Fun with simple analysis problems I: the rest of the story

In an earlier post with the same title (and without the subtitle) I introduced some thoughts that were triggered by this simple problem:

Suppose that

$|\frac{df}{dx}(x)| \leq \lambda |f(x)|$                    (1)

for all $x$, that $f$ is continuous and differentiable, and that $f(0) = 0$.

Prove that $f(x) = 0$ everywhere.

In that post (which you can find here  Fun with simple analysis problems I ),  I started by presenting three solutions and then generalized and explored further.

What I did not reveal in that post, was that writing it, gave me an idea for a more advanced problem. Not too long afterwards, Laramie Paxton joined my group and I gave him this problem to work on for his dissertation. We collaborated in solving the problem, since that is how I mentor all my students — their dissertations are collaborations with me. This resulted in a paper we wrote together: A Singular Integral Measure for $C^{1,1}$ and $C^1$ Boundaries that can be found here.

Laramie Paxton arrived at WSU quite naive with respect to analysis, having completed an online masters in mathematics that did not give him a good foundation in analysis. But he very quickly he adopted habits that led to rapid progress. He started by studying intensely the summer before arriving and passing the qualifying exam on his first try.  Then he took my challenging undergraduate analysis course (I used Fleming’s Functions of Several Variables), pushed through courses in advanced analysis, and geometric measure theory, and worked on applications in image analysis (generating papers he actually led) and finished his dissertation, all in the space of two years. After a year of postdoc, he landed the job he is about to start, at Marian University in Wisconsin. I believe that both the University and Laramie are lucky to have each other.

In general, I believe that small universities are good places to be nowadays, but from everything I hear, this place is better than good — it is perfect for Laramie’s talents and skills. (In addition to his impressively growing mathematical skills, he was already phenomenally skilled in logistics and organization which can be seen in his highly effective help in making the events listed here, from April 2017 to July 2018, a reality.)

A major point of both the original post on the problem and this present post, is that the paper with Laramie, as well as the results in the first post, flowed from taking time to think about a simple analysis problem that would usually be viewed as a not-too-hard exercise, not worthy of more thought than it takes to find one solution.

While I am sure that there are other undiscovered aspects of the problem that launched these two posts and Laramie’s dissertation problem, I believe that what has been explored illustrates why it makes sense to treat simple problems as invitations to playful exploration and creativity.

The Colors of Memory and Wisdom

Reading Zeyn Joukhadar’s novel, The Map of Salt and Stars, has taught me once again that fiction can be more truthful than non-fiction. And even though the vast majority of fiction is to reading what junk food is to eating, there are novels that inspire even the pickiest of readers, with the highest (or most peculiar) standards for what is inspiring or illuminating.

What we know is a such a minuscule particle in a vast infinite universe of what could be known, that the skeptical inquirer is doomed to a rather poorly illuminated reflection of tiny bits of what is known. But skepticism is not the only option. Those willing to use all the tools at the disposal of an aware, enlightened human being, can embark on a voyage filled with light and a rich, ever-unfolding life.

In the living experience and fable woven together in Zeyn’s novel, the human spirit and the Infinite meet in an explosion of life and color and light and dark, moving us to a place where we can see and feel far beyond the narrow confines of overly rigorous, reductionistic thinking and experience. The deeper truths in the stories, sometimes stated very plainly, other times only seen in the wholistic experience of the story, are profound, demanding a stillness and quietness before they open to our view.

The overwhelming energy moving through the story, illuminating my response, was one of light and color and memory and feeling, reinforced by the synesthesia of Nour, the little girl through which we see the story. While a few might consider Nour’s synesthesia to be an unnecessary device, I found it completely natural, even essential. For me it was a door anyone can enter if they will but take the time to listen to the music and feel the color in the stillness and quietness, to see the light shining through the broken places, to experience the infinity between the ticks and tocs of a clock.

When I taught 7th and 8th grade science during graduate school, I used to take my students out into nature with notebooks in hand and ask them to see and feel and hear, and then write. Most had a very difficult time finding the stillness necessary to do this. I know they had a hard time connecting with my descriptions of what happened on my walkabouts, when I moved into that living path mode of seeing and hearing. It was also my first time trying to describe this mode and inspire others to try it for themselves. After those experiences I often simply shared the insights I found in that state, realizing that it is a very hard thing to actually move somebody into that mode of being.

Nevertheless, over the years, I remained as hopeful as I was when I tried guiding the students, that this mode of seeing and hearing and feeling is open to anyone willing to listen to stillness.

Lately, though, I had started losing hope in the power of words to actually enlighten or inspire or even prompt others to begin a journey. I could find lots of examples that supported my growing doubt. But when I finished this book I was struck by the strong sense that I was wrong. Some written words are still very powerful, inspiring and healing, opening readers to that infinity I began to experience so many years ago in my walkabouts in the desert and later in the forests. Immersing myself in this story, I find again, in yet another form, that stillness containing infinity.

I was also reminded that when you have passed through extreme crisis, you learn what is important and what is not, you learn to choose the simple life and connections with those that love you and those that can benefit from your simple help. You remember that so many things in our surroundings, considered so important, cannot compare with the song of an insect, or connection with a friend, or peace of encompassing sunshine. You realize there is nothing to prove, that simple things contain everything you need because they are doors to infinity. You see helping those who struggle, easing the path of those who have very little and seek simply to live in peace, is an integral part of finding and sharing the depth and beauty we are wired to seek, to explore. One cannot truly have depth and beauty without the healing and compassion.

