Mathematical sciences is a discipline that is based on mathematics and applies mathematics to understanding the real world. Mathematics, as a common language for describing and understanding natural and social phenomena, is a fundamental discipline that supports the entire science. Therefore, mathematical sciences is endowed with "abstraction" and "universality," which abstract and generalize the subject, as well as a "logic" that supports deductive thinking. Based on its characteristics, it is one of the roles of mathematical sciences to understand data-driven and inductive research methods such as artificial intelligence (AI) and the results so that humans can handle them. Therefore, the presence of mathematical sciences is increasing in the promotion of modern digital transformation and cyber-physical systems.

In Japan, the report "Mathematics as deserted science in Japanese S&T policy" (National Institute of Science and Technology Policy (NISTEP) 2006) pointed out that there is a shortage of applied research and human resources responsible for it compared to overseas developed countries. This has had a strong impact on the mathematical community, such as the Mathematical Society of Japan, and there are indications, albeit gradually, of an increase in activities with applied research. In fact, the report "Mathematical Innovation Strategy" (2012/2014) and "Measures Required to Promote Mathematics Innovation" (2016) by the Mathematics Innovation Committee established by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) were published. These reports strongly pointed out that there is a need for "mathematical innovation" to fully utilize the power of mathematics, which is a common language of various sciences, to develop various scientific discoveries and technological inventions, and to create innovations that create new social and economic value. They also pointed out that there is a need to establish a network system (such as consultation desk, information sharing, human resource development, and career path building) to promote it.

In parallel with them, many measures have been taken to contribute to mathematical innovation. The Strategic Basic Research Programs of the Japan Science and Technology Agency (JST) were carried out as research fundings since 2007. In addition, Institute of Mathematics for Industry (IMI), Kyushu University, which is the base for industrial mathematics, was established in 2011, and has achieved results such as efforts on a smart factory in ROHTO Pharmaceutical Co., Ltd. that implements cyber-physical systems. In 2017, the "Advanced Innovation powered by Mathematics Platform (AIMaP)" (2017-2021) was implemented as a network system by institutions involved in mathematical sciences.

Since 2007, these efforts have promoted applied research on mathematical sciences, industry-academia collaboration, and the development of human resources involved in them. However, the interim evaluation of AIMaP pointed out that management such as the horizontal deployment of industry-academia collaboration knowledge and activities for collaboration between companies, other academic societies and the mathematical community were insufficient, although it was evaluated to a certain extent. In other words, human resource development, industry-academia collaboration, and brain circulation in the mathematical sciences continue to be limited in Japan.

In order to contribute to the further development of "Mathematical Innovation" promoted in Japan, this report aims to summarize (1) historical background, (2) mathematical scientific approaches that contribute to real problems, (3) the relationship between the mathematical sciences and industry in Japan, and (4) trends in the United States and Europe, and to identify future issues. In addition, this report focuses on proposing a new ecosystem of mathematical sciences as one of the next steps.

• (in Japanese) Mathematical Sciences and its innovation in Japan: Past, Present, and Future Prospects