Interview with Prof. Dr. Masahiro Yamamoto—Winner of the Mathematics 2022 Best Paper Award

We wish to congratulate Prof. Dr. Masahiro Yamamoto for winning the Mathematics 2022 Best Paper Award.

Name: Masahiro Yamamoto
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan
Interests: inverse problems in mathematical sciences; parameters in evolution equations; determination of shapes of domains from overdetermining data

Masahiro Yamamoto has been a full professor in the Graduate School of Mathematical Sciences of The University of Tokyo, Japan, since 2010, and he is now a Project Professor and Emeritus Professor at the University of Tokyo and a professor at Zonguldak Bulent Ecevit University in Turkey. In 1983, he obtained the degree of Master of Science from the University of Tokyo. He received a Ph.D. from the University of Tokyo for a thesis entitled “Inverse Spectral Problem for Systems of Ordinary Differential Equations” in 1988. His research interests include inverse problems by Carleman estimate; inverse boundary value problems; determination of shapes; fractional partial differential equations and optimal control.

The following is from an interview with Prof. Dr. Masahiro Yamamoto:

1. Could you give a brief introduction of yourself to the readers? Could you introduce your current research direction and provide an update on your progress?
I have been working on control theory, inverse problems for partial differential equations and time fractional partial differential equations. In particular, I have started mathematical research for fractional differential equations, motivated by the analysis of the anomalous diffusion of radioactive nuclide, which was seriously emitted by the Fukushima nuclear disaster in 2011. In order to maintain reasonable shapes of my own theoretical research, I, a mathematician, am confident that it is so important to maintain contact with more applied aspects. Moreover, through such contact, one can find significant topics from the real-world which are awaiting mathematical studies. In other words, in this field, mathematicians must always pursue not only theories but also possible practice of the developed theories.
Holding such views, I have recently been working on the uniqueness and stability of various inverse problems for time-fractional differential equations, as well as for transport, parabolic, hyperbolic, Schrodinger equations.

2. Could you please briefly introduce the main content of the winning paper?
There have been many important works on time-fractional differential equations such as the unique existence of solutions to initial value problems and initial boundary value problems. However, aiming at more consistent convenient applications to inverse problems, I proposed a framework based on the operator theory. This approach is not purely new, but I have integrated existing results and expertise such as the operator theory and the theory of evolution equations, and I have demonstrated the effectiveness of the framework in various problems for time-fractional differential equations.
In the winning paper, I have tried to establish foundations of the theory of time-fractional differential equations, which is directly applicable to inverse problems.

3. Could you describe the difficulties and breakthrough innovations in this research field?
Fractional calculus is similar to classical calculus treating derivatives of natural number orders, but not the same. For example, formulae of integration by parts for fractional derivatives do not allow us to use energy estimates and Carleman estimates, which are strong tools for inverse problems. Such a difference is a main difficulty. Thus, it is definitely a breakthrough that we can completely clarify which properties can still hold and which are essentially different among the properties of parabolic equations and hyperbolic equations. Through such complete clarification, we can also understand the essence of anomalous diffusion.

4. What appealed to you about the journal that made you want to submit your paper? How was your experience submitting to Mathematics?
The wide range of audience is appealing, and the referee procedure was excellent.

5. Which research topics do you think will be of particular interest to the research community in the coming years?

The inverse problems for fractional partial differential equations;
The establishment of the theory of time-fractional differential equations which may generalize the classical theory of partial differential equations.

6. Do you have any advice for aspiring young researchers looking to make a meaningful impact in their respective fields?

Do not aim too much at currently popular research fields. Also, we should keep in mind that we try to find rough gemstones which will open up rich and important research fields;
It may take a long time until your achievements are well recognized and evaluated; please be patient and continue to work;
In order to maintain your mathematical research, please grasp wide scopes not only involving mathematics but also real-world problems. Remember that mathematics looks transcendental and far away from reality, but it can solve real world problems and potentially even save the world.

7. As the winner of this award, is there something you want to express or someone you wish to thank most?
The late Professor Dr. Rudolf Gorenflo. He introduced me to the world of fractional calculus and has supported my research. It is a pity that now I cannot directly convey my gratitude to him, and I am surely indebted to him.

8. What is your opinion of the open access model of publishing?
Open access is fine, especially for mathematics if used relevantly, because mathematical achievements are fundamental and so are more similar to common properties, especially among natural sciences.