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.

Beginning Again

I am beginning to see tiny, yet brilliant slivers
of something that happened at the Cross that was so
enormous, so full of wonder, so full of illumination,
so powerful that it brings a deep stillness to everything
it touches.

A defeat of death and sin and all that is evil, a defeat
so complete that time held its breath, so awesome that
everything changed in that instant.

The tearing of the veil was the beginning of a tearing,
of a cracking, a disintegration of all the separates us
from God.

We were at that moment in a completely new era.

Time had begun its transformation, back to where it

Face to face, in the silence that sings and heals, with
words that speak without words, nothing is withheld.

Nothing is withheld.



An Invitation to Geometric Measure Theory: Part 1

While there are a variety of article-length introductions to geometric measure theory, ranging from Federer’s rather dry AMS Colloquium Talks to Fred Almgren’s engaging Questions and Answers to Alberti’s Article for the Encyclopedia of Mathematical Physics, I will take a different approach than has been taken in any of these and introduce geometric measure theory through the vehicle of the derivative.

The Derivative, Geometrically

The derivative that is encountered for the first time in calculus is defined as the limit of a ratio of the “rise” over “run” of the graph of a function. For y = f(x), this becomes

\frac{df}{dx}(a)=\lim_{x\rightarrow a}\frac{f(x) - f(a)}{x-a}.

This is visualized as the slope of the secant lines approaching a limit – the slope of the tangent line – as the free ends of those lines approach (a,f(a)). This is illustrated in the first figure.


The derivative as \hat{L}_a, the optimal linear approximation to f at a, is another, very useful way to think about the derivative. Here, we focus on the fact that the tangent line at (a,f(a)) approximates the graph of f(x) at (a,f(a)) as we zoom in on the graph. More precisely, writing x = h+a,

f(x) = f(h+a) = f(a) + \hat{L}_a(h) + g(h)h,

where \hat{L}_a is linear in h, g(h)\rightarrow 0 as h \rightarrow 0, and the tangent line L is the graph of the function y = f(a) + \hat{L}_a(x-a) .

Exercise: use the facts that (1) linear \hat{L}_a:\Bbb{R} \rightarrow \Bbb{R} have the form h\rightarrow sh, s a scalar, and (2) g(h) \rightarrow 0 as h \rightarrow 0, to rearrange this last equation for f(x) into the original definition of a derivative.

Using the equation above to get

\left|f(x) - (f(a) + \hat{L}_a(x-a))\right| \leq (\sup_{|s|\in[0,\epsilon]} |g(s)|)|h| for  h\in[-\epsilon,\epsilon],

we are able — after some work (see the exercise below) — to get this nice geometric interpretation:


The figure illustrates the fact that the graph of f(x) lies in cones centered on L, whose angular widths go to zero as we restrict ourselves to smaller and smaller \epsilon-balls centered on (a,f(a)). Inside the \epsilon_1-ball, the graph stays in the wider cone, while in the smaller, \epsilon_2-ball the graph stays in the narrower cone.

Let’s restate this. Defining

  1. p \equiv (a,f(a)),
  2. B(\epsilon) to be the ball of radius \epsilon centered on p,
  3. F\equiv\{ (x,y) | y= f(x) \},
  4. C_L(p,\epsilon) to be the smallest closed cone, symmetrically centered on L, with vertex at p such that F\cap B(\epsilon) \subset C_L(p,\epsilon), and
  5. \theta (\epsilon) to be the angular width of C_L(p,\epsilon),

we have that

f is differentiable at a \Leftrightarrow \theta(\epsilon) \rightarrow 0\text{ as }\epsilon \rightarrow 0

Here is a figure illustrating this:


Exercise: provide the missing details taking us from the above inequality bounding the deviation from linearity to the above statement that {f is differentiable at a \Leftrightarrow \theta(\epsilon) \rightarrow 0\text{ as }\epsilon \rightarrow 0} using the facts that (1) the above inequality defines cones that are almost symmetric about L and (2) the \epsilon-ball centered at p is contained in the vertical strip (x-a,y-f(a)) \in [-\epsilon,\epsilon] \times(-\infty,\infty)

With this shift to a geometric perspective, we are now in a position to take a step in the direction of geometric measure theory.

Note that in our definition the cones contain all of the graph as they narrow down and we zoom in. But what if all we know is that a larger and larger fraction of the graph is in a narrower and narrower cone as we zoom into p? That is precisely the idea that approximate tangent lines capture. We will introduce two different versions of the concept.

Densities as a path to an approximate tangent line

Tangent Cones

The tangent line discussed above is also the tangent cone. The tangent cone of a set in \Bbb{R}^n can have any dimension from 1 to n. For nicely behaved k-dimensional sets, the tangent cone will also be k-dimensional. In the case of the usual derivative of functions from \Bbb{R} to \Bbb{R}, we are working in the graph space \Bbb{R}^2 with 1-dimensional sets. Moving to tangent cones, we can approximate one dimensional sets which are not graphs or, more generally, arbitrary subsets of \Bbb{R}^n.

