representations, and modular representations of reductive groups.
Ring theory is the research area in which rings, a generalization of number systems, are studied. The theory of commutative rings can be regarded as the local theory of algebraic geometry. In contrast, the study of noncommutative rings focuses on analyzing the structure of module categories and derived categories. In that context, categorical methods such as representation-theoretic approaches using quivers and tilting theory are employed, with applications in the theory of cluster algebras. In our research group, we study commutative ring theory in positive characteristic and the representation theory of orders.
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.
May 2025
