**Remembering **

Hermann

Flaschka

Hermann

Flaschka

Hermann Flaschka was a distinguished member of the faculty at the University of Arizona from 1972 to 2018. These were times of fundamental change, not only for mathematics and our department, but also for society at large. Hermann’s mathematical career is intertwined with the emergence of nonlinear science, an interdisciplinary enterprise centered on the fusion between modern and classical approaches to dynamics, which was motivated by technological advances in computation. This was a watershed moment for mathematics in the mid-20th century and Hermann was very much a part of that moment.

His pioneering work in nonlinear lattices, soliton theory, Backlund transformations, theta function solutions of integrable dispersive partial differential equations, and related algebro-geometric approaches to perturbation theory broke down walls between traditionally unrelated areas of pure and applied mathematics and built scientific bridges between fields. Not only have these bridges lasted, but they continue to evolve to this day.

Hermann’s contributions to nonlinear science were multifaceted. One of the most visible examples was his leadership in founding Physica D: Nonlinear Phenomena, together with Joe Ford, Alan Newell and Al Scott. This was the first, and for many years the only, journal in the new discipline. It is noteworthy to recall the early vision in some of what they wrote in 1980, in the inaugural volume:

"The editors are aware that the term nonlinear attempts to define a field of study by what it is not. Nevertheless there appears to be in every quantitative area of science an implicit understanding of the word, perhaps both because there is a common appreciation that many phenomena cannot be analyzed by linear models and that there is much common ground between the role and effect of nonlinearity in otherwise unrelated areas of science. … Our principal aim is to publish reports of experiments, techniques, and ideas which - although they may be derived and explained in the context of a particular field - advance the understanding of nonlinear phenomena in general."

Such a reflection may sound commonplace nowadays, in an era of interdisciplinarity strongly influenced by the pioneers of nonlinear science; but, one should not underestimate how strikingly novel these words were 40 years ago.

Hermann’s pioneering spirit extended to novel educational endeavors related to the Department and the University. He was centrally involved in the early development of the Graduate Interdisciplinary Program in Applied Mathematics. His graduate text, *Principles of Analysis**,* was specifically designed for students in the Program. His aim was to guide readers to an intuitively rigorous but practical understanding of the increasingly central contribution of modern analysis to scientific applications. This was a great success. The text, which has been a part of the Program’s core courses for over 30 years, has received repeated encomiums from former students, who continue to refer to it and make use of it in their own curricula at other institutions.

Hermann also gave back to the Department by serving as its Head from 1996 to 2001. He worked to maintain the strengths of our academic profile in applied mathematics, number theory, geometry and mathematics education. This reflected his broad perspective on the connectedness and unity of all mathematics. He also strongly supported new curriculum developments in undergraduate mathematics education and the founding of our Undergraduate Math Center. His efforts laid the groundwork for the Department’s first successful securing of a broad (VIGRE) infrastructure grant.

In 1995, Hermann received the Norbert Wiener Prize in Applied Mathematics, in recognition of his “deep and original contributions to our understanding of completely integrable systems.” This award is made every three years and is jointly sponsored by the *American Mathematical Society* and the *Society for Industrial and Applied Mathematics*. In his own (characteristically humble) words upon receiving this Wiener Prize, Hermann reminded us that curiosity drives discovery in mathematics:

"I thank the Societies for complimenting my work—but the prize really recognizes a rich area more than anything I might have added to it. It is difficult to ask a question about the Toda lattice that has a boring answer. Professor Morikazu Toda has made a wonderful contribution to mathematics. His “Toda chain” reaches back to the great classic studies of Jacobi and Stieltjes, while constantly making contact with the newest ideas. It led me to think about such diverse topics as discrete shock waves, modulation equations, Riemann surfaces, theta divisors, toric varieties, momentum maps, semisimple and affine Lie algebras, Szegö’s theorem, and measurable rearrangements; as an interested spectator, I have seen the Toda chain appear in many more mathematical settings too numerous to try to list. I have enjoyed using integrable systems as the vehicle to learn a little about interesting fields with which I was not familiar."

Hermann was the first to establish and understand the integrability of the system of nonlinear oscillators known as the *Toda lattice*. Since then, the Toda chain has played a role that extends well beyond integrable systems theory per se. Indeed, one could supplement Hermann’s list of “ideas” with current and inter-related developments in representation theory, symplectic and Poisson geometry, and stochastic dynamics. But Hermann will forever be associated with this fascinating area within nonlinear analysis and geometry through the fundamental change of variables, known as the *Flaschka transformation*, that put him on the path towards deep insights.

A memorial event was held in Hermann’s honor on April 17, 2021. Due to the COVID-19 pandemic, this was virtual. More than 90 former colleagues, students and family members came together to commemorate Hermann’s legacy.

Any commentary/memories about Hermann that you would like to communicate would be welcome. Please send these to remflaschka@math.arizona.edu We will be happy to share those comments with his family and colleagues in the Department.