We now define the tangent cone of F\subset\Bbb{R}^n at p.

To obtain the tangent cone, begin by translating F by -p. (This moves p to 0.) Define F(\epsilon) \equiv (F\cap B(\epsilon))\setminus p. Use a projection center at 0 to project the translated F(\epsilon) onto the sphere of radius \epsilon centered on 0. Take the closure of the resulting subset of the \epsilon-sphere. Finally take the cone over this set. Call this set T_p^\epsilon(F). That is,

T_p^\epsilon(F)=\{\Bbb{R}\geq 0\}(\text{Closure}(\cup_{x\in F(\epsilon)}\frac{x-p}{|x-p|})).

Now define the tangent cone of F at p to be the intersection of T_p^\epsilon(F) at any sequence of \epsilon_i‘s going to zero; \epsilon_i = \frac{1}{i} will do. Thus the tangent cone of F at p, T_p(F) is given by:

T_p(F)=\bigcap_i T_p^\frac{1}{i}(F).

Here is a figure illustrating the key idea:


Note: the tangent cone is centered on the origin, 0, but I will be plotting it as though it were centered on p. Similarly, the tangent lines will sometimes be thought of as linear subspaces (i.e. centered on the origin 0, and other times as the shift of that linear subspace to p.

In the case of a differentiable function f:\Bbb{R}\rightarrow\Bbb{R}, this tangent cone is the usual 1-dimensional tangent line.


Now we need \theta^k(\mu,F), the k-dimensional density of F at p.

Define \omega(k) such that it agrees with the volume of the unit ball in \Bbb{R}^k when k is an integer (there is a standard way to do this using \Gamma functions). Let \mu measure k-dimensional volume. Typically this will be k-dimensional  Hausdorff measure, \mathcal{H}^k. Whatever intuitive idea you have of k-dimensional measure is good enough for our purposes. (At the end of this post I also define Hausdorff measures more carefully.)

Now, \theta^k(\mu,F) is given by

\theta^k(\mu,F)=\lim_{\epsilon\rightarrow 0}\frac{\mu(F\cap B(\epsilon))}{\omega(k)\epsilon^k}

when this limit exists. When the limit does not exist, we work with the limsup and liminf of the right hand side which are called upper and lower densities of F at p and are denoted by \theta^{*k}(\mu,F) and \theta^k_*(\mu,F) respectively.

Approximate Tangent Cones

We now define the approximate tangent cone at p to be the intersection of closed cones whose complements intersected with F have density zero at p:

\tilde{T}_p(F)=\bigcap\{\text{closed cones }C\text{ with vertex }p|\theta^k(\mu,(\Bbb{R}^n\setminus C)\cap F)=0\}

Originally (in this section), we were aiming at having a definition of approximate tangent line that was invariant to (small) pieces of the set F outside the sequence of cones, provided those pieces got small enough, quick enough. Now we can make that more precise. We want a definition of approximate tangent line that ignores such excursions of F provided these excursions have density zero at p. Rather anti-climatically then, here is the definition we have been waiting for (though you might have already guessed it!)

A 1-dimensional set has an approximate tangent line at p when the approximate tangent cone is equal to a line through p.

When the curve is an embedded differentiable curve, the tangent line and the approximate tangent line are the same.

Remark: in general, when we are dealing with k-dimensional sets in \Bbb{R}^n, we will get approximate tangent k-planes.

Exercise: can you create examples of one dimensional sets which have a (density based) approximate tangent line at p but not the usual tangent line at p?

Exercise: prove that a tangent line to a continuous curve is also the (density based) approximate tangent line at p.

Integration as a path to an approximate tangent line

There is different version of approximate tangent k-plane based on integration. (The one dimensional version is of course an approximate tangent line.)

We start with the fact that we can integrate functions defined on \Bbb{R}^n over k-dimensional sets using k-dimensional measures \mu (typically \mathcal{H}^k). We zoom in on the point p, through dilation of the set F:

F_\rho(p) = \{x\in\Bbb{R}^n | \;\;x=\frac{y-p}{\rho}+\text{ p for some }y\in F\}.

We will say that the set F has an approximate tangent k-plane L at p if the dilation of F_\rho(p), converges weakly to L: i.e. if

\int_{F_\rho} \phi d\mu\rightarrow_{\rho \rightarrow 0} \;\; \int_L \phi d\mu

for all continuously differentiable, compactly supported \phi:\Bbb{R}^n \rightarrow \Bbb{R}.

In the next two figures, we illustrate this for the case of 1-planes – i.e.lines: in the first figure, L is the weak limit of the dilations of F, while in the second it is not.

Note: solid green lines are the level sets of \phi while the dashed green line indicates the boundary of the support of \phi. Note also that the \rho‘s of 0.4, 0.1, and 0.02 are approximate.



