Author Archives: Kevin R. Vixie

About Kevin R. Vixie

Mathematician, teacher, artist, inventor, observer, poet, father, husband, brother, runner, climber

Finding and Following Your Own Path

When my brother succeeded in getting me to join an Alanon group in 1994 or 1995 I had little understanding of the healing for mind and spirit I would find among the gentle, powerful souls that opened the door to healing simply by listening to me and speaking of their own paths in a way that made it clear I could take what I found healing and leave the rest.

What I did know was that I was deeply afraid of others trying to tell me how to think, how to live, or even who to be. What I did not know was that the boundary violations I had experienced when I was young had created some very large trauma that was only increased by living through the slow deaths of both of our parents when my brother and I were teenagers.

Healing and a deep inspiration flowed from a combination of those inspiring spaces for listening and walkabouts in the forests and mountains, first of Orgeon and then of New Mexico. And what I found was an understanding that no one had the answer for me, no one had the right to tell me what I should do, that only by that personal walk with God in those places of quietness and stillness could I hear and see and feel and find my own muse, my own path, my own unique way of creating and connecting.

I have begun to understand the enormous power of words, the extent to which the misuse of words has created large swaths of humanity and society with seriously reduced capacity for sensing reality, sensing the power of the words and images that are thrown around so carelessly.

So much of what we say to each other is either powerless, without inspiration or filled with power to damage and limit those who accept the words. This comes either as a result of ignorance (the most common case) or intentional malevolence. I have been guilty of using words in ways that were not respectful of the need for others to find their own way. Phrases like “you should do …” are rarely helpful or useful and are often damaging. I would now argue that they never belong in print because, when they are appropriate, it is always very situation dependent. When they do appear with well intentioned people, I believe it is most often due to enthusiasm for discovered insights that have worked well for them.

We discover something that works for us and we immediately evangelize others, certain we know the way, that we have the answer for them as well. This is most pronounced in those that have an undeveloped gift for teaching, but it seems to effect everyone who has made discoveries they think others might need. So often we speak these words and add force of our own, lest those listening (or who are forced to listen) not get the importance of what we are saying.

Yet this betrays a misunderstanding of the power of truth and inspiration. It shows we are not sufficiently aware of how others find their muse, their path of creativity and connection.

Balanced, wholistic truth contains in itself all the power needed for it to take root and grow. In growing, it adapts to the soil it finds, encouraging the uniqueness it finds, illuminating the creativity that results. The creativity that results in turn illuminates new and original facets of that truth. A simple telling of a narrative containing that truth, without force, transmits that truth in a way that is most likely to be accepted, though perhaps laying, like seeds in the ground,  awaiting just the right conditions for to take hold and grow.

The encouragement to the listeners to find, explore, hear the stillness themselves, to become adventurers and participants in that deepest conversation with their Teacher — this becomes the most powerful thing we can do for others.

In their seeing what we have learned, what we have found, what is part of our story, they are inspired to begin their own journey, to find their own muse.


It is arrogance, blind as it always is, that leads us to think we can find the path for others. Sometimes that arrogance is a subtle, cultural type of arrogance. Other times it is overt and obnoxious.

When we begin to see clearly and deeply, the humility that must accompany this leads us to get out of the way of others in their quest to discover who they are, where they can go and what they are privileged to create.

We discover that the path of the creative teacher and mentor, collaborating alongside those involved in the joy of finding and following their own paths, is an experience full of living energy and fresh discovery.

 

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 the following 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 that you can find here — Fun with simple analysis problems I  — I started off by presenting three solutions and then generalized and explored further.

What I did not reveal in that post was that writing the post gave me an idea for a dissertation problem for someone that could be perhaps pushed a ways. Not too long afterwards, Laramie Paxton joined my group and I gave him this problem. We collaborated in solving the problem, since that is how I mentor all my students — their dissertations are collaborations with me — and 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 . The paper explores that idea that was suggested by thinking about the original post I wrote in response to that simple analysis problem that started everything.

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 admittedly challenging undergraduate analysis course (I used Fleming’s Functions of Several Variables) and courses in advanced analysis, geometric measure theory, applications in image analysis (papers he led) and 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. While I believe that, in general, small universities are the places to be nowadays, from everything I hear, this place 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.)

While I am sure that there are other 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, that 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 even healing.

The problem with a lot of  non-fiction is that those that write it often seem not to recognize that what we know is 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, while those that are willing to use all the tools at the disposal of an aware, enlightened human being, can in fact embark on a voyage filled with light and a rich, ever-unfolding life.

