I spent my career (if one should call it this, since I do not consider pursuing scientific research a career but a calling) working on problems that piqued my interest at one point or another due to several factors: 1) the challenge, the complexity of it 2) the impact, if solved, and 3) the social relevance. I cannot say that my path to the topics that I ended up working on was always guided by a “quantitative” assessment of these factors – it was more of an intuitive feel for them. I think this is appropriate when the level of uncertainty is high. Namely, in my view if the problem is “right”, then the probability of success is small…but if success is to be had, then the impact can be big. Those are the kind of problems that I like working on.

A case in point is the interest in ergodic theory that I developed early on, as an undergraduate. It certainly was not a required part of my education. I was browsing books at the local bookstore (yes, in the 1980’s we could still buy science books in bookstores!!!), and this one chapter in a book on theoretical physics spoke about ergodic theory as the basis for statistical mechanics. The mathematics of it seemed challenging, and the idea was fascinating: the robustness of the large-scale, thermodynamic properties of our observable world come about due to the complexity (ergodicity) of individual trajectories in the phase space describing the system. Then I went to Caltech to study dynamical systems, and learned about utilizing geometrical methods to describe their properties. But something was missing – there were systems that we knew possessed invariant sets that were much rougher than the smooth ones covered by the theorems I learned. I did not know that the solution to my question “how can we discover all the invariant sets, no matter whether they are smooth or not” had an answer within ergodic theory. But, still being fascinated by it, and doing some bedtime reading of Mane’s book led me right to the answer. I kept coming back to these types of questions, and ventured into operator theory through the book of Lasota and Mackey…found some more answers to good questions…kept going….and found myself today coupling ideas in operator theory, geometry of dynamical systems, and machine learning.

What helped immensely along the way was finding the problems and needs stemming from applications and matching these problems with my intuition on which field of mathematics could be the base for solving them. Early on, it was fluid mechanics. A beautiful field, with mostly complex nonlinear equations, of massive importance in engineering (e.g. aerospace, and microfluidics for biomedical applications) and large social impact (e.g. climate change studies). In the early 2000’s, I turned my “applied” attention to energy efficiency in buildings. This taught me about needs for data-driven analysis, since we do not have precise equations describing all of the phenomena, and correctly modeling the complexity of interactions might be more important than modeling of individual components…Today I am quite interested in applied problems such as those in network security. This is a wonderfully challenging field where discrete meets continuous…sometimes. Finding how to effectively treat problems such as anomaly detection in this hybrid setting is difficult. And yet, those dynamical systems, ergodic theory and operator theory ideas show up again and help chart the path.

On the institutional side, it was interesting to me to observe how research entities can differ in nature…I was a graduate student at Caltech and a faculty member at Harvard. Those are venerable schools, that built their reputations over a hundred or more years of existence. In contrast, I was a postdoctoral fellow at the Mathematics Institute of the University of Warwick and am currently a Professor at the University of California, Santa Barbara. Those are the schools that rose quickly through the ranks - by focusing tightly on areas of excellence and insisting on bringing in people that lift the average quality of the place. It is not easy to do these things right, but with the right leadership and vision, it is possible. Research is often a solitary discipline. But when high quality researchers get their minds together for a common good of an institution, great things can happen. It is important to lift our heads off of mathematical problems for a moment to realize that potential.

The Figure at right shows the support, in yellow and green, of an exponentially decaying mode of sea ice thickness in the Barents Sea. This has recently been confirmed and emphasized in media by another research group (See https://www.cnn.com/2022/08/11/us/arctic-rapid-warming-climate/index.html). The important part of this discovery is that the reduction of sea ice cover in that zone has been happening exponentially over the last decade. Such a decay, unfortunately, represents a so-called "tipping point". While this is localized to the Barents Sea and so there is still hope that larger effects can be mitigated by upcoming measures, it is entirely possible that the extreme events at lower latitudes that we have experienced this summer are causally related to the tipping point event in the Barents Sea.

Some of the best parts of the professional life in applied mathematics are associated with the people in it. After the COVID-19 years, the conferences are starting again. I recently went to the Fields Institute and had some wonderful dinner conversations with friends and colleagues. It is true, when we go for these trips I feel we never stop working. But again, is it really work when it is also fun? SIAM provides great venues for these kinds of interactions. Another big part of this is working with students and postdoctoral fellows, creating together, finding new avenues, and solving old problems. And then seeing how some of these solutions achieve a life of their own in other capable people’s hands.

It is somewhat odd to write these pages and realize that the same core intuitions that drove my research interests in my 20s still somehow hold up and help me see many problems through their lens. I see their utility in problems as disparate as those in relativistic quantum mechanics (just posted an arXiv paper on it) and artificial intelligence for learning of dynamical systems (I feel there is no artificial intelligence WITHOUT dynamical systems, as causality detection – the key part of intelligence - is impossible without the time arrow) that I worked on with a number of collaborators of over the last decade. But perhaps most satisfying is the fact that this journey enabled me to see my own field, dynamical systems, through a different, operator-theoretic, perspective. The perspective that retains and utilizes all of the beauty of the geometrical approach, while providing an algebraic basis for computation of its key objects from data.

It is great to have such utility of our algorithms and methods. But there is something to be said about the aesthetics of the work that we do as well. I have done music all my life. Practiced, composed, performed. To me, music contains the ultimate abstract charm. I pondered the question: what makes a piece of music interesting? The music that I find appealing has themes that repeat in form, and elements that vary and surprise me. Almost any composition by Bach is a good example. And then I realized that the equation I base a lot of my work on – the Koopman Mode Decomposition that I discovered in a 2005 paper – recognizes both the quasi-periodicity - almost repeatability - and the surprise element contained in the continuous part of the Koopman operator spectrum. Thus my passion for music somehow met my passion for mathematics. Thinking of that makes me happy.