The Start

This is Yulin Pan, an assistant professor in the department of Naval Architecture and Marine Engineering at the University of Michigan. Throughout my education I was in engineering departments, receiving all my degrees in engineering. I do have a minor degree in mathematics from MIT, but to achieve that I only needed to take three courses from the math department there. And for me, I took all three courses from the very applied side of mathematics. In summary, my college education of mathematics is highly limited to engineering mathematics, numerical methods and some applied asymptotic analysis to nonlinear problems. 

In spite of the lack of training in rigorous mathematics, I do keep a keen interest in mathematics over my research career. During my Ph.D., I got interested in the field of wave turbulence, based on which I later developed my Ph.D. thesis. The field of wave turbulence is quite technical in terms of the derivation, which is usually considered as part of theoretical classical physics. I would also mention that when I started research in wave turbulence, I had no background in theoretical mechanics as well. For example, I had no idea what a Hamiltonian system is. Starting everything from scratch, it took me months and years to understand and master the major techniques on the physics side of the wave turbulence theory. However, up to that point I have never encountered anything about the pure side of mathematics, in particular about the rigorous analysis of PDE.

After starting my faculty position at the University of Michigan, I had more chances to work with researchers from different fields. I have also been fortunate enough to work with several pure mathematicians, who I knew from the "Simons collaboration on wave turbulence". At the beginning, I found it very difficult to understand what mathematicians are working on, like some people described that "they live in a different world as physicists". However, with more time spent with mathematicians and some devotion of my efforts, many concepts became more natural to me and I started to be able to construct some simple results in the pure analysis field. While I always have a desire to learn mathematics systematically, at my career stage (with many duties/interests in different applied areas) this hope seems not very realistic. Even if I can collect some amount of time to read mathematics books, without using the knowledge in research I would soon forget them. Hence, I think that the best way for me to keep exploring the pure math side is to learn things while working with my collaborators in mathematics. This is what the blog is about!

More specifically, this blog is established for three purposes: (1) It will serve as my own study notes. I believe that presenting things in writing will no doubt strengthen my own understanding of the knowledge; (2) It will motivate myself to keep learning new things, in some sense, to continue this blog; (3) It can help (at least I hope) other researchers with a similar engineering/physics background as me who hope to understand something on the pure side of analysis. Before starting the technical stuff, I would also like to warn the readers that many materials on the blog may not be that rigorous, and perhaps erroneous in mathematicians' eyes. This is of course limited by my current background, which I have to keep learning to improve (and in the process to further modify the blog). On the other hand, however, I think such materials may be more accessible to engineers/physicists, because with my background I know exactly where the gap is for understanding, and these gaps will be emphasized in presenting with some sacrifice of rigor. So with regards to the third point above, I do hope that this blog has some unique advantages.

With this, I wish myself enjoy the journey, and we can get started.

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