The Journal of Geometr ic Analysis

Volume 8, Number 5, 1998

Recollections of Fred Almgren By Frank Morgan

While visiting Princeton for the 1997-98 year I have had the privilege of using Fred's office and helping Jean to go through his papers. It has been moving to see the magnitude of Fred's helpful correspondence with students and colleagues, some of which Jean and I returned to them with an invitation to contribute short recollections. This article gives some selections from their responses. For more recollections see Mackenzie's article on "Fred Almgren: Lover of Mathematics, Family, and Life's Adventures" (Notices AMS 44, 1997, 1102-1106) and the memorial issue of Experimental Mathematics (Vol. 6, 1997, 1-12).

Wendell Fleming

Fred Almgren and I arrived at Brown at the same time (fall 1958), he as a beginning graduate student and I as a new assistant professor. Fred took my real analysis course. While it was clear from the start that Fred had an excellent intuition and original ideas, he was not yet trained to think like a mathematician.

Fred's PhD thesis was a brilliant one. In his excellent article, Brian White mentioned the curious episode in which the Brown Graduate School hesitated to accept it, on the grounds that the thesis had already been accepted by the journal Topology. A very firm stand by Herb Federer persuaded the Dean to withdraw his objection.

My thesis advisor L. C. Young had expressed the need for a kind of Morse theory in terms of multivariable calculus of variations. Soon after the thesis, Fred provided such a theory in terms of what he called varifolds. Varifolds are very similar to Young's generalized surfaces, but the name varifold is much more appealing.

A lot was happening in geometric measure theory during the years 1958-62 when Fred was at Brown. He and I ate lunch regularly in the cafeteria. During these lunches, Fred found out more or less all I knew and of course I learned a great deal from him in return. It was clear even then that the regularity problem for varifolds which minimize k-dimensional area (or some other geometric variational integral) was going to prove extremely difficult in codimension more than one. I have the greatest admiration for Fred's determination and persistence in wrestling with these regularity problems through ten years, culminating in his massive three-volume regularity proof.

After Fred left Brown we saw each other only occasionally. One such occasion was in summer 1965 when we were both visiting De Giorgi at the Scuola Normale in Pisa (see photographs).

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I still have very pleasant memories of excursions with Fred to Lucca and Siena, which are interesting towns nearby. He knew how to enjoy life during the times when he was not immersed in mathematics.

We were very pleased to have Fred as an honored guest at the 1988 Brown Commencement, when he received a Distinguished Graduate School Alumnus Award. Each year this award is given to two or three of Brown's most distinguished former PhD graduates.

Jim Eells

I cherish the memory of Fred's and my times together. There was much hilarity! Once we tried serious collaboration (on the Variational Scene, as I called it in one of my letters) - - but, to our mutual surprise, without success: our styles were too different.

Brian White

Some random memories about Fred:

1. Though he was, in general, a great lecturer, when I first knew him he would, when teaching grad classes, speak extremely quietly. The first day, students would be scattered throughout the room. The second day, everyone would be in the first two rows.

If Jean was present, she would tell him to speak more loudly. He would stare at her blankly for a few seconds, then begin speaking again at what he thought was an exaggerated volume (but was actually normal). However, after a sentence or two he would be right back to barely audible.

2. Some of Fred's advice:

(a) When giving a lecture, never go overtime. Of course many people are aware of this principle. If I remember correctly, Fred always abided by it.

(b) When you're working on a problem, you should focus most of your effort on the part where things are least likely to work out.

I would guess that most successful mathematicians follow this principle consciously or uncon- sciously. However, from time to time there's a case of some mathematician wasting years of effort that could have been avoided by following Fred's advice.

3. Of course to do the work that he did, Fred must have had tremendous powers of concentration. One time I got to see that first hand.

It was on one of those Thursday evenings when he invited students and sometimes a visiting lecturer to his house for dinner. As usual, after several glasses of wine, he was at the head of the table telling his anecdotes. On this particular evening, Karen (who was two or three years old and sitting next to him) started tugging on his arm, perhaps because she wanted to interrupt. But Fred kept on talking. Karen then crawled into his lap and started poking his cheek. Fred kept on. By the end of his anecdote, she had crawled onto his head, was batting his face, and saying something like "la-la-lee-la-lee-la" over and over again. Fred never gave any indication that he noticed a distraction.

