Roughly speaking, Analysis is a major field of mathematics where the main tools are differentiation and integration. Namely the main tool of Analysis is calculus. The idea of calculus is already used in the rigorous formulation of intuitive concepts appearing in our daily life. For example, differentiation which deals with infinitesimal quantities is necessary in the formulation of the constantly changing speed of an object in free-fall. In the definitions of the area/volume of the curved bodies such as circles and spheres, integration appears as the sum of the infinitesimal quantities. Moreover, when dealing with the density of materials and flow of fluids that continuously change depending on position, calculus is needed to describe relations between these infinitesimal quantities. The spirit of modern science, which subdivides objects into components, analyzes them and, as necessary, resolves to analyze the nature of the whole like this, is supported theoretically by Analysis. And in the development of modern science, Analysis offered a great number of models and techniques, achieving great success.

The courses in the College of Mathematics of University of Tsukuba are composed of an efficient curriculum to study the continuously developing field of Analysis from the basics and be able to use it in truly advanced research. From the first school year to the fourth, you will study necessary techniques in common—calculus, vector analysis, theory of differential equations, complex analysis, theory of the Lebesgue integral, probability theory, Fourier analysis, and functional analysis. While studying these, you will attend special seminars from the final semester of the third year through the fourth year. Through these seminars you will be able to learn advanced mathematical theories and the way of mathematical thinking. In University of Tsukuba the courses of Analysis can be classified as follows.

### Theory of Partial Differential Equations (Tomoyuki Kakehi, Tamotsu Kinoshita)

Partial differential equations (PDEs) are basic tools that describe relations between rates of variations of physical quantities, and hence appear in every aspect of science and technology. Accordingly their study forms a significant part of Analysis. In the classes of the College of Mathematics, we have a complete teaching staff concerned with PDEs, and you will learn a lot about theory and practice of PDEs at every opportunity through lectures, exercises, and seminars in both undergraduate and graduate courses.

In the graduate school, you can choose your topic without restraint. Major examples are the hyperbolic equations, the Schroedinger equations, etc. You can learn techniques of functional analysis, Fourier analysis and microlocal analysis, and apply them to the study of the well-posedness of the Cauchy problems, the spectral and scattering theory, etc. Problems with variable coefficients on a domain of the Euclidean space such as the half space, or more generally, on manifolds are of particular interest of the institute. Furthermore, for the last decade, we have been enthusiastic about research also about applications—the numerical analysis, the inverse problems and the wavelets—and have achieved a great success. The class of PDEs in the college provides an environment that allows you to choose and research subjects from a wide variety of themes.

### Algebraic Analysis (Yoshihiro Takeyama, Toshiro Kuwabara)

Algebraic analysis is a field that uses algebraic techniques for the study of Analysis. For example, the study of solution spaces of linear partial differential equations and their transforms by D-modules, that is, modules over the non-commutative ring of partial differential operators, and Sato’s hyperfunctions based on cohomology theory became the starting point of further remarkable development in algebraic analysis.

Algebraic analysis also relates with other fields of mathematics. For example, analytic structure over some holomorphic symplectic manifolds has deep relation with some noncommutative algebras, infinite-dimensional Lie algebras, and vertex operator algebras. The study of such relation introduces new ideas into the field of representation theory.

Moreover, the methods of algebraic analysis also have applications to the study of solvable model in the area of mathematical physics. For the study of solvable model, representation theory of some noncommutative algebras and special functions such as Gauss' hypergeometric functions play important roles. The technique used here also works for a certain area of number theory and probability theory, and some interesting results have been obtained.

### Probability Theory (Yuji Hamana, Ryoki Fukushima, Kouhei Matsuura)

The College of Mathematics at University of Tsukuba offers several lectures and exercises in measure theory and in probability as well. Through these classes students will be trained so that they understand basic concepts and ideas in the modern probability that is based on measure theory. Then they will be lead to two important theorems; the law of large numbers and the central limit theorem. In the third and final years, students are introduced to more advanced fields of probability under the supervision of professors.

In the graduate course, students will study analysis of natural phenomena accompanied by randomness, which is one of main subject of modern probability theory. Examples include: stochastic differential equations that describes a motion of particle subject to random noises, random motion of particles in a space containing random impurities, and the Dirichlet form describing the energy of the random motion of a particle.