Algebra produces sophisticated new notions from observing various phenomena in all areas of mathematical sciences; the notions are then incorporated into new theories, and finally, the obtained results are reduced to the original areas. For example, the geometrical notion of smoothness of spaces is clarified by its translation into the smoothness of commutative rings, as Grothendieck defined: a commutative ring *R i*s said to be smooth if any ring homomorphism onto *R* with nilpotent kernel necessarily splits. Such a method of algebra opens the door to drastic developments that were impossible if one stayed within differential geometry, for example.

We *the algebra team at Tsukuba*, roughly divided into three groups presented below, work together to educate leading researchers of the new generation. Energetic people are strongly encouraged to join us. Let's together pursue universals which lay in phenomena and develop a new world!

**Analytic Number Theory (Hiroshi Mikawa, Hajime Kaneko, Shigeki Akiyama）**

Analytic number theory is an area to study various algebraic structure of integers. Integers are naive and fascinating in themselves. Although many problems are described in terms of addition and multiplication, they are often long-standing and the progress on them is very beautiful. Surprising unpredictable applications to other areas have been discovered. Contrary to the adjective ``analytic", any available mathematical tools can be used, and all branches of mathematics could be involved. However stress may be put on analytic methods (complex analysis, harmonic analysis, dynamical system, etc.), information theory, and computer science.

**Hiroshi Mikawa: ** In recent years, expositions on *prime numbers* (and their *distribution*) are pretty easy to find in the literature. On the other hand, somewhat awkward explanations have also become widespread. Please refer to the standard textbook ``Yoichi Motohashi, Analytic Number Theory I, Asakura (2009). You may also check my essay ``Twin Prime Problems" Mathematical Sciences no.651 (2017).

**Hajime Kaneko: **Number systems (such as decimal expansion of real numbers) are naive objects with many attractive unsolved problems. For example, in the decimal expansion of algebraic irrational numbers, the digits from 0 to 9 are expected to appear uniformly, which is a wide open problem. I am researching *uniform distribution theory* including number systems by applying inequalities related to algebraic numbers (*Diophantine inequalities*) and dynamical systems (especially *symbolic dynamical systems*).

**Shigeki Akiyama: **Mathematics seems self-contained and static but this is far from the reality. Actually it develops through conversation with nature and society in depth. Number theory is a discipline that deals with individuality of discrete objects. *Ergodic theory* describes global statistical properties of natural/social phenomena. My interest lies in interaction between these two fields having very different characters. This cross-cultural exchange is really fruitful and extends to adjacent fields such as *dynamical systems*,* aperiodic order*, and *computer science*.

**Representation Theory, Mathematical Physics ****(Daisuke Sagaki, Scott Huai Lei Carnahan)**

Mathematical physics aims to rigorously prove various theories and phenomena discovered in physics from the viewpoint of mathematics, and to return the results to physics again. Representation theory aims to study vector spaces (called representations) on which algebraic systems, such as groups and rings, act linearly. Both fields are actively studied not only from algebra but also from various viewpoints such as analysis and geometry. In addition, mathematical physics and representation theory have an influential relationship with each other; in fact, representations for some special algebraic systems, such as Lie algebras, appear naturally in physics.

**Daisuke: Sagaki:** Sagaki studies the combinatorial representation theory of Lie algebras and quantum groups. In particular, he studies *crystal bases* for representations over quantum groups (which are ``bases" for representations having some good combinatorial properties) and their combinatorial realizations as *crystals*. One of his goals is to obtain combinatorial descriptions of representation-theoretic quantities (such as characters) in terms of these *crystal* *bases* and *crystals*. For a detailed introduction on Sagaki, see ``Tsukuba Suugaku Tsuushin (in Japanese)".

**Scott Huai Lei Cranahan:** Carnahan studies *Moonshine*, which is about phenomena that connect the areas of automorphic forms and representations of finite groups. Many of these phenomena appear to be best understood using concepts from theoretical physics, such as *conformal field theory*. Exploring these connections involves work in vertex algebras, infinite dimensional Lie algebras, and algebraic geometry.

**Ring Theory, Algebraic Geometry and Arithmetic Geometry (Kenichiro Kimura, Ryotaro Sakamoto,Tomoki Mihara, Kazuhhiko Yamaki, Akira Masuoka)**

Algebraic geometry confirms geometric intuition with rigorous arguments of algebra. Grothendieck revolutionized the research area in the 1960s. This greatest mathematician of the 20th century believed: it is not true that given a space, there arises a commutative ring of functions on the space, but spaces and commutative rings exist on an equal footing from the beginning. He re-founded algebraic geometry on the solid basis of ring theory.

**Akira Masuoka:** A group is a space equipped with some operation. Equipped with the corresponding operation a commutative ring turns into a commutative Hopf algebra. Algebra is so generous that we can define the notion of *Hopf* algebra without the commutativity assumption. Akira studies such Hopf algebras that are not necessarily commutative, applying the results to *Galois Theory of differential equations* and *super-geometry*.

Being aware of the famous name of *Teiji Takagi*, many of you may know that algebraic number theory is Japan's forte. The country was also one of the places where in the 1980's there arose the movement of combining number theory with algebraic geometry. The resulting area of research has developed into a core of modern mathematics, which is called *arithmetic geometry*. We have four precious colleagues who are leading the research area.

**Kenichiro Kimura: **The famous Riemann zeta function is the zeta function of the field of rational numbers. More generally, a projective smooth algebraic variety over a number field has its zeta function, which is called the Hasse-Weil zeta function. *Beilinson's conjecture* asserts that the values of the zeta function of a variety over a number field at integral arguments are closely related to the algebraic K-groups of the variety. I have been interested in this conjecture since I started research in mathematics. Related to this, I also work on *algebraic cycles* and *period* *integrals*.

**Ryotaro Sakamoto:** Sakamoto is working on p-adic properties of Selmer groups and L-functions. In particular, he is interested in the Bloch--Kato conjecture, which describes special values of L-functions using arithmetic invariants (Selmer groups). The Bloch--Kato conjecture is a generalization of the Birch and Swinnerton-Dyer conjecture, one of the millennium prize problems, and is being actively studied worldwide.

**Tomoki Mihara:** Mihara is working on* p*-*adic analysis*, *p-adic geometry*, and *p-adic representations* in order to verify general properties of *p*-adic algebras. One of the main features of the study is the cross-disciplinary framework over algebra, geometry, and analysis. Another feature is the deep insight to focus on analogies between the extremely different number systems: real numbers and *p*-adic numbers.

**Kazuhiko Yamaki:** Yamaki has been working on arithmetic aspects of algebraic varieties which are defined over "arithmetic fields" such as number fields and function fields. Through this research, he cotributed much as well to nonarchimedean geometry, which includes analytic geometry over nonarchimedean valued fields and tropical geometry. This geometry has been in great progress recently, and many young mathematicians are very active in the research area. **Remark by the editor: **Prof. Yamaki is a winner of Algebra Prize of the Math. Soc. of Japan, 2021.

Here is the course menu which we, *the algebra team*, provide for you. At our college of mathematics for undergraduates, you can attend lectures together with exercise classes, first on linear algebra through to its advanced topics, and then on algebraic systems such as groups, rings and modules. In the final year, you can attend lectures on more advanced topics such as Galois theory or Lie theory. In addition, you will join seminars to do your graduation research, choosing topics you like.

After proceeding to our graduate school you can attend lectures which discuss advanced number theory or representation theory, using universal methods which are applicable to other areas of mathematics, as well. Experiencing every day mathematical life with your adviser and other members of our algebra team, you will acquire practical research methods 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.