Collaborative Research Group (1) (Mathematics Group)

Research subject

Mathematical theory and construction of analytical methods for operator-theoretic data structures

Research Overview

To analyse data based on operator-theoretic data structures, which is the proposal of this research, it is important to study the mathematical properties of the nonlinear dynamical systems that generate the data structures. Two factors are particularly indispensable to the creation of a technology that can withstand practical applications. One is defining and examining the properties of computable metrics for comparing different data structures. The second factor is building a stochastic model that can flexibly consider the noise that appears in real data. In a research study of this type, in addition to the theory of the dynamical system, which has a direct relation with the phenomenon, and the functional analysis and the theory of operator algebra that are working in the background, a number of other mathematical techniques are also applied. With respect to issues such as rigorous mathematical proof of the boundedness of Koopman operators that are actually used to represent data structures and the clustering and classification of various invariants defined for operatortheoretic data structures, we attempt the application of cohomology theory and category theory, which are used in arithmetic geometry and are the areas of specialization of the main collaborators, in addition to the theory of operator algebra, the theory of differential geometry, and optimal transport theory. With this objective, we conduct our research from a global perspective as well as a mathematical perspective, focusing on the research items stated above.

Positioning in the research plan and its necessity

To lay the foundation for operator algebraic data structures, our technique must work well, and it is also essential to have a mathematical basis to prove that the analysis method is appropriate. Diverse mathematical methods are required in this study. Therefore, our pure mathematics and mathematical physics group includes researchers from a wide range of mathematical sciences including algebraic systems such as number theory and arithmetic geometry, geometric systems such as differential geometry and optimal transport, and analysis systems such as partial differential equations and operator algebra. Our group also counts mathematical physicists as members. All group members work towards addressing this issue by seamlessly integrating various mathematical and mathematical scientific methods. In operator-theoretic data analysis, it is extremely important to define the invariants that extract information accurately from the data. Noise is an essential factor when dealing with real data, and hence it is important to incorporate stochastic elements into the current model. Moreover, checking for the boundedness of an operator that is being used is an exercise that tends to be neglected in the field of information systems. Although it initially appears to be a matter of interest only in pure mathematics, it is an important factor in ensuring that the results obtained by numerical calculations converge to the actual model. It is thus important for determining the effective range of this analysis method. To perform a more detailed analysis by classifying these data structures, more advanced mathematical theories such as category theory and cohomology theory are applied. This is indispensable to advancing the research concept and contributes directly to the goal of this area, namely the “construction of theory and technology that contribute to the creation of innovative information utilization methods incorporating mathematical concepts.”