The world of mathematics is strictly governed by logic—if it can be proved, any approach is acceptable. Attracted by that freedom, singularities are my daily challenge.
Algebraic geometry is a discipline that bridges algebra, which deals with symbolic expressions, and geometry, which deals with figures. As Professor Takagi describes it, “algebraic geometry is the field of study in which figures are analyzed algebraically.” It is mainly concerned with investigating the nature of algebraic varieties, special types of figures that can be represented by polynomials.
One thorny problem that arises in the study of algebraic varieties is that of the so-called singularity. Singularities are points on algebraic varieties that are “sharp or twisted, but not smooth.” The properties of an algebraic variety without a singularity can be analyzed by differentiation, but this doesn’t work for those with singularities.
Professor Takagi studies algebraic varieties in the unique world of “characteristic p,” where p is a prime number, in which adding 1 to itself p times yields a result of zero.
”Algebraic varieties are classically considered in the framework of complex numbers. In the world of characteristic p, however, a “pathological” phenomenon happens, something that isn’t seen in the case of complex numbers, and this makes analysis of figures difficult. The world of characteristic p might seem artificial, but it actually has real life implications, as it’s used in encrypted communications on the Internet.”
Algebraic geometry is something of a Japanese specialty. Three Japanese mathematicians have been awarded the Fields Medal, sometimes referred to as the Nobel Prize of mathematics, and all three studied algebraic geometry. Kunihiko Kodaira, the first Japanese to win the Fields Medal in 1954, graduated from and was later appointed to a professorship in the Department of Mathematics at the University of Tokyo. The influence of Kodaira still resounds throughout the department today.
Professor Takagi says that the freedom of mathematics appeals to him. “Ultimately, if you can prove something, you can choose any approach you wish, no matter how outrageous. The exciting part is that your thinking is completely unfettered.”
He was fascinated by this freedom of thought since he was an elementary school student. “What I liked doing at that time was the kind of math question called a word problem. Although they might have been contrived, I enjoyed thinking about real events in terms of numbers.”
When he was in high school, his interest in the application of mathematics to real problems grew, even to the extent of reading an introductory text on operations research. When it came to choosing his path in higher education, he had a hard time deciding between the Department of Mathematics and the Department of Mathematical Engineering and Information Physics, which teaches applied mathematics.
“I was interested in applied mathematics, but I was also concerned about whether I could keep up with abstract discussions of pure mathematics. At that time, I sought advice from Professor Junkichi Satsuma, who was in charge of a class on ordinary differential equations, and he recommended that I enter the Department of Mathematics. ‘In the Department of Mathematical Engineering and Information Physics, students learn the basics of mathematics at first, and the basics are the same in both pure and applied mathematics. If you think you can’t keep up with the pure mathematics in the Department of Mathematics, you can go on to the Department of Mathematical Engineering and Information Physics in graduate school. If you do so, what you learned in the Department of Mathematics will be useful.’ I followed that advice and entered the Department of Mathematics, where I soon found my niche in algebraic geometry.”
“The attraction of the Department of Mathematics is the excellence of its students,” Professor Takagi says.
“We often think of mathematics as a solitary activity, but it's very important to discuss your work with others in order to get your ideas into shape. There are so many wonderful people around me, it’s an excellent working environment.”
Interview and text: Masatsugu Kayahara
Photography: Junichi Kaizuka
Originally published in The School of Science Brochure 2020