## Research

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#### Promoting the Interdisciplinary Development of Science and Engineering by Pioneering New Mathematical Perspectives

Institute for Mathematical Science (WISE Research Initiative)

Hideo Kozono

【Commentary: Waseda Research Institute for Science and Engineering, University Research Initiative】

Waseda University established Vision 150 to meet a variety of objectives for the 150th anniversary of its founding (in 2032), with 13 core strategies in development. One of the core strategies that has been laid out is “promotion of original research and enhancement of the ability to delivery this internationally.” In order to drive this core strategy forward in the field of science and technology, seven research initiatives for responding to social challenges were newly established with the Waseda Research Institute for Science and Engineering (WISE) at its core. A research institute will be established for each initiative in April 2018 with the aim to pursue globally first-class research. In order to pursue interdisciplinary research and to strengthen international research in each field of focus, this group of research institutes, called the cluster institutes, will work cooperatively. There is also a plan to launch the “Waseda Earth Restoration School” (WERS) as a place where visions of future research can be formulated.

The seven cluster institutes established for each research initiative

Commemorative symposium for the establishment of the research initiatives, where research representatives and members of the seven initiatives gathered (December 22, 2017)

We asked Professor Hideo Kozono of the Faculty of Science and Engineering to talk about the aspirations and vision of the soon-to-be-launched Institute for Mathematical Science, with a focus on the mathematical science research initiative (one of the seven initiatives).

##### Building a Base for Delivering Information by Gathering People

Waseda University’s Faculty of Science and Engineering has maintained a tradition of integrating science and engineering in advanced research, on which the faculty’s education is based. Mathematics is the foundation that supports the “science” component mentioned above. Waseda University has mathematics professors with a wide range of expertise, including algebra, geometry, mathematical analysis, foundations of mathematics, as well as applied mathematics such as mathematical modeling and analysis, computational science, and statistics. Applied mathematics is a field of research that analyzes natural, physical, and social phenomena using mathematics. It serves as a bridge between mathematics and engineering, as well as between mathematics and social science. This is characterized by interdisciplinary exchanges surrounding education and research between faculty members of mathematics and applied mathematics, within the wider field of mathematical science.

“The Faculty of Science and Engineering at our university has traditionally promoted cooperation and collaboration between researchers in various fields, with the aim of pursuing interdisciplinary and empirical education and research. As a cornerstone of this, Waseda’s rich tradition of mathematical science, including mathematics and applied mathematics, has served as the focal point that connects empirical and theoretical research. Although mathematics does not directly impact the world the way engineering does, the elaborate theories of mathematics are essential in the development of engineering.” (Prof. Kozono)

Research related to mathematical science does not require large-scale experimentation equipment the way particle physics and engineering do. To put it simply, all you need for math is a brain and a pencil, and in recent years, desktop computers and laptops are capable of performing extraordinary things with computer simulations of various phenomena studied in applied mathematics. The most important factor in this field is people—to educate people is to train them, and to build a network of people is to cultivate the ability to disseminate information around the world and engage in international exchange. This provides the traction needed to advance research capabilities.

"People naturally gather in places where you have strong researchers. We will gradually create a research base where people grow and will continue to invite more talented people. I believe that the fact that this university is capable of building a research base for mathematical science is a testament to the excellent research we have conducted by attracting ambitious pioneers who share our unique vision of science and engineering. In addition, it is also important to communicate the results of our research and to engage in internal exchange. Having discussions with researchers around the world in settings such as international conferences and research societies is essential for the advancement of research in mathematics, and in order to conduct leading-edge research, we need to set up bases worldwide to constantly keep up with research trends. (Prof. Kozono)

Discussions with the leading researchers around the world through international conferences on cutting-edge themes is essential for the development of mathematical science.

##### Aiming for Leading-Edge Global Research Through Three Initiatives

In the first term (five-year period) of the Institute for Mathematical Science, the plan is to focus on the current era, in which massive volumes of data known as “big data” are being handled, and to take an approach that applies mathematical methods to face complex and unsolved natural and social phenomena. In addition to producing academic results for each field of specialty, we aim to produce research results that can contribute to the field of engineering and to the real world.

