Algebra
From time immemorial, “solving equations” has been one of the most fundamental problems for human beings. I believe that everyone has learned formulas (methods of solving) for solving linear and quadratic equations in junior high and high school. In which case, would you believe that there are formulas for solving cubic equations and above? There are, in fact, known formulas for solving cubic and biquadratic equations (although they are quite complicated). However, when they become fifth-level, no formulas for solving such equations exist. This was proven in the 19th century by the Norwegian mathematician Abel (1802–1829).
Afterwards, the mathematician Galois (1811–1832) discovered “groups,” and used it to investigate the existence or nonexistence of equation-solving formulas. This was the beginnings of modern algebra. By the way, the theorem Galois proved was a very difficult one, that “the necessary and sufficient condition that an equation can be solved algebraically (which is almost the same as saying that a formula for solving the equation exists) is that the Galois group of the equation is solvable.” This is currently studied in the third or fourth year of university. Galois passed away at the young age of 20, so one can understand from this subject just how much of a genius he was.
Starting with the groups Galois discovered, various algebraic structures (algebraic systems) are researched within modern algebra. Among them, the most fundamental structures are groups, rings, and fields. To give a rough explanation,
Field: rational numbers are still rational numbers even after addition, subtraction, multiplication or division between them. Similarly, real numbers are still real even after addition, subtraction, multiplication, or division between them. In other words, the set of rational numbers and the set of real numbers are closed under “addition,” “multiplication,” and their opposite operations “subtraction” and “division.” These kinds of sets are generally called “fields.” As another example, the set of all numbers of the form a+b\sqrt{2} (where a and b are rational numbers) is closed under the four operations above, and become a field.
Ring: the set of all integers that is closed under the three operations “addition/subtraction” and “multiplication.” (Note that integers are not closed under division.) These kinds of sets are generally called “rings.” As another example, the set of all numbers of the form a+b\sqrt{3} (where a and b are integers) is closed under the three operations above, and become a ring.
Group: the set of all real numbers except 0 is closed under the two operations “multiplication/division.” Also, when dealing with all integers, the set is closed under “addition/subtraction.” These kinds of sets, closed under an operation and its reverse operation, are generally called “groups.”
At the College of Mathematics, University of Tsukuba, you will study linear algebra (general matrix and vector theories) as the basics of all mathematics. After that, from your second to third school years you will study group theory and ring theory. From the third through the fourth school year, you will study field theory. One of the features of the curriculum at our College of Mathematics is the “Introduction to Algebra” class in the second year, offering concrete examples for sometimes difficult, abstract concepts of group theory and ring theory designed to allow digestion at a relaxed pace. In the third year, you will learn the basics of field theory while studying more general theories of groups and rings. And in the fourth year, you will combine all of this knowledge by learning the Galois theory mentioned in the beginning.
In the fourth year at the College of Mathematics, as well as the (Master’s and) Doctoral Program in Mathematics of the graduate school, lectures and seminars are offered regarding more complicated algebraic systems: for example, Lie groups and algebraic groups that are groups which have analytic and geometric structures as differentiable manifolds and algebraic varieties, respectively; Lie algebras that are obtained through “approximating” them by vector spaces; algebras that are rings which another ring or field acts on; and quantum groups (which are not groups in fact but are a type of algebras) that were developed in the context of statistical thermodynamics, and Hopf algebras, which generalize them; and vertex operator algebras, possessing two completely different backgrounds, which are physics and finite group theory (a theory of groups consisting of finite elements). In particular, fourth year mathematics courses are organized to allow the study of Lie algebras and algebras, because they are essential in modern mathematics next to groups, rings and fields. In addition, we also offer lectures and seminars on theories related to actions of these algebraic systems on vector spaces (representation theory); number theory that freely uses various techniques of algebra, analysis and geometry; and algebraic geometry, which studies geometric objects defined by algebraic equations using algebraic systems and geometry.
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.