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Pertti Mattila

I first met Fred in the ICM-78 in Helsinki. At that time I had been working in geometric measure theory for about five years, and I was rather frustrated. I got a good start from Bill Ziemer and John Brothers by spending the academic year 1972-73 as a graduate student in Bloomington, but after that i had been pretty alone in this field in Helsinki. I did not believe much in what I was doing and I was looking for a new area. Talking to Fred changed this completely. He was extremely kind and encouraging. He made me believe that my work had some value and that it had connections to other things.

Joel Hass

My first encounter with Fred was when I was a graduate student, working on some problems in topology and minimal sm'face theory. I wrote Fred a letter asking about the status of the "Bridge Principle." He was extremely generous in his response, sending me a very encouraging letter and showing great interest in my efforts. He even offered to write up a proof of the result I needed.

Mort Gurtin

We were graduate students together, he in math and I in applied math, and we shared a fence. We would often walk home together and talk about our work; unfortunately, at that time I did not have the background to really understand what he said; little did I know that his ideas would later form a basis for much of my work. He was a fine person and a great mathematician.

Penny Smith

Everyone knows Fred Almgren was a great mathematician, but he was also a great soul, kind and supportive to young talent. In order to solve an elasticity problem, I reinvented some geometric measure theory (with R. Mallek Madoni) and the new tool of either/or regularity. Fred and Jean very kindly encouraged this work, gave me copies of their unpublished book on GMT, and invited me to their GMT workshops.

Joe Fu

I first met Fred Almgren as a junior at Princeton. Despite my strong attraction to mathematics, at that time I was struggling with doubts about its purpose and meaning. Around then Fred and Jean's Scientific American article on soap films appeared. Here was a kind of mathematics that promised both technical depth and direct application to physical reality. For my independent work Fred directed me toward crystal growth, in particular the problem of finding a satisfying explanation of the intricate symmetry of snowflakes. He gave me papers to read, and suggested established mathematics to learn about, but above all he displayed for me the confidence that all these things fit together and could make some sense out of some directly perceptible aspect of the real world. Typically I would enter his office in a state of excitement over some insight, and Fred would deftly

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find the right counterexample to put my achievement in perspective. He encouraged me to complete a senior thesis of which I was (and to some extent still am) egregiously proud.

Fred led me to one of the great treasures of my life, namely the hope that in mathematical striving our aesthetic interest can help to connect us to the God-given world.

Bob Kohn

Fred determined the "initial conditions" for my career when he pointed me toward optimal design. It was a radical suggestion, one whose mathematical content neither he nor I fully appreciated at the time. Roughly, he said "Engineers have to choose the shapes of things; that means they must be solving some sort of geometry problem when they optimize a structure. Let's find out what kind of geometry problem this is, then study it." We were both starting from scratch, of course, so the first task was to learn about mechanics of solids. Here too he had some radical advice: "Forget about linearized theories; let's concentrate on the real thing." Natural enough from a specialist in minimal surfaces, where the linearized problem is Laplace's equation and the most fundamental estimate (monotonicity) comes from a fully nonlinear construction. Natural, but far from conventional wisdom in the mechanics community.

Fred's suggestions set me on the trajectory that has become my career: Fritz John's theory of deformations with uniformly small strain; quasiconvexity and relaxation of nonconvex variational problems; optimal bounds on the effective moduli of composites; martensitic phase transformation and shape memory materials; and so on. I feel lucky to have been the beneficiary of his remarkable courage and vision.

Seiki Nishikawa

I first met Professor Almgren on September 19, 1977, when I attended the Japan-United States Seminar on Minimal Submanifolds in Tokyo. After the conference, while visiting several temples and shrines with his wife Jean and me, he was enchanted by "Shi-Shi," the sculpture of a mythical lion guarding sacred places. A few months later I sent him pictures of a replica of "Shi-Shi" from my father, who was an antique dealer. He wrote, "Jean and I have made full size sketches of Shi-Shi and placed them at the entrance stairway of our home. We also have inquired of metallurgist friends about the durability of bronze and received assurances that Shi-Shi would survive several centuries !"