“As major objectives of our research institution, we aim to 1) create new fields of research in mathematics and new mathematical concepts, and 2) explore mathematical perspectives and methodologies for the challenges facing society. Rather than going straight into engineering and interdisciplinary research, the first term will focus on advancing research within the field of mathematical science, and to produce research results that will later serve as the foundation of interdisciplinary research.” (Prof. Kozono)

Research efforts will start by organizing the three core research fields of nonlinear analysis, computational science, and mathematical statistics, each organized into research teams (see Figure 1). The Nonlinear Analysis Team will focus on fundamental mathematical research centered on nonlinear partial differential equations, and applied mathematical research based on nonlinear analysis methods in areas such as mathematical fluid dynamics, fluid engineering, and nonlinear dynamical systems. The Computational Science Research Team will aim to decipher innovative computational science technology by using computers to help decipher complex phenomena, focusing on fields such as numerical computation with results verification, nonlinear mathematical models, and applied integrable systems. The Mathematical Statistics Team will aim to develop optimal inference theories and analyses of phenomena by focusing on areas such as financial engineering and data science, as well as by constructing feedback for modeling and mathematical statistics theories with knowledge gained in actual on-site applications.

“In each of these fields, there are researchers engaging in some of the world’s most cutting-edge research. For example, my expertise is in nonlinear partial differential equations, a field that is essential in analyzing and describing natural phenomena. It is a field of research that is showing remarkable advancements in recent years, and our university boasts the most advanced research capabilities in Japan. Every year, we hold international conferences on Navier-Stokes equations, which is my area of expertise. This event attracts researchers from all over the world. Similarly, with the other two initiatives, we plan to first focus on the areas where our greatest strengths lie.” (Prof. Kozono)

Figure 1: The three research initiatives of the Institute for Mathematical Science

In the field of computational science, the university has a history of remarkable achievements by Professor Shinichi Oishi and other researchers, who, in the research field verified numerical computation, have advanced the Core Research for Evolutional Science and Technology (CREST) at Japan Science and Technology Agency (JST) for over 10 years. Other research themes include contraction algorithms used in image compression and frequency domain filtering with the Fourier transform. Today, streaming movies on demand on the internet has become the standard. The development of information contraction algorithms, which take only the essential parts of data to create accurate simulations by performing less calculations, has made it possible to watch movies without interruption on tablets and smartphones, and to accurately predict the trajectory of typhoons in weather forecasts.

As its name suggests, mathematical statistics is a field of research that analyzes phenomena with optimal statistical inference. In finance, the fundation of derivatives is derived from the probability theory and statistics. Research into the theory that serves as the basis for financial engineering, which enables the design of these derivatives and market analysis, as well as calculation methods that serve as the basis for the currently trending big data analysis, are pillars of this field. Also within the scope of objectives is the training of data analysis specialists, or data scientists, who have the knowledge of methods to derive meaningful information out of massive amounts of data.

##### Collaboration Between Theoretical and Experimental Research

Navier-Stokes equations, which Professor Kozono is an expert in, are equations that underlie the nonlinear behavior of fluids such as water and wind, and is known as one of the complex problems that have not been solved mathematically to this day. When dealing with Navier-Stokes equations, models can be drawn from initial conditions as long as the phenomena are relatively simple. However, there are many unresolved questions, such as just how complex a phenomenon can be drawn, and whether the solutions exist in eternity, or if it is restricted to a finite amount of time, just how far it extends. This is one of the Millennium Prize Problems in mathematics, offering a million-dollar prize for anyone who solves it.

A kick-off symposium held on March 6, 2018

“In our attempts to solve difficult mathematical problems like this, we can gain clues through coordination with the Computational Science Research Team’s numerical analysis research. In numerical analysis, phenomena are analyzed using numeric operations, while we model phenomena and express them logically using numerical formulas. Numerical analysis tries to analyze the actual phenomena themselves, which limits it to analyzing them in their finite states. However, we analyze abstract models using numerical formulas, making it possible to express infinite states.” (Prof. Kozono)

For research surrounding computational science, the theoretical research of nonlinear analysis offers crucial clues for development. Conversely, for nonlinear analysis, the empirical research of computational science offers such hints. The objective is to pursue interdisciplinary research within the scope of mathematical science, and then further develop into interdisciplinary research within the various fields of engineering. On March 6, 2018, a kick-off symposium was held for the Institute for Mathematical Science, to which well-known researchers from around the world were invited. Please look forward to the remarkable results produced by some of the most-cutting edge research in the world.

###### Related Links

Mathematical Science Research Initiative

Waseda Research Institute for Science and Engineering (WISE)

Faculty of Science and Engineering, Waseda University