Fundamental Science and EngineeringCreative Science and EngineeringAdvanced Science and EngineeringⅠ FeaturesⅡ History and ProfileⅢ RequirementsCONTENTS◆ AlgebraAlgebra contains the following research themes: algebraic number theory, diophantine equation, automorphic function theory, commutative algebra, homological algebra, arithmetical geometry, geometric code theory, algebraic geometry, and algebraic combinatorics.◆ GeometryGeometry consists of the two pillars, “Analysis on Manifolds” and “Topology.”The first pillar, Analysis on Manifolds, has made remarkable progress under the influence of relativity and quantum field theory, and has developed into a huge field which can be called the core of modern mathematics. The research themes in this area consist of differential geometry, complex and hyperbolic geometry, Lie groups and representation, algebraic analysis, and integrable systems.The other pillar, Topology, is an active area centering on the theory of three-dimensional manifolds and dynamical systems. The research themes in this area are (a) geometry of knots, (b) dynamical systems, (c) theory of three-dimensional hyperbolic manifolds, and (d) applied singularity theory.◆ AnalysisIn the research area of Analysis, studies are conducted mainly on functional analysis, real analysis, and theory of functional equations.In functional analysis, studies on application of the theory of function algebras to complex analysis, function spaces appearing in probability theory, etc. are conducted. In real analysis, studies on various function spaces such as real Hardy spaces, application of interpolation theory to partial differential equations, etc. are conducted.In the theory of functional equations, partial differential equations, especially non-linear equations, are the main topic of study, and a wide variety of problems such as nonlinear evolution equations, optimal control problems, hyperbolic equations, parabolic equations, elliptic equations, fluid equation systems, and variational problems are studied. Therefore, students in this field need to select their research theme from a wide range of options with a clear awareness of the issues. A variety of methods including orthodox differential and integral calculus, functional analysis, theory of nonlinear semigroups, variational problems, mapping degree, viscosity solutions, Fourier analysis, bifurcation theory, and computer assisted proof are used as study methods. Accordingly, even among instructors working on similar themes, the research methods and means employed may vary widely. ◆ Mathematics of PhenomenaThis research area aims to examine various phenomena which appear in natural science and 40
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