Mathematical Statistics (Makoto Aoshima, Kazuyoshi Yata, Nao Ohyauchi)

In real society and the natural world, there exist a great number of phenomena governed by randomness. Humans have been troubled with uncertainty generated by randomness for a long time. Statistics was born as a science to deal with uncertainty. Statistics provides statistical tools to analyze scientific data and it allows decision-making to take uncertainty into account. Nowadays, statistical methods have been applied in various fields of natural science, social science, humanities, and applied science.

Mathematical statistics is the study of statistics from a mathematical standpoint. In this day and age, it is essential for the people engaged in scientific data to have a sufficient knowledge of mathematical statistics and plenty of exercises on statistical analysis.

At the College of Mathematics, University of Tsukuba, we have a substantial amount of lectures and exercises on mathematical statistics. In the lectures, you will study distribution theory and statistical inference that provide the mathematical underpinning for statistical methods. You will gain a thorough grounding in mathematical statistics, together with generic skills in estimation and tests of hypotheses. In the exercises, you will acquire programming knowledge in statistical data analysis. We also offer a substantial amount of seminar-style lessons, helping you assimilate knowledge of data analysis through both theories and applications together with concrete examples.

In the Doctoral Programs in Mathematics, University of Tsukuba, you can learn theories and methodologies to analyze high-dimensional data such as DNA microarray, medical imaging and financial data that appear in modern science. You can discover new knowledge of statistical inference for non-regular cases. We offer a substantial amount of seminar-style lessons on these topics. Regarding those studies, the pursuit of the cutting-edge of modern science is one of the major characteristics of our school.

Our division of mathematics, which is responsible for our doctoral programs, consists of splendid professors doing a variety of research such as the types of research mentioned above, and you can study various areas of mathematics through specially arranged seminars, lectures and workshops on all sorts of mathematics. The principal objective in our master’s program is to enable graduates to contribute short talks at semiannual conferences of the Mathematical Society of Japan, and the objective in our doctoral program is to let them publish papers in academic journals of international renown and to give talks in both domestic and international workshops. All in all, our doctoral programs in mathematics aim to raise people shouldering the burden of the next generation of mathematics or contributing to the development of society with their acquired expertise in mathematics.

  Mathematical Logic (Masahiro Shioya, Kota Takeuchi)

Concerning the problem of “what is valid reasoning”, logic has been known to exist since ancient times as a branch of philosophy. Relatively recently, however, logic has come to be recognized as a branch of mathematics. This goes together with the fact that as mathematics becomes more complicated/abstract (for example, the calculus as developed since the 17th century), an intuitive explanation becomes all the more difficult, and a correct proof based on valid reasoning (for example, ε-δ reasoning formulated in the 19th century) has become important. Finally in 1930, Gödel proved the Completeness Theorem that combines axiomatic systems and mathematical structures, and a study of logic began that embraced modern mathematical logic as a branch of mathematics.

In undergraduate mathematics classes in University of Tsukuba, we have arranged courses for the purpose of a full-fledged study of mathematical logic, one of the few nationwide. First you will understand the Completeness Theorem, studying one of the definite solutions to the problem “what is a correct proof in mathematics?” In seminar-format graduation research, you will stand on the foundation of the Completeness Theorem, looking back down on and compiling your four years of university mathematics from the standpoint of mathematical logic.

At the graduate school, you will conduct research among a variety of specialized development fields from Gödel’s pioneering research, particularly model theory and set theory. Model theory is a branch of mathematical logic that deals with various mathematical structures (groups, rings fields, graphs, etc) from a unified perspective. Set theory, on the other hand, establishes a strong axiomatic system that can develop all modern mathematics, making it a field that researches the notion of infinity within that framework.

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.

  Computational Mathematics (Issei Oikawa, Akira Terui)

Although modern mathematics is a science that is centered on argumentation, with the development of computers from the 20th century to the present, experimental methods also play a major role in the development of mathematics. When solving various problems in mathematics and science and engineering, not only numerical computation but also various algorithms and/or data structures reflecting mathematical models are implemented and used on computers. Computer mathematics is a field that studies theories, systems, and applications that are linked to actual computations by computers from the standpoint of mathematics.

In computer mathematics, the emphasis is on solving a given problem in a realistic range of time and computer resources. Therefore, in addition to theoretical correctness, it is important to discuss the design of efficient algorithms and computational complexity. Furthermore, in some areas of computer mathematics, such as numerical computation, real numbers are often approximated with errors, thus research on numerical analysis is also developing for evaluating and/or certifying the computed results. These research results also contribute to the development of mathematical software for solving various problems on a computer.

At our College of Mathematics, you can learn elements of computer mathematics from the very beginning to the intermediate level. As a sophomore, you can take a course on how to solve mathematical problems using computers, together with rudiments of computer programming with a computer algebra system (CAS) or some other computer software handling mathematics. As a junior, with a focus on the Euclidean algorithm, you will learn elements of mathematics to use in practical computation, such as algorithms and computational complexity for solving problems in mathematics. You will also learn elements of numerical computation and how to solve mathematical problems occurring in the real world using computers.  As a senior, in seminar-style courses, you will study the theory and practice of computer mathematics through program development, deepening our understanding of computer mathematics.

In our Doctoral Programs in Mathematics, you will carry out your research under supervising advisors on various topics, such as “numerical analysis” for computing high-precision approximate solutions of partial differential equations by making use of mathematical theories and "computer algebra" for studying algorithms and software handling algebraic formulas, and for actually running them on computers to solve mathematical problems. We have a substantial computing environment as well.

Our department of mathematics, which is responsible for our doctoral programs, consists of splendid professors doing a variety of research such as the types of research mentioned above, and you can study various areas of mathematics through specially arranged seminars, lectures and workshops on all sorts of mathematics. The principal objective in our master’s program is to enable graduates to contribute short talks at semiannual conferences of the Mathematical Society of Japan, and the objective in our doctoral program is to let them publish papers in academic journals of international renown and to give talks in both domestic and international workshops. All in all, our doctoral programs in mathematics aim to raise people shouldering the burden of the next generation of mathematics or contributing to the development of society with their acquired expertise in mathematics.