Exercise: can you create an example of a one dimensional curve which has the usual tangent line at p but not an (integration based) approximate tangent line at p?

Closing Note On Hausdorff Measure

We would like a notion of k-dimensional volume or k-dimensional measure. In many cases, the right notion turns out to be k-dimensional Hausdorff measure. We already know what 1,2, and 3-dimensional measure is as long as the objects we are measuring are regular enough, like subsets of lines, rectangles, and cubes. It does not seem too much of a stretch to think that we can extend these measures to things that are somewhat wiggly. That is, we can still easily imagine measuring the length of a subset of a smoothly turning curve, or the area of a piece of a surface that undulates slowly. Hausdorff measure permits us to measure not only such smooth sets (giving the same result as any reasonable extension of the usual Lebesgue measures to the nice cases), but also to measure very wild sets (like fractals).

How to compute the k-dimensional Hausdorff measure of A\subset \Bbb{R}^n:

  1. Cover A with a collection of sets  \mathcal{E}= \{E_i\}_{i=1}^\infty, where diam(E_i) \leq d \;\; \forall i. Here, diam(E_i) is the diameter of E_i.
  2. Compute the k-dimensional measure of that cover: \mathcal{V}_\mathcal{E}^k(A) = \sum_i\omega(k) (\frac{diam(E_i)}{2})^k
  3. Define \mathcal{H}_d^k(A)=\inf_{\mathcal{E}} \mathcal{V}_\mathcal{E}^k(A) where the infimum is taken over all covers whose elements with maximal diameter d.
  4. Finally, we define: \mathcal{H}^k(A)=\lim_{d\downarrow 0}\mathcal{H}_d^k(A).

Remark: Suppose that for any  \epsilon > 0, there is a cover \{E_i\}_1^\infty of A, such that \sum_i diam(E_i) < \epsilon. Then for k \geq 1, \mathcal{H}^k(A) = 0.

Here is a figure illustrating Hausdorff measures:


Clearly, this can be difficult to compute. It turns out though that in \Bbb{R}^k, \mathcal{H}^k = \Bbb{R}^k. And by use of mappings, this can take us quite a ways in computing \mathcal{H}^k(A) for integral k and rather general A.

Exercise: Show that if 0 < \mathcal{H}^\gamma(A) < \infty then \mathcal{H}^\alpha(A) = \infty and \mathcal{H}^\beta(A) = 0 for \alpha < \gamma < \beta.

Thoughts on receiving a negative review

This is a slightly edited version of something I wrote in 2009, not long after arriving at WSU from Los Alamos. It remains as pertinent now as it was then. Coincidentally, Gaza is again in the midst of increased mayhem.

Today I received a copy of a review of a paper I am an author on. Needless to say, the reason I am writing about it here is that the review was negative in a way that was not helpful. While the reviewer did make some good points, and we will address those points, it was done in an unfriendly way.

Have I seen worse reviews? Of course. So why write about this review? I suppose because it comes at a time when I am being reflective and when I am thinking about such things more carefully. The error that reviewer made was in not reading the paper carefully enough. Of course we can improve the paper and make it less susceptible to misinterpretation, and we will, but I think that the acceptance of this status quo of negativity and a cultivated attitude that looks for errors and ignores insights, ends up robbing our society of a great deal of original, creative productivity.

In my new position in the mathematics department at Washington State University, as I look around and get the intuitive sense for this university and put that in context of what I have observed at other universities, I see a pattern. And that pattern is tradition and conservatism and narrowness that has its roots in narrow self interest. It inhibits interdisciplinary work. It makes people far more apt to see the mistakes in other work, rather than finding the insights and innovations.  It makes people timid and afraid of adventure, of risk.

Do I like it in academia? Yes. There is still a decent amount of good and potential for a great deal more. There is freedom to develop truly new initiatives. And there are some students and colleagues who are inspired and inspiring.

But the threads I am disturbed about are simply local expressions of global states of human consciousness that we all observe in their horrific consequences: Gaza, the economic crisis, epidemics in Africa, etc.  Underlying everything are multiple threads, but the one that I see everywhere is an unconsciousness, a blindness that is deeply disturbing.

In this state, humans think there is no connection between their personal negativity and selfishness and the atrocities in Gaza. It is acceptable or even good to inflict inhuman atrocities on your enemy, but evil for those “terrorists” to strike back in the ways they can. The unconscious see a great gulf between them and the “terrorists”. They believe that some people are intrinsically good and some intrinsically bad. And the end justifies the means. The work of Chris Hedges — see for example, “I don’t Believe in Atheists” or “War is the Force that Gives us Meaning” or his columns in — feels and proclaims aloud the absurdity of these inconsistencies.

So what can my response be — be it to the reviewer, or the critical, narrow nature of some in academia, or the unmotivated, narrow minds of some students, or the unthinking, unconscious state of some people I run into in my daily life? Certainly, becoming negative and critical is not the answer.

