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The Beauty, Power and Future of Mathematics I Am Thrilled and at The The Beauty, Power and Future of Mathematics I am thrilled and at the same time humble to be honored by the Doctor Honoris Causa of the University Politehnica of Bucharest. Romania is a country with a very excellent tradition of mathematics at the very best. When I was in high school in Hong Kong, I read with care an elementary book by Popa with lucid explanation and colorful illustrations of basic mathematical concepts. During my undergraduate studies at Sir George Williams University in Montreal, a professor introduced me to Barbu’s works on nonlinear differential and integral equations. I should also mention that he introduced me to the book on linear partial differential operators by Hörmander. When I was a graduate student at the University of Toronto working under the supervision of Professor Atkinson, I learned Fredholm operators from him and the works of Foias and Voiculescu from the operator theory group. I was also influenced by the school of harmonic analysts led by E. M. Stein at Princeton. When I was a visiting professor at the University of California at Irvine, I learned from Martin Schechter of the works of Foias on nonlinear partial differential equations. This education path of mine points clearly to the influence of Romanian mathematics on me, the importance of knowing in depth different but related fields and my life‐long works on pseudo‐differential operators and related topics in mathematics and quantum physics. I chose to specialize in mathematics when I was in grade 10 at Kowloon Technical School in Hong Kong dated back to 1967, not least inspired by an inspiring teacher. In him you could see the characteristics of a dedicated mathematician. They are the search for truth, beauty and most important of all, understanding. Understanding is the most important thing in life. We need understanding between parents and their children, between husbands and wives, between different cultures, between countries, between people from different parts of the world. The occasion today has mandated me to talk about understanding in mathematics. What is understanding in mathematics? Knowing isolated facts, memorizing names of mathematical objects and notable mathematicians, knowing how to follow the proof of a theorem, quoting big and grand general theory without knowing what it means to claim that a mathematical statement is true or false is not understanding in mathematics. Mathematics understanding means knowing the general picture of which isolated facts are special cases, the nuts and bolts of a mathematical theory, important examples upon which the mathematical theories are built. Understanding in mathematics is very satisfactory and pleasing as all colleagues in mathematics know. This is due to the fact that seeing order from seemingly chaotic and unrelated facts is a long process of frustration and despair to revelation and complete understanding. Mathematics is usually and mistakenly considered to have two separate but related parts: pure mathematics and applied mathematics. To me, there is only one enterprise known as mathematics. Pure mathematics with no regards to societal needs and other branches of science is a meaningless game. Applied mathematics without any real and rigorous mathematics built into it is an empirical science, which can be considered experimental mathematics at best. It is only through understanding that new and beautiful mathematics can be created and then turned into power through science and technology transfer. Thus, beauty and power are both necessary ingredients of the enterprise that is dubbed mathematics. The current trends in mathematics include many disciplines unheard of when I was a student of mathematics in the 1960s and 1970s. To wit, I can name mathematical biology, the mathematics of climate change, disease modeling, financial mathematics, mathematical imaging and the list goes on. I can see that the great majority of bright young students nowadays go into these new areas and the traditional areas like algebra, analysis, topology and geometry have appeared to wither away. Needless to say, experts can easily see that the traditional branches of mathematics underpin the modern and fashionable applied areas listed above. Underlying the applied areas listed above are partial differential equations, dynamical systems, topology, Fourier analysis and many much more abstract mathematics such as algebraic geometry, arithmetical geometry, number theory and others. The most challenging intellectual pursuit today is perhaps an understanding of who we are, what the universe is or the multiverses are? Here I am talking about theoretical physics as it stands today. Theoretical physics today is an elitist study embracing nearly all branches of pure mathematics encapsulated by the modern developments of mathematics. Here I only mention the Atiyah‐ Singer index theorem, the Langlands program and Alain Connes’ noncommutative geometry. The index theorem blends analysis and topology in a miraculous way. The Langlands program entails the study of number fields, representations of Lie groups, harmonic analysis, automorphic functions and quantization. And the noncommutative geometry of Alain Connes finds applications in quantum physics and sheds some light on the resolution of the Riemann hypothesis. Fields that I have not touched on such as theoretical computer science and discrete mathematics should not belittled. They are now very much connected with artificial intelligence and robotics, and as such, are very much connected with the question of whether conscience should be part of a new quantum mechanics of which the quantum mechanics that we know of today is only a special case. On a personal note, I have mentioned that my works in mathematics have been focused on pseudo‐differential operators and its ramifications. To me, it is mathematics that embodies both beauty and power. The beauty comes from its connections with many branches of mathematics like partial differential equations, harmonic analysis, operator theory and algebras, differential and integral geometry, which are deep and broad. The power of pseudo‐ differential operators first comes from giving explicit solutions of partial differential equations. To give an example, the Green’s functions are the kernels of pseudo‐differential operators. Pseudo‐differential operators and their relatives are versions of quantization, namely, the geneses of turning observables in classical mechanics into observables in quantum mechanics. Much of the filters in signal analysis and image processing can be modeled by pseudo‐ differential operators and such related operators as Weyl transforms and localization operators. As filters, pseudo‐differential operators and related operators can be and are being used in imaging in biomedical, geophysical and atmospherical sciences. So, the subject of pseudo‐differential operators is a subject on big applications per se. The mathematics of the 21st century as I can see these days is interdisciplinary in nature that requires collaborations among people across the globe. Mathematics, after all, is a basic science that aims at understanding, finding new links and forming new perspectives. Most mathematicians have at most a general understanding of other sciences and engineering. Then how can a mathematician excel at technology transfer like practical and visionary engineers do? In fact, very few people can do two things well. Those who can do one thing well, like mathematics, should do it extremely well, but at the same time, maintain dialogs with chemists, physicists, biologists, medical doctors, engineers and other professionals in order to have a good overview of what the societal needs are and then develop their expertise accordingly. In this way, applied scientists learn from novel mathematics to provide innovative products, and mathematicians have a compass to guide them to produce new, beautiful and powerful mathematics. Last but not least, we, the mathematicians, have an obligation to educate young men and women to become the next generations of mathematicians. The public and the government of a country should understand the importance of mathematics in a knowledge‐based or science‐ based economy. This is our responsibility because research in mathematics requires funding from the government who generates these resources from tax‐payers. We do hope that decent and secure jobs be made available to young and brilliant highly qualified personnel. Each country has its own system of priorities, funding, education, jobs and opportunities, but the bottom line is that a young man or woman after nearly twenty years of education to become a mathematician is entitled to a decent job. How to achieve this is what the higher administration of a university and the government should think about. A society with a huge mathematically‐ literate population is a prosperous society. This brings us to the topic of outreach, which is very popular nowadays in universities in North America. We should not live in an ivory tower isolated from the society at large. We should go about finding partnerships with the industrial sectors. We should explain what we do in precise, concise, clear, understandable and interesting ways to the public, to the students from kindergarten to high schools to universities so that we can attract some if not all of the best students to mathematics. In the province where my university, York University, is situated, the university curriculum, the high school curriculum and the elementary school curriculum have been and are being scrutinized carefully in order to meet the needs of today. Programs termed mathematics education are created in order to produce highly qualified teachers to teach our children. The hope is to help people understand the importance and the pleasure of doing mathematics, which helps scientists do better science and engineers with better technology transfer. Thank you. .
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