When I met Professor Almgren next in Arcata, California, in 1984, he told me, "We keep the Shi-Shi in a prominent place in our living room!" Soon after I returned from the United States that year, my father died [at the same age as Professor Ahngren].

These memories of Professor Almgren with my father are still vivid and precious in my heart.

Aaron Nung Kwan Yip

It was the combination of the words - - Geometric Measure Theory - - that attracted my attention as (I thought) I knew something about each word. During the first class of this graduate course, a professor energetically described some open problems. The one I remembered vividly was the "opaque square problem." This and other questions can be simply stated and yet the answers often involve high power mathematical techniques, with each step of the proofs relying on interesting

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geometric intuitions. I was so amazed by this subject and the professor's enthusiasm for mathematics, I said to myself, "This is it: I am going to be a student of THIS professor." That was still my first week of class at Princeton.

Throughout my five years of interaction with Professor Fred Almgren, his encouragement and persistence were important factors for the completion of my thesis. When I worried too much about my progress, he would say, "If you believe in the axiom of hard work, I can assure your graduation. All it takes is just my signature." If I became too distracted by other interests, his friendly advice would be, "You have only 24 hours a day."

Professor Almgren's geometric intuition and precision were extraordinary. When I came to him with complicated ideas and unpolished proofs, within minutes he would ask, "How do you do this step?" already understanding the most crucial obstacle of the proofs. Through such encounters, I began to learn how to simplify problems and focus on the essential difficulties.

Professor Almgren encouraged me to take every opportunity to give presentations. He would listen to my rehearsals and make critical suggestions. Once he even brought his video camera to record the lectures. His favorite question was, "What do you want the audience to get out of your talk?" He helped me think over the purpose of the work, its difficulties and the key steps of the proofs. I changed from feeling nervous about giving talks to enjoying it.

I feel grateful to have such an advisor who not only guided me through times of uncertainty but also made mathematics an enjoyable subject. I was saddened by his passing away. He still had a lot to offer to the younger generations.

Steve Strogatz

[Fred's] work on soap films and minimal surfaces was famous, even among us undergraduates. One of the topics [which he suggested] - - the geometry of supercoiled DNA molecules - - seemed like exactly what I was looking for. This was 1979, several years before the topology of DNA became a hot topic, but Fred sensed that there were many fascinating mathematical issues here. He gave me a few papers to look up and sent me on my way.

The following fall, we began to work together, meeting about once a week in his office. Fred had the idea that a long, thin molecule like DNA could easily get itself tangled, unless evolution had developed some mechanisms to prevent this. But what does it mean for a curve to be "tangled?" He suggested that tangling must be more than a topological notion - - it must have something to do with how close some parts of the curve pass by each other, and probably it also involves elasticity (stiffer curves should be less prone to tangling) and friction (an infinitely slippery shoelace would never get tangled). The goal would be to develop a theory of tangling, and maybe even to prove that DNA could not get tangled, given its geometrical, elastic, and frictional parameters.

As the months slipped away, I didn't succeed in finding the fight definition of tangling, and certainly didn't have any theorems about tangled curves. This made me embarrassed to talk to Fred, and I started seeing him less often.

One day in mid-January, I screwed up my courage and entered Fred's office, to ask his permission to give up the quest for a theory of tangling. I remember that my heart was beating fast. But instead of the disapproval I had feared, Fred was gentle and completely understanding. He told me that he now realized that tangling was much too hard for a senior thesis p rob lem- - he said it was more appropriate for a Ph.D. thesis, or perhaps a research program for a professional mathematician. (That made me feel better!) And when I described some alternative problems that were both biologically important

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and solvable by an undergraduate, he beamed at me with that unforgettable Almgren smile. From then on, Fred became my cheerleader as I entered into a collaboration with Abe Worcel, a molecular biologist at Princeton and an expert on DNA supercoiling. That collaboration was a thrilling research experience for an undergraduate, ultimately leading to a new model for the higher-order structure of chromatin, the mixture of DNA and protein found in chromosomes (A. Worcel, S. Strogatz, and D. Riley, Structure of chromatin and the linking number of DNA, Proc. Natl. Acad. Sci., USA 78, (1981), 1461-1465).

Department of Mathematics, Williams College, Williamstown, MA 01267 e-mail: Frank.Morgan @ williams.edu