What remains for me, as I write these words in the afterglow of the story, is a sense of living stillness and remembering and color, and the deep peace when we remember the intense richness of knowing what is important.

Cultures of Creativity and Innovation

Books reliably inspiring enthusiastic conversations are books worthy of close attention. When Beata read and recommended Daniel Coyle’s book, “The Culture Code”, encouraging me by reading bits and pieces of it to me, it was not long before I knew that I had been introduced to just such a book. Soon I was buying copies and giving them away. Over the course of 2-3 months I gave away a bunch of copies and organized an evening in the top floor of the Monarch Motel in Moscow, Idaho devoted to discussion of the book.

The present article is part of my evolving reaction to the stories and theories in Coyle’s book, prompted by the reading-inspired, barehanded combat with those ideas.

The stories of remarkable environments for creativity and productivity, as well as the stories of studies and research aimed at understanding cultures of creativity and productivity, are brilliantly chosen. For this reason alone, I can, and do, recommend the book to everyone.

If those stories are listened to, and felt and thought about, and experimented with, the effect on the reader is large.

When I get a (non-mathematical) book and read it carefully, it means I have chosen to engage rather deeply. Usually I write in the margins, in a sort of hand to hand combat with the details and nuances.

There is a fair bit now written in the margins of this book.

While I sometimes have issues with the theories used to explain things – mostly nitpicky like the fact that nonlinearilty does not equal non-logical, see the story of the Allen Curve – the quality of the inspiration affected by the book completely outweighs any concern about the book’s shortcomings.

The core of this book is the threefold cord of (1) safety, (2) vulnerability and (3) purpose which, expanded a bit, becomes:

1. Safety and belonging – taxing existential questions are never the lot of individuals in highly creative, productive environments. The growing scarcity of safety and belonging in many workplaces should be a source of deep concern. The gig economy is an indication that we are eating our seed corn and have ceased to pay even lip service to wisdom and a sustainable future.
2. An empathetically evolved environment enabled by vulnerability powered connection. A status flat environment in which creative energy flows easily is an environment in which truth and kindness (together!) are common, even foundational. Empathy, in its nuanced and expanded incarnations, is at the root of all highly effective, sustainable environments.
3. Purpose and vision – a bold, omnipresent clarity on the deeper foundational laws of being as well as the aims, the goals and the lofty visions that drive everything. An environment filled with signals keeping these principles and visions in constant view, is an environment whose vision is sustainable. Opposing the natural trend towards higher organizational entropy, these signals are an energy that enables the culture to remain inspired and organized for innovation and collaboration.

These three threads are the pillars of environments that have no trouble retaining those entering their influence. We visit and never want to leave – quite literally. In fact, Daniel himself admitted that when he was doing the research for the book, he found himself making excuses why he needed to stay in the environments he was investigating, even after he had the information he needed for his book.

Of course, some of the research was historical, visible only through the stories of those who were lucky enough to be part of those past places. Take for instance, Bell Labs in its heyday and Harry Nyquist.

In trying to understand the smaller group of super-innovators at Bell Labs, every possible factor was eliminated until it was discovered that all of these super-innovators ate lunch with Harry. He would draw out and listen to his lunch-mates with interest and curiosity, quietly giving them inspiring ideas and questions to go away and think about. Though Harry was also well known and influential because of his own research and innovation, neither this fact, nor his ability to spark innovation in others, seemed to effect his gentle, fatherly demeanor or tranquil reliability. In fact, these characteristics seemed to be significant part of the reason for his power. Disarmed by his demeanor, they opened up to his relentless curiosity.

At IDEO, the design company responsible for a large number of design innovations,  Roshi Givechi plays a similar role, roaming from one design group to another, helping them to overcome obstacles and find new creative grooves through a powerful ability to listen and ask questions. In fact, when Daniel Coyle told her the title of the book he was doing the research for, it was not long before he had a new subtitle after she asked a question about his choice of subtitle.

The other stories and anecdotes are very well selected and wide ranging. Some illustrate principles of collaboration. The Allen curve, showing that effectiveness of collaboration is inversely proportional to the distance between desks of those collaborating, is another striking story of discovery that is both surprising when you hear it for the first time and sensible, even intuitively reasonable, when you take it in and think about it for awhile. While it is not an illogical relationship, as Coyle asserts, it is a non-linear one that will nonetheless make sense to anyone whose intuitions include some instincts for physics and chemistry and interactions and reactions.

Other stories are rich with insight, a sort of living book waiting to be read more and more deeply. Coyle starts his book with such a story, of kindergartners outdoing, by a factor of two, groups of business students and professionals in a challenge to build the highest tower with a piece of tape, a string, a few dried spaghetti  and a single marshmallow. And for me at least, this set the tone of the book.

As noted above, I ended up with a book full of marginal notes (in pencil!) and a lot of thoughts that were discussed with others. If I had to select a phrase that captured the influence of the book on me I think it would be:

… brilliantly selected stories and simple principles that were even more compelling because they were validated by my own experiences in trying to build highly effective teams of innovators …

And the effect of the book does not end with the sharing of the book and discussions.

The histories of places like Bell Labs and Xerox PARC and Los Alamos and the Rad Labs in Boston were already part of my own context, either through direct experience or through careful histories that I had read and internalized, but something about the combination of this book and my own struggles with getting groups together that were sometimes partly or mostly successful and other times were pretty clear failures, created in me a deeper openness and readiness to put these principles into action.

While the experiment that is now underway is a topic for another article, I can say that the timing for the discovery that Beata made and passed on to me was remarkable.

I give the book my highest recommendation.