News
The 2025 Abel Prize is awarded to Professor Masaki Kashiwara in RIMS Kyoto University for his contributions to algebraic analysis. Congratulations ! One result among Professor Kashiwara's many achievements is the rationality of the roots of the Bernstein-Sato polynomial, which are essentially the same as the real log canonical thresholds that play an important role in singular learning theory. Professor Kashiwara's mathematics has relations to many mathematical fields, which also connects to today's artificial intelligence studies.
What algebraic analysis is :
When we talk about "analysis," it is generally understood as the branch of mathematics that rigorously deals with limit operations such as differentiation and integration, and this understanding is not incorrect. However, "algebraic analysis" studies the algebraic properties that appear in analysis, independently of those limit operations. For example, if we define X as the operation of "multiplying by x" and ∂ as the operation of "differentiating with respect to x," the commutation relation ∂X−X∂=1 holds. Algebraic analysis investigates the properties of sets that satisfy such algebraic relations. The module over a ring of differential operators is called a D-module. Algebraic analysis is sometimes referred to as the "representation theory of D-modules." Research topics include questions like "What algebraic characteristics do solvable nonlinear differential equations have?" or "Does there exist a differential operator that has the complex power f(x)^z of a polynomial f(x) as an eigenvector?" Addressing these questions is known to extend mathematical world and to provide solutions to many unresolved problems in mathematics. In this subpage, we introduce the relation between algebraic analysis and statistical learning theory.
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(d^2/dw^2)(w^(2z+2))=(2z+2)(2z+1) w^(2z).
M.F. Atiyah, Resolution of singularities and division of distributions. communications on pure and applied mathematics, pp.145-150, 1970.
This is an example.
By using Hironaka resolution map, any analytic function can be made a normal crossing function in any local neighborhood.
This is an example of a zeta function.
The integration of a normal crossing function can be concretely calculated, which is equal to a meromorphic function whose largest pole is the minus real log canonical threshold.
This is an example of Mellin transform.
The inverse Mellin transform of the meromorphic function is equal to a product of t and log(t). This relation is universally employed in the zeta function theory.
This is an example of Laplace transform.
When n tends to infinity, the main term of the Laplace transform is equal to
( log n)^(m-1)/n^{lambda} . This result will be continued to the marginal likelihood using empirical process theory.