Professor, Doctor of Science
Address: Department of Mathematics and Statistics, Wakayama Medical University, 580 Mikazura, Wakayama-City, 641-0011, JAPAN
Primary Research Filed: Algebraic Geometry
(in particular, Pathology on Algebraic Varieties in Positive Characteristic)
In Abstract Algebraic Geometry at the present day, we research the algebraic varieties not only over the field of complex numbers but also over an arbitrary field of positive characteristic. Accordingly, we have obtained the Weil-Deligne theorem, which is a monumental achievement in Pure Mathematics. Moreover, we can apply Abstract Algebraic Geometry to Coding Theory and to Cryptology, e.g., Goppa Code, Elliptic Curve Public Key Cryptosystem. Namely, at present, the investigation of algebraic varieties in positive characteristic forms the essential foundation for the modern social infrastructure. On algebraic varieties in positive characteristic, however, there are singular phenomena, which are called "pathological" phenomena and which are regarded as exceptional phenomena from the viewpoint of Classical Algebraic Geometry over the field of complex numbers. Meanwhile, by virtue of recent research findings, they are not rare phenomena so that the remark "exceptional" is improper. To put it differently, another aspect of algebraic varieties is required in the circumstances both of Pure Mathematics and of Technology.
From this point of view, the purpose of my investigation is to propound a new classification of algebraic varieties agree with positive characteristic. For that reason, I also investigate the Tango structures, which have a close relation to pathological phenomena. I expect that they gives a new approach to algebraic varieties over the field of complex numbers.
Articles Published after 2010
Groups of Russell type and Tango structures (pdf) in "Affine Algebraic Geometry: The Russell Festschrift", CRM Proceedings & Lecture Notes 54 (2011).
Examples of non-uniruled surfaces with pre-Tango structures involving non-closed global differential 1-forms (pdf) in "Affine Algebraic Geometry: Conference held at Osaka in 2011", World Scientific (2013).
Secondary Research Field: Mathematics and Complex Systems
(in particular, Indefiniteness of Mathematics, Natural Numbers, Finite and Infinite)
It is not debatable that modern formalistic mathematics bring enormous results, as well as other modern sciences. Indeed, I take the stance of formalism when I study the above-mentioned main research field. The fact remains, however, that the formalism is mere one method for studying mathematics. For example, let us consider Life Science. It seems that their manner and methods are incompatible with formalistic mathematics. Meanwhile, the recent research findings of Complex Systems shadow some beautiful mathematical principle. In other words, it suggests the existence of New Mathematics arising from Life Science, corresponding to present Mathematics arising from Physics.
The purpose of my investigation is restructuring the notion of the natural numbers and of finite and infinite by making positive use of Philosophy and Life Science etc., which currently stay away from Formalistic Mathematics.