Geometry
Geometry concerned with the properties of shapes has fascinated people ever since the dawn of history. Some of you may remember the joy you felt when you could find the area of an equilateral triangle inscribed in a unit circle, or prove the congruency of two seemingly different triangles. Geometry, always beloved by people, has developed under the influence of natural sciences, for example physics, as well as other fields of mathematics. Nowadays, geometry deals with abstract shapes that cannot be perceived visually, and it covers a wide range of mathematics, and it has great possibilities.
Modern geometry studied at university is generally classified into topology and differential geometry; the former concerns the connectivity of shapes, and the latter concerns the curvature of shapes. Let us explain the differences between them by using a surface as an example. From the side of topology, one can understand how a surface stretches by using an invariant called the Euler characteristic. One can calculate the Euler characteristic of a surface by approximating the surface by a polyhedron, and by enumerating the numbers of the vertices, the edges, and the faces. From the side of differential geometry, one can grasp how a surface curves by using a function called the Gaussian curvature. One can find the Gaussian curvature at a point in a surface by assigning a neighborhood of the point to a smooth coordinate map, and by appropriately calculating the second partial derivatives of the map. The Euler characteristics and the Gaussian curvatures explained here are defined based on their respective distinct points of view. On the other hand, according to the theorem of Gauss-Bonnet, for a given closed surface, the value of the integral of the Gaussian curvature over the surface is equal to 2π times the Euler characteristic of the surface. In this way, there exists a deep connection between topology and differential geometry.
In our college of mathematics (undergraduate school), you can learn all the basics of modern geometry. From our lectures of topology, starting with the basics of general topology, you can learn algebraic topology such as the theories of homology and homotopy, and the fundamentals of geometric topology. From our lectures of differential geometry, based on the knowledge of vector analysis, you can learn the theory of surfaces, the theory of manifolds, and the fundamentals of differential geometry of manifolds. In our exercise classes of geometry, by solving exercises that deepen your comprehension, you can acquire the methods of calculation and proof, and the abilities to understand a variety of geometric notions. In our seminar-style courses, you can study the details of various topics on geometry while receiving guidance from the teaching staff.
In our doctoral program (graduate school), you can increase your expertise while learning advanced theories of modern geometry. Furthermore, under individual seminar guidance, you can study geometric subjects of your personal interest. In our geometry group, we focus on the research fields outlined below.
In our topology group, we focus on general topology (point-set topology and geometric topology), and low-dimensional topology (low-dimensional manifold theory and knot theory). In general topology, we geometrically consider abstract topological spaces, and all shapes of Euclidean spaces (including spaces with wild topology). In this area, we study topological properties of abstract spaces and complicated shapes based on dimension theory, the theory of dynamics, and various theories related to continuous maps. In low-dimensional topology, we consider knots and handle bodies in 3-dimensional or 4-dimensional manifolds. In this area, we study topological properties of knots and others by using a variety of invariants.
In our differential geometry group, we focus on submanifold theory (including the theory of surfaces) and Riemannian geometry (including geometric analysis). In submanifold theory, we consider surfaces and submanifolds inside specific spaces. In this area, we study various properties of surfaces and submanifolds geometrically, analytically and in terms of representation theory, based on the theories of homogenous spaces, symmetric spaces, and harmonic maps. In Riemannian geometry, we consider Riemannian manifolds and metric spaces with some structures similar to Riemannian manifolds. In this area, we study various structures of Riemannian manifolds and metric spaces metrically, topologically, and analytically.
In the mathematics major, we gather a teaching staff performing such research, leading to all sorts of special mathematics seminars, lectures and workshops and allowing the study of various types of mathematics. By the end of the master’s degree course, the goal is that of contributed talks at such places as the Mathematical Society of Japan; by the end of Ph.D. program, the specific goal is the publication of articles in foreign languages at a level to be published in scholarly publications and academic journals for domestic and foreign workshops. The math major shoulders the burden of the next generation of mathematics, and strives to use mathematics to cultivate talented individuals to contribute to society.