In the fable and living experience woven together in Zeyn’s novel, the human spirit and the Infinite meet in an explosion of life and color and light and dark that moves 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 that moved 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 to be not only completely natural, but in fact a door that anyone can enter if they will but take the time to listen to the music and feel the color to be found in stillness and quietness, to see the light that shines 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 and probably 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 usually simply shared the insights I found in that state.

Nevertheless, I am no less hopeful today than 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. Immersing myself in this story, I find again, in yet another form, that stillness containing infinity.

Lately 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 laid this book down, I was struck by a strong sense that I was wrong about that, that some written words were still very powerful, inspiring and healing and opening to that infinity I began to experience so many years ago in my walkabouts in the desert and later in the forests.

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 that there is nothing to prove, that the simple things contain everything you need because they are doors to infinity. You see that helping those that struggle, easing the path of those that 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 the 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 that comes when we remember the intense richness of knowing what is important.

While it is of course up to you to read the story and see what beauty and stillness resonates with you, I believe that you too will find that this story is one of those stories that eases the pain of living and even more, that opens the heart to the hope that heals.

 

 

 

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. And there is a fair bit written in the margins of this book. But 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 keeping those that enter 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.

Median Shapes

When I wrote the paper with Simon Morgan pointing out the L^1\text{TV} functional was actually computing the flat norm for boundaries, we suggested this gave us a computational route to statistics in spaces of shapes. While earlier work certainly touched on this idea of using the flat norm for inference in shape spaces — see this paper on shape recognition, it was not until my student Yunfeng Hu collaborated with myself and Bala Krishnamoorthy (my collaborator, also a co-mentor of Yunfeng’s), that we started addressing the idea of statistics in shape spaces in the original paper with Simon Morgan.

The results can be found here: https://arxiv.org/abs/1802.04968 , in a paper with the title Median Shapes, with authors Yunfeng Hu, Matthew Hudelson, Bala Krishnamoorthy, Altansuren Tumurbaatar, Kevin R. Vixie. Tumurbaatar wrote the first complete version of the code used, and Matthew Hudelson contributed a pivotal new result on graphs inspired by a problem in the paper, while Bala led the computational end of things and I led, in collaboration with Yunfeng Hu (and Bala keeping us honest!), the theoretical parts of the paper. It was a fair bit of work.

We went over the more difficult results a few times, finding improvements and corrections. Of course, there may be a few things here and there to improve, but for now, it is done.

Yunfeng probably spent the most time writing up the piece proving that near regular points on the median, the collection of minimal surfaces meeting the median have a tangent structure we describe as a book.  While this is clear to experienced geometric analysts,  there are lots of little details and we wanted most of the paper to be more accessible to a wider audience. There are lots of other pieces here and there that took time to think about and write up (and rewrite). For example, when showing the set of medians need not contain any regular members,  the part where we show that we need only consider graphs when searching for a minimizer was not easy. And of course, as in most all of geometric analysis, there are problems you solve without too much effort at a high level, but find that writing down is tedious, though at times enlightening due to the fact that those little details turn out to be hard and illuminating.

Because the problem of computing the median reduces to a linear program, while the mean reduces to a quadratic program, we focused on the median problem. Some parts of the paper are a bit long winded, for the reason that we wanted it to have more details that would usually be in a paper communicating to others that understand geometric analysis.

Anyway, have a look. If you find yourself interested, there is already code you can use to compute medians, though we hope eventually to have faster code.

 

A Silence, Rich with Inspiration

Glynne Robinson Betts’ 1981 Writers in Residence is for me a lyrical invitation to quietness. The photo illuminated essays on places, in time and space, where writers lived and wrote,  invoke a strong sense of life lived with wide spaces for thought and creativity. The sections on Carl Sandburg, Anne Dillard, Robinson Jeffers and several others, recall daily rhythms friendly to depth.

These passages somehow bring back my life in the late 60’s and 70’s, when computers were rare and time to think was not hard to find. In college in the early 80’s, there was still time to think, time to read through a book on a weekend or master complex ideas in quietness, with a devotion to deep comprehension. I wonder, how many now feel that call to stillness, to a silence that is rich in inspiration?

Where do we find the wide spaces today? And who lives there? Those that tasted the thrill of illumination through immersion in quietness remember those spaces, but what of those addicted to their mobile devices, what of those who believe social media connects, wikipedia illuminates and TED talks are the pinnacle of inspiration? While wikipedia is useful, TED talks are sometimes narrowly inspiring, the intense illumination of bare-handed, personal discovery leaves you changed, forever.

Seeking the simplicity of those wide spaces, listening till we hear the quietness sing, we find the same places the writers found, the same illumination that is never forgotten … we walk through an open door to the infinity that lives between the ticks of the clock, between the words on a page, between the breaths we breath.