  Historically, probability theory has been studied for a long time, and
Kolmogorov, Levy Ito Kiyoshi, and others have mathematically developed modern
probability in conjunction with measurement theory. On the other hand,
commutative algebra and algebraic geometry have historically been the subject
of interdisciplinary research led by Grothendiek. Many Japanese, notably
Matsumura, Hironaka, and Kodaira, have contributed to this field. This paper is
an attempt to focus on the research theme of Professor Sumio Watanabe of Tokyo
Institute of Technology, "Algebraic Geometry and Probability Theory," from my
own perspective. The mathematical theory development starts from Kolmogorov's
axioms, and the proof and introduction of "Probabilistic Algebraic Variety" are
given. Problems in computation and applications, analysis by computational
homology, and unsolved problems in regression problems will be introduced as
applications to statistics.