It seems to me that the only thing I can do is to spend all my energy creating a personal atmosphere of rich, creative productivity and connection based on love. Generating happy beauty and a vibrant, living atmosphere, beckoning to those in the sphere of my influence to cooperate in creating little bits of heaven on earth, even if only locally, is the only real evidence there is for the existence of love or heaven … or God.

Geometric Measure Theory by the Book

There are an armful of texts that I have used to learn and teach geometric measure theory. In this note, I will give a review of these texts, which are:

  1. Herbert Federer’s Geometric Measure Theory
  2. Frank Morgan’s Geometric Measure Theory: A Beginner’s Guide
  3. Krantz and Parks Geometric Integration Theory
  4. Lin and Yang Geometric Measure Theory – an Introduction
  5. Leon Simon’s Lectures on Geometric Measure Theory
  6. Pertti Mattila’s The Geometry of Sets and Measures in Euclidean Spaces
  7. Evans and Gariepy’s Measure Theory and Fine Properties of Functions
  8. Ambrosio, Fusco, and Pallara’s Functions of Bounded Variation and Free Discontinuity Problems
  9. Enrico Giusti’s Minimal Surfaces and Functions of Bounded Variation

My two favorites are Leon Simon’s Lectures on Geometric Measure Theory and Evans and Gariepy’s Measure Theory and Fine Properties of Functions. Before I dive into comments on each of the books, here is a bit of history concerning my path into the subject.

The Backstory

I was first turned on to geometric measure theory by David Caraballo, the last student to finish with Fred Almgren before Fred died. (Fred who was famous for his deep results in geometric measure theory, was a student of Federer and a professor at Princeton.) I met David at the Nonlinear Control Theory Short Course, organized by Hector Sussmann and Kevin Grasse, at the 1999 Joint AMS-MAA meetings in San Antonio. David and I became instant friends and I was soon swept away by David’s passion for geometric measure theory, realizing that this field was also particularly well matched with my mathematical muses.

In particular, David talked up Evans and Gariepy’s text, so the first thing I did when I got back to Los Alamos was hibernate in my office and immerse myself in that text. It was beautiful and exhilarating — and I was hooked.

At the time I was working on inverse problems and dynamical systems. The inverse problems involved images and it was the image analysis that drew me further into geometric measure theory. The biggest influence in this migration was of course the rising prevalence of total variation regularization in image analysis methods. The first papers I read were David Strong’s. These inspired me to introduce total variation regularization into the sparse tomographic reconstructions methods we were working on. (There were other people dabbling with these methods at the same time at Los Alamos, but Tom Asaki and I took these methods and ran with them. Abel inversions were the tomographic workhorse at Los Alamos, so this was one of the first targets of our work. The first papers can be found here: Abel inversion using total-variation regularization and Abel inversion using total variation regularization: applications)

Through my work in image analysis, I met Andrea Bertozzi, and while visiting Duke at her invitation early in 2003, I met Bill Allard. Bill and I started a close collaboration on data analysis work at Los Alamos, and I began picking up pieces of geometric measure theory from him. Bill was a student of Fleming’s at Brown, but he had also very carefully read and commented on Federer’s entire text as Federer was writing it. (If you know Federer’s book, you can’t help but be impressed by this.) After graduating from Brown, Bill moved to Princeton where he did his seminal work on varifolds.

So why does image analysis lead rather naturally to geometric measure theory? For the simple reason that edges are a big deal in images, while functions of bounded variation (BV) are a very natural class of functions to use when representing functions with discontinuities (edges). And functions in BV are particularly nice, because they are wild, but not too wild — we can still make sense of derivatives (they are nice measures), and all sorts of other nice properties are still at our disposal. For example, sets, whose characteristic functions are in BV, still have usable, generalized outer normals and as a result, we still have the divergence theorem in such regions. Analysis is still nice, or at least possible. (De Giorgi used this class of functions to solve the minimal surface problem in one of the three papers in 1960 that solved this problem. Another was written by Federer and Fleming, the third by Reifenberg. All three used different methods.)

Comments on the Books

I have studied, referenced or taught out all the above monographs. I now give more detailed comments on those texts. But I am not going to introduce the subject of geometric measure theory here — that will be the subject of other posts (coming soon), nor will I simply outline the contents of the texts, since I don’t think that adds much value. Instead, I will add the things you can’t get from a perusal of the table of contents.

