In my research I develop mathematical methods to understand complex systems, such as, neural networks, living cells, or ecosystems. These systems consist of many constituents that interact in a complicated manner with each other. The interactions between the constituents are often described in terms of large networks and scientists are interested to understand how the structure of these networks influences the dynamics of complex systems. For example, the human brain consists of about 100 billion neurons, each of which interact through electrical signals with about ten thousand other neurons. To understand how the network architecture of the brain affects its information processing and learning properties, we need to develop novel mathematical methods and that is what I am contributing to. My main contributions are in describing the spectral properties of networks but explaining what that means would take us a bit too far.
Scientific progress is often driven by novel advances in mathematics. For example, Isaac Newton developed differential and integral calculus to write down the Newton laws of motion that form the basics of classical mechanics.If I think about important contemporary challenges, then I would say that these include, among others, understanding how the human brain works, what causes an economic crisis, or how to interpret the human genome. Progress in these research areas is hard because from a mathematical point of view they involve many constituents interacting with each other in a complicated manner leading to specific dynamics. Unfortunately, so far little mathematical progress has been made in understanding complex systems, but I hope this will change in the future with the advent of more advanced mathematical methods.
Mathematics can provide quantitative answers to problems that the society is facing, which go beyond the qualitative insights that scientists have developed through experience. For example, virologists know that a lockdown helps to contain the spread of a virus. However, if you want to know how many days of lockdown are required, then you need to talk to a mathematician that can predict the outcome of a pandemic. Mathematics provides us also with a framework to think about problems and formulate precise questions. For example, if you want to know whether a drug has a certain side-effect, say the occurrence of a heart disease, then you need to rule out the possibility of naturally occurring heart diseases. Mathematicians can formulate such a question into a specific mathematics problem. Another interesting feature of mathematics is that it is an abstract language that establishes connections between problems that appeared a priori to be unrelated. That is why a mathematician can collaborate at the same time with a biologist and an economist.
The best way to develop a new collaboration is to visit a research institute for several months. If you discuss with someone daily and exchange ideas about mathematics, then naturally you will start working together on a problem. Currently it is unfortunately not possible to travel because of travel restrictions due to the pandemic. This has made it difficult to develop new collaborations as establishing contacts through email or video chat is difficult. The development of collaborations can also be stimulated by government funding. Certain funding schemes require collaborations between researchers from different research areas or countries. Such funding schemes push researchers to get out of their comfort zone and establish collaborations that they otherwise would not have considered.
In my opinion, the quality of research in China has significantly improved over the last decade. Research is mainly about developing new ideas that no one has thought about before or solving critical problems that researchers considered very hard to solve. Nowadays, I can often find such contributions in papers from Chinese research groups, and I think that the impact of research of Chinese mathematicians will increase further in the future.
About half of the students in my class are from China. I like very much to interact with them as they are motivated, hardworking, and interested in mathematics. Sometimes I wished they would be a little bit less shy and ask more questions during the lectures. But, I must admit that I was also shy to ask questions when I was a student…