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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: , 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 that is Rich in 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.




Other Planets

you set


free from this 

to narrow orbits

the third
that opens 

the living 
me to flights
without limit




bright with 
the light

by love



other planets
on which 
to live
to create
to unfold

no longer
for a few shekels
for mere crusts of bread

for enough
is a feast
opens infinity

in the light 
of other 

on other planets 


Connection vs Attention

At our fingertips, in the present, in the place we find quietness, we may find boundless inspiration for a rich, creative life. When this is focused and refined under the influence of our own uniqueness, our own particular genius, we find illumination and a deeply satisfying flow.

Far too often we surrender who we are in an effort to gather attention, when what we are seeking is connection with others, connection with a richly creative life. Human society has almost completely abandoned the cultivation of truly individual genius for the pursuit of attention. As a result we are infinitely poorer.

Yet this choice is completely under our control — we may refuse this lopsided bargain. We may instead choose an abundance that more than makes up for whatever loss of fame or fortune our choice entails. Those that turn away from that obsession, towards quietness and life, find a healing, restoring force, gently coaxing them back to playfulness, to a place of freedom, to originality and creativity.

Immersion in nature, connection to the life that surrounds us, communion with quietness that speaks and engages us with Infinity — here we find sustenance for a life that never loses freshness or originality.

Boldly choosing connection rather than attention, such a life does not sacrifice its own brilliant originality to the temptations of fame or fortune, nor does it hide from the face of fear. Instead, that life enriches everyone and everything it touches, and in so doing finds connection.

In a present quietness cultivated, we find inspiration for a rich, creative life.



Using Photography

I am building a website for the Analysis + Data Group that I am helping establish at WSU. I am trying something new, in order to communicate to potential student recruits much more than a usual mathematics website communicates. I want students who visit the website to begin to get a feel for how we think, who we are, even what it is like to think with us and learn with us. To do this I am partly using non-standard (for mathematics) photography: no mugshots allowed!

(A little about the group: there are five principal members — myself, Bala Krishnamoorthy, Haijun Li, Charles Moore, and Alex Panchenko — and about 12 Graduate Fellows in it — that number will firm up this fall. We will focus both on pure analysis and applications of insights from analysis to data problems.)

Here are a few of the photos I have already taken (or my son Levi has taken of me), that I will be using in the new website.


Alex Panchenko, who combines insights from nonlinear analysis and statistical physics in his work in analysis/applied analysis.


Chuck Moore, who works on problems in PDE and Harmonic Analysis.


KRV, about to explain 4.2.25 to Levi (and Obi).

This project has revived my interest in photography. As a result, I have also been doing some macro work, on my walks with Obi.



The last picture, of the aphid farm being tended by the ants, pushes the limits of the Samsung S-4 phone I have been using to get these pictures. So it has prompted me to get some new equipment. In that process I discovered Mathieu and Heather’s wonderful blog on  mirrorless cameras and photography MiirrorLessons. They introduced me to the electronic photography magazine Inspired Eye  which I also recommend without reservation. The two founding editors of the magazine — Olivier Duong and Don Springer — have precisely the right attitude/philosophy. That philosophy results in an environment that is rich and generous, a fact that is made abundantly clear once you read and  experience the resulting publication. They get art, in a fundamental way. That might seem like a funny statement since the magazine is about photography, and mostly street photography at that. But I stand by what I said — they get art in a way that very few do.

And life is about art, be it creative work in mathematics, or the way one makes food, or the way we (can) relate to others, or how we think and write. While it doesn’t upset me any more, I still protest when people express the idea that mathematics is somehow a non-creative non-art. Those kinds of statements are my cue for very gentle, non-forceful illumination.

I will post a link to the new group website when it is released in a few weeks.

First Post

As indicated in the “About” page for this blog, I am using this blog as a place to put things I have written.  As also noted there, though I have comments turned off, I welcome email conversations with anybody interested in respectful conversations. For that purpose, you can email me at