Here are the comments:

  1. Federer’s 1969 Geometric Measure Theory: To a very large degree, this is still the ultimate go-to reference for the contents of the first 4 (of 5) chapters. This is not to say that that content has not evolved, but rather that it is still the foundation for current work. (For example, Solomon and White have enabled us to avoid the difficult structure theorem in getting existence for minimal surfaces, but that structure theorem is still very important.) This text is also rather notorious for it’s density and difficulty. Some of that difficulty is due to the (sometimes understandable) impatience that readers bring to their reading, but it also seems that the lack of pictures in this particular book is a rather eloquent statement about its accessibility. I have learned that it is much easier to read when you translate nearly everything into pictures (because you can!), though its terseness encourages me to use it as a reference, not a text. But it is a completely indispensable reference! Every students should own a copy and read pieces as needed.
  2. Frank Morgan’s Geometric Measure Theory: A Beginner’s Guide: Frank wrote his highly successful text as a path into, and an inspiration for the study of, Federer’s book. In contrast to Federer, Frank draws lots of pictures, many of them very enlightening. Almost everybody in the younger generations — people who could have used Frank’s book first — have first read Frank’s book before using other texts. This is as good a place as any to explain that there have been two main branches of effort in GMT, one focused on variational problems like the minimal surface problem, and the second focused on the geometry of sets and measures, with a particular focus on harmonic measures. Morgan and Federer come out of the first branch, while Mattila’s text, which is commented on below, comes from the second branch. (I will comment below on the two new branches of GMT — GMT in metric spaces and GMT with a view to data analysis.) I always use Frank’s book as an illuminating reference for newcomers. I recommend it very highly as a first exposure and an evangelical tool.
  3. Krantz and Parks’ Geometric Integration Theory: At first I was skeptical because I am into nice figures (I am an xfig devotee) and some of Krantz and Parks’ figures were very bad. But then I used the text as a reference when proving the deformation theorem for simplicial complexes (see Simplicial Flat Norm with Scale for the paper) and I was impressed with the exposition. Last year I used it as a text for a graduate class. While there are aspects of the text I still don’t like, I do recommend it as a reference that every student should own and consult. (By the way, Federer originally wanted to name his book Geometric Integration Theory, but didn’t because Whitney had already written a book with the same name. Whitney’s book is relevant for those interested in geometric measure theory, and it is now available from Dover books!)
  4. Lin and Yang’s Geometric Measure Theory – an Introduction: I have not used this as a text. I was discouraged from using it by the typos, which it is important to note were also very irritating to Fanghua Lin because he had tried to get the publisher to correct them! But it does contain a very nice selection of topics, maybe the broadest selection in the above texts.
  5. Leon Simon’s Lectures on Geometric Measure Theory: As I said above, this is one of my two favorites. I am currently (Fall 2012) teaching a graduate class from this text. The book is not so easy to get, but if you are willing to persist, you can get it from the Centre for Mathematical Analysis at Australian National University. (Though the last time I ordered a bunch of copies, they all had the first page of the index missing and they were clearly photocopy’s of the original print run of the book.) I used Leon’s text one other time, when I taught a short course on GMT at UCLA in the spring of 2007. Why do I like the text so much? There are flaws, like typos and a typewriter type font that takes getting used to and things I would change here and there. But Leon’s selection of topics (not too many!), his versions of theorems, the way in which he gives enough details in proofs, but not too many (leaving many implicit exercises and problems for the reader), and the way in which he puts everything together has generated a book that I really like to study and teach from. I recommend it very highly as a primary text, after or alongside Frank’s book. As a side note, Leon is working on a second edition of this text. It should be very good!
  6. Pertti Mattila’s The Geometry of Sets and Measures in Euclidean Spaces: As mentioned above, this text comes from the harmonic analysis branch of the subject. As a result, it does not deal with currents, which were developed for their use in GMT (on minimal surfaces) by Federer and Fleming. Mattila’s book is well written and challenging, but not so challenging that students don’t like it. It has explicit problems, which many students like. (I always preferred implicit, fill-in-the-details or what-if-I-weaken-this-assumption type problems.) It covers lots of material, including Marstrand and Priess’ results (about which I would recommend De Lellis’ notes on Rectifiable Sets, Densities and Tangent Measures ), fractals and connections to singular integrals. It contains things that none of the other books I am commenting on have, and it is the only representative of the harmonic analysis branch of GMT in my selection of books. And it does have a different flavor, as one might expect. I consider it something that every student of GMT should own.
  7. Evans and Gariepy’s Measure Theory and Fine Properties of Functions: As noted in my story above, this was the first book I saw on the subject. It deals with the same subjects that the first part of Federer and the first part of Simon deal with. It does not delve into currents. The writing is very clear, the proofs are complete, and the amount of filling in, in the proofs, is consistently small enough to make it fairly fast to study, but often enough to keep you very engaged. I found it inspiring when I first read it and it is still usually the first book I have my students buy. The chapters cover measure theory and integration, Hausdorff measure, Radon measures, area and co-area formulas, Sobolev spaces, BV functions (including detailed development of the structure theorem for sets of finite perimeter), and a final chapter on things like Radamacher’s theoorem and extension theorems like Whitney’s. CRC even lowered the price from 180$ to 90$ (it has crept back up a bit) in response to our complaints about the price! As far as prerequisites are concerned, most students find this more accessible after a first course in graduate analysis, but some might be happy with it as an introduction to analysis, as long as some other text, like Royden or Folland is also on hand. I recommend it very highly as a text and reference.
  8. Ambrosio, Fusco, and Pallara’s Functions of Bounded Variation and Free Discontinuity Problems: David Caraballo was also the first to tell me about this book. It prepares the reader to deal with the existence results for the Mumford-Shah functional, which is an image analysis functional for used for image segmentation. Ambrosio and De Giorgi proposed the space of special functions of bounded variation (SBV) for use with free boundary problems in 1987. In a 1989 paper by De Giorgi, Carriero, and Leaci, these ideas were used to prove the existence of minimizers for the Mumford-Shah functional. Later Ambrosio developed the theory of SBV more fully and this book is a logical follow-on to these works. I have not used this book, though I have read a small bit here and there. I think it is well liked by students, but it is unreasonably expensive at 250.00 (list price). Shame on Oxford University Press! They should not be following the example of the Dutch profiteers! This is an object lesson in why, if you are a mathematician, you should publish your own book and not give it to some publisher to exploit. OK. Done with my soapbox. If you can afford it, get this book! If you can’t, write to Oxford and complain bitterly about their crazy prices and the fact that they are limiting access to an excellent and fascinating book!
  9. Enrico Giusti’s Minimal Surfaces and Functions of Bounded Variation: I read much of Giusti’s book and liked it a great deal. I recommend it as very a good source for the subjects it covers. This book is inspired by De Giorgi’s path to a solution for the minimal surface problem, though it contains more material since it was written over 20 years after that work by De Giorgi. Here is review of Giusti’s book by Fred Almgren. It contains a very detailed account of the contents of the book and some nice history as well. Again, this book is overpriced at 183.00 for the paperback but luckily, Springer (who owns the book’s publisher Birkhauser) sells it for 91.50! So buy it directly from Springer. And I do recommend buying it. You will enjoy studying it if you have any interest in this subject.

