Faculty of Engineering

# A short insight into mathematics

In this edition of “Tell Us Teacher,” we interviewed Associate Professor Martin SERA from Germany, who specializes in mathematics and mathematical sciences at the Faculty of Engineering’s Department of Mechanical and Electrical Systems Engineering.

When I was nine or ten years old, I realized I liked mathematics very much.

Yes. From my very first year of undergraduate studies, I got interested in Complex Analysis. Within Mathematics, Complex Analysis is an important area that is applied in other areas of Mathematics, Physics, Engineering etc. I started to conduct research properly for my master’s thesis and continued to work in this area for my PhD studies (both at the University of Wuppertal, Germany) and a post doc in Gothenburg, Sweden.

I came to Japan two years ago when I started working at KUAS. Before that, I had had the opportunity to visit Japan many times for research cooperation in addition to some private trips. There are not many researchers in the world working in my research area. Yet, there are many Japanese among them and the field is pretty active here. Therefore, I had been considering working in Japan for a long time.

To be precise, my research interests involve something called Complex Geometry, which is the study of complex manifolds and vector bundles. In this area of study, my focus was on singularities, which I studied through complex spaces and sheaves.

Complex geometry is the study of geometric structures based on complex numbers. Simple examples of such structures are the sphere (that is, the surface of a ball) or the torus (the surface of a doughnut). One easily spots the difference between these two examples: the sphere does not have a hole like the torus. However, this is not easy for more complicated structures that can be found in String theory, for example. The understanding and classification of such general structures (so-called “complex manifolds”) is the main goal of Complex Geometry.

Recently, I am working on a project which is supported by a Kakenhi grant. Smooth Hermitian metrics are crucial tools in Complex Geometry, as they are used to characterize the curvature of complex manifolds. Unfortunately, this does not work for all manifolds. Therefore, my project aims to extend such characterizations by using singular Hermitian metrics in combination with so-called generalized Monge Ampère products.

A Hermitian metric is a tool to measure the length of curves. This is useful as information about the shortest available curve between two points tells us a lot about the geometry. On a sphere, we know that these are on great circles (a cut made by a plane through the centre of a sphere). Thereby, the sphere and torus are structures in space, and we can easily compute the length of curves. Yet, if we consider more complicated structures, then these are not necessarily in space. This is what causes so many problems in string theory. Therefore, we use Hermitian metrics so that we might measure the length of curves on arbitrarily complicated structures.

As is often the case in fundamental research, my current research aims to improve scientific theories and develop new methods of analysis that will have an impact on society.

By starting collaborations with my colleagues in the near future, I am looking forward to contributing to their research with the mentioned theoretical expertise.

There are many possibilities. Let me pick just one example: computer simulations to understand the thermodynamics of materials and their electromagnetic behaviour. These are needed to develop better and more efficient devices such as power modules.

Yes. For me, mathematics is very clear and if you just follow some simple rules, then you can understand very complicated concepts. Unlike literature, you can answer questions very clearly.

Yes, starting this semester I teach lectures on advanced mathematics, for example, about ordinary differential equations.

That is right. My message is that they should not be afraid. Math is always difficult at the beginning and looks unsolvable. Take some time and try to understand, you will get it eventually, which can be very rewarding. I will try to help my students as much as possible.

My hobbies are photography, city trips and their combination.

I very fondly remember Miyajima. There is a torii in the sea that you can walk up to when the tide is low. Nothing like it exists in Germany.

Learn more bout Dr. Martin Sera

**Q:When did you realize you were interested in mathematics and science?**When I was nine or ten years old, I realized I liked mathematics very much.

**Q:I have heard that you studied complex analysis in university. What is that?**Yes. From my very first year of undergraduate studies, I got interested in Complex Analysis. Within Mathematics, Complex Analysis is an important area that is applied in other areas of Mathematics, Physics, Engineering etc. I started to conduct research properly for my master’s thesis and continued to work in this area for my PhD studies (both at the University of Wuppertal, Germany) and a post doc in Gothenburg, Sweden.

**Q:What were your reasons for coming to Japan, and when did you first come?**I came to Japan two years ago when I started working at KUAS. Before that, I had had the opportunity to visit Japan many times for research cooperation in addition to some private trips. There are not many researchers in the world working in my research area. Yet, there are many Japanese among them and the field is pretty active here. Therefore, I had been considering working in Japan for a long time.

**Q:Please tell us about your area of specialization.**To be precise, my research interests involve something called Complex Geometry, which is the study of complex manifolds and vector bundles. In this area of study, my focus was on singularities, which I studied through complex spaces and sheaves.

**Q:What is complex geometry?**Complex geometry is the study of geometric structures based on complex numbers. Simple examples of such structures are the sphere (that is, the surface of a ball) or the torus (the surface of a doughnut). One easily spots the difference between these two examples: the sphere does not have a hole like the torus. However, this is not easy for more complicated structures that can be found in String theory, for example. The understanding and classification of such general structures (so-called “complex manifolds”) is the main goal of Complex Geometry.

**Q:What areas of these studies do you wish to emphasize?**Recently, I am working on a project which is supported by a Kakenhi grant. Smooth Hermitian metrics are crucial tools in Complex Geometry, as they are used to characterize the curvature of complex manifolds. Unfortunately, this does not work for all manifolds. Therefore, my project aims to extend such characterizations by using singular Hermitian metrics in combination with so-called generalized Monge Ampère products.

**Q:What is Hermitian metric?**A Hermitian metric is a tool to measure the length of curves. This is useful as information about the shortest available curve between two points tells us a lot about the geometry. On a sphere, we know that these are on great circles (a cut made by a plane through the centre of a sphere). Thereby, the sphere and torus are structures in space, and we can easily compute the length of curves. Yet, if we consider more complicated structures, then these are not necessarily in space. This is what causes so many problems in string theory. Therefore, we use Hermitian metrics so that we might measure the length of curves on arbitrarily complicated structures.

**Q:In the future, how would you like to give back to society through your current research?**As is often the case in fundamental research, my current research aims to improve scientific theories and develop new methods of analysis that will have an impact on society.

By starting collaborations with my colleagues in the near future, I am looking forward to contributing to their research with the mentioned theoretical expertise.

**Q:What kind of research does joint basic research fall under?**There are many possibilities. Let me pick just one example: computer simulations to understand the thermodynamics of materials and their electromagnetic behaviour. These are needed to develop better and more efficient devices such as power modules.

**Q:Did you decide then that you would spend your life studying math? What attracted you to it?**Yes. For me, mathematics is very clear and if you just follow some simple rules, then you can understand very complicated concepts. Unlike literature, you can answer questions very clearly.

**Q:Dr. Sera, are you going to teach the advanced mathematics classes?**Yes, starting this semester I teach lectures on advanced mathematics, for example, about ordinary differential equations.

**Q:It seems to me that advanced mathematics are very difficult to understand. What message would you like to share with students who might feel the same way?**That is right. My message is that they should not be afraid. Math is always difficult at the beginning and looks unsolvable. Take some time and try to understand, you will get it eventually, which can be very rewarding. I will try to help my students as much as possible.

**Q:If you don’t mind, please tell us a bit about your private life and your hobbies.**My hobbies are photography, city trips and their combination.

**Q:Under COVID-19, it is not so easy to travel, but what places have you enjoyed most in Japan that you have visited so far?**I very fondly remember Miyajima. There is a torii in the sea that you can walk up to when the tide is low. Nothing like it exists in Germany.

Learn more bout Dr. Martin Sera