Finally, as promised above, a few words about the other two branches of GMT. As mentioned above they are GMT in metric spaces and GMT with a view to data.

As far as I know, GMT in metric spaces had its genesis with Ambrosio and Kirchheim’s paper, Currents in Metric Spaces. It is a very active area of research, including for example, analysis on the Heisenberg group, where paths are allowed tangents in proper subspaces of the tangent space instead allowing the path to have arbitrary directions in the tangent space. (These are called sub-Riemannian spaces.) As far as I can tell, this branch was inspired by the growing area of analysis in metric spaces.

By GMT with a view to data, I mean both GMT applied to data and new GMT inspired by data. That the flow of ideas goes in both direction is very important and the reason why this is a very exciting, productive, high-potential place to work. This is where I am working. It includes the work at the intersection of image analysis and BV/SBV functions, like the work of Rudin, Osher and Fatemi which introduced TV regularization to image analysis and the segmentation functional of Mumford and Shah. Other examples include the work of Jones, Lerman, Schul and Okikiolu on Jones’ beta numbers, the multiscale flat norm work I am doing with collaborators (inspired by the connection between the L1TV functional and the flat norm), as well as the applications of curvature measures (and things like curvature measures) to data. Examples of this last item include the work of Adler, Taylor and Worsley at the intersection of statistics and integral geometry and the work of Chazal, Cohen-Steiner, and Merigot on boundary measures. Many more examples exist, but these examples give a flavor for the kinds of things happening at the intersection of GMT and data.

Actually, the GMT data branch has the other three as subbranches for the simple reason that all three have very useful insights to offer data and because data suggests problems leading to new ideas in each of the three areas.


As mentioned above, I like figures. But, while not having figures in a geometric measure theory text doesn’t make so much sense to me, it is the case that students should be drawing their own figures anyway. This is part of the work involved in making the subject yours, in internalizing the ideas and techniques. One might even argue that books without figures are better for students who must then draw figures for themselves in order to grasp what is going on more fully. But I would not go quite that far. I think that drawing pictures, as many and as often as you can, should be the part of the GMT culture. Figures should be common, and they should also appear in books. I do believe that authors should not try to draw pictures for everything, but should draw just enough to get the students going themselves, help students avoid pitfalls and inspire them as they struggle to master the ideas. At the very least, this discussion helps you see why I can still like Leon Simon’s book so much even though there are no figures in it!

It should also be noted that even though I talk about the two older and the two newer branches of GMT, it would be silly to insist on a bold demarcation of the boundaries between branches and a subsequent classification of everything and everyone. Part of this is because the intersections between branches are very large. Another part of this is because the two newer branches are by their nature agnostic, caring only about generalizable (in the case of metric spaces) or useful (in the case of data analysis) ideas or developments in GMT. Using Federer to supply another example, even though Federer’s most famous paper (the one with Fleming that solved Plateau’s Problem) was focused on calculus of variations, he also established other significant pieces of the foundation for the entire field. It would therefore make no sense at all to try to assign him to a single branch. So thinking about the field as characterized by branches is useful only if you do not take it very seriously, or worse yet, turn the branches into fences impeding travel!

Higher Education: the real problem is not the cost

Even though there is a lot of talk about money and cost, the real problems in higher education are the mistaken ideas that have gained respectability through the motion of the cultural herd we all live in. Here are four such ideas:

(Mistaken Idea 1) Everybody should go to college after high school because most jobs actually justify requiring a college degree as a qualification.

I do believe that everybody should have the opportunity, when they have real desire, to learn more deeply, to go to college and interact with mentors, etc. But the current way in which college is a knee jerk path to jobs is just silly. Giving people something they really don’t want in response to ill-founded ideas on what is useful to them is a recipe for very unhappy situations.

(Mistaken Idea 2) Adding computers and the Internet to the educational enterprise makes it better.

In fact, the way most people use the Internet changes the way they think, making it very hard for them to focus, think independently or deeply about anything.See for example the research cited by Nick Carr in his (possibly too) provocative book “The Shallows”.

This constant tweeting, emailing, texting, surfing, video-gaming that college kids are immersed in is seriously degrading their ability to focus deeply. I come to that conclusion through a combination of direct observation of the lower level classes I have taught — differential equations, business calculus and linear algebra — as well as the results reported in Carr’s book, as mentioned above. Another very interesting talk to listen to is David Levy’s Google Tech talk “No Time To Think” which can be found on youtube here.

 The illusion of greater knowledge hides the fact that less is being internalized, reasoning powers are weaker, and that the capacity for depth has been seriously degraded.  While it is true that a disciplined use of the Internet can be very helpful in research, the ways it is used by the vast majority has little to do with these positive uses.

(Mistaken Idea 3) Grant money for research has improved the overall quality of education.

In fact, the addiction that universities have to grant money has driven them to pay lip service to education while in reality, by any organic measure, they have relegated education to a least important role.

Grants have evolved from a luxury to a necessity in the sciences (and many other areas of academia). Universities are now completely dependent on that funding, and the drive to get that funding takes precedence over everything else. In particular, while professing a great focus on teaching, a close look at status/promotion/pay/etc at institutions of higher education (other than the teaching colleges) will show that teaching is most definitely not a top priority. Of course, an important part of the blame for this is the decreasing revenue from states, which necessitates an even greater emphasis on grant getting and an increase in class size. This last trend – bigger class sizes – has a seriously negative impact on teaching quality. Teaching is inherently a one-on-one or one-on-few exercise. Mentoring and training on the large scales that is being adopted to deal with the lack in funding is very difficult unless one agrees to the (large) drift in the definition of education that accommodates the growth in class size.

(Mistaken Idea 4) Education is what happens in a lecture class, by reading a book, or by searching the Internet.

Education is about helping students discover and follow their passion, in collaborations with teacher/mentors. It is also about training the students to develop and exercise their moral muscles. Independence of thought and action and the exercise of compassion and generosity results in an education that is robust and versatile, forming a foundation for long term creativity and happiness. It leads to a sustainable society.

So, books and lectures and Google Scholar can all be tremendously valuable to the process of education, but education is very far from the simple mastery of some subject of study. Even if the lectures are very inspiring, the books extremely well written and all the papers that are needed can be found and downloaded, these are merely a small to medium sized piece of an education.


Education — which should be about helping students discover and follow their passion, in collaborations with teachers/mentors — is severely hampered by the mistaken ideas outlined above. The acceptance of those ideas greatly disadvantages students and professors, and in the long term, our entire society. Independent and creative (yet disciplined) thought is a critically important ingredient of a happy society. When there is a shortage, as there currently is, everything from economics and technology to moral strength and community suffer. But because the time scale at which this happens is longer rather than shorter, it is invisible to politics as we know it. As a result, positive progress will come from the bottom up, from the grass roots.

The main task at hand, is the fueling of a grass roots movement by helping people realize they are operating on mistaken assumptions, that the real crisis in higher education has nothing to do with the cost and debt, but rather with the nature and substance of education.


moving through me,
flies away, having drawn me to a deeper stillness. 

A vision, 
flowing like music  
compels me to continue on. 

fresh in its originality, 
luminous and living in action and influence, 
brings release.

    I find God. 

The pain 
is transmuted 
into the awe of companionship 
with Him in whom I live and move and have my being.

The Power of solitude … and Social Connection

Awhile ago, Eric Blauer blogged this:

“Of this there is no doubt, our age and Protestantism 
 in general may need the monastery again, or wish it 
 were there. ‘The Monastery’ is an essential dialectical 
 element in Christianity. We therefore need it out there 
 like a navigation buoy at sea in order to see where we 
 are, even though I myself would not enter it. But if 
 there really is true Christianity in every generation, 
 there must also be individuals who have this need. […]”
  —Kierkegaard’s Papers and Journals:  A Selection,
 translated and edited by Alastair Hannay, 
 47 VIII I A 403, pg. 275

in response to something I had written him. In turn, that prompted me to write the following.

The monastery in its essence has always been there.  At least in its original, unpolluted version of time in solitude with God, it has always been accessible. The solitude of walks with God in nature, the quiet seclusion in which we hear and see, is closer than most think.

In fact, we are invited to find it by waiting:

“But they that wait upon the Lord shall renew [their] strength; they shall
mount up with wings as eagles; they shall run, and not be weary; [and]
they shall walk, and not faint” Isaiah 40:31

“… in quietness and in confidence
shall be your strength …” excerpted from Isaiah 30:15

Which my walk has combined to:

“They that wait upon the Lord shall renew their quietness and confidence”

The monastery, as an ideal, is flawed. In pursuit of this ideal, a culture is robbed. For it is fundamentally wrong to view communion and union with a mate, interaction with the world, and social flow as distractions from a deeper walk with God. Acted upon as a model for spiritual depth, such views lead to an impoverished life, an impoverished culture.

Yet the simple, solitary walk with God is a powerful experience leading to deep insights and fresh originality. Spiritually, creatively, we are drawn to the greatest depths by an existence constantly moving between a walk with God and a walk with others.

The monastery impulse, stripped down, reduced to its essence of deep communion with God, is a powerfully transformative impulse. Enlarged by communion with others, it grows generous. Freed from the burden and unnatural restrictions of tradition, it becomes the source of such a rich profusion of creativity and connection that observers are constrained to recognize that something extraordinary is at work.

In such an atmosphere, where love and depth, generosity and creativity flow freely, no arguments are needed to persuade others that we have good news, for it is self evident.

Who we are becomes the only argument we ever need.


   I rage with a lonely rage

   against isolation, blindness and a smiling callousness,
   against the denial of our nakedness, our need

   against the illusion of goodness


sing to me connection, sing to me life, flowing,
quietly moving me to vision

sing to me a fountain of companionship


   what will these gods do for you? 

   ... these gods of all false comfort, 
   taking credit for gifts not of their making, 
   these gods who rob us, yet remain barren ...

   ... what will they do for you?

   I rage with a white hot rage


Sing to me songs of comfort
songs from silence
silence ... singing


   I rage

   against cleverness,
   cleverness masquerading as depth,
   against sophistication, imprisoning the wounded soul

   against a proud intellect, withering the spirit


sing to me


Prelude to Stillness

Many years ago a cousin of mine suggested that I make a habit of walkabouts in the woods and forests where I lived. He cited his observations of the deep peace that he always saw in me after I went on some wandering exploration, often after having started the walk in some agitated state.

I took his suggestion.

Those walkabouts in nature,  started many years ago now, have opened me to the rich, tangibly living nature of stillness, of quietness. My talks with God and connection to life, to creative flow, to the infinite, living universe we inhabit, all flow out of that quietness. Each of these pieces of Isaiah:

“In remembering and rest you are healed, in quietness and confidence is your strength”

“They that wait upon the lord shall renew their strength. They shall mount up with wings as eagles, they shall run and not be weary, they shall walk and not faint”

has become personal and tangibly real.

The Path of Quietness

“Trust in the lord with all thine heart” is the “peace, be still” state. From here flows the quietness and confidence that is my strength. The “peace be still” state releases any need for final, absolute statements or a capturing or encompassing of universal laws. In particular, it accepts the abundance of apparent paradoxes and conundrums which are actually teachers of deeper truths. It relaxes in the finite approach to the infinite.

The struggle to encompass, to make whatever finite statements of truth we can muster the focus of our trust, is a struggle to not have to abandon ourselves to the naked freedom of trust in God.

I now realize that the invitation to abandon an encompassing of final absolute truth, is an invitation to infinity. Accepting that I cannot hold it all within my being, I have opened to an infinite exploration. While God is infinite in being, we are infinite in potential. And trust releases us to experience these infinities.

By the Light of the Moon in Broad Daylight

Sometimes a piece of music resonates so deeply it seems to be singing from inside you. The music is your own — you are confident it is music you would have written, had you been in the habit of writing music.

The movie starts with music — Benjamin Britten’s A Young person’s Guide to the Orchestra being played on a child’s portable record player. Delicate, yet robust — reanimating things past, painting pictures with innocence (and a little bit of sad, jaded reality), Moonrise Kingdom is a tone poem that will stay with you long after the movie is over. The story of two 12 year olds, running away together into the wilder parts of a small island on which the entire story unfolds, is captured with a simplicity and joy that defies words. But listening to The Heroic Weather-Conditions of the Universe again, I am drawn back into the story. Clearly inspired by Britten’s piece and the story unfolded in the movie, the simplicity of Desplat’s song almost without words, captures the tale completely, vividly.

It was at the end of a day with disappointments that I went to see the movie “Moonrise Kingdom”. Letting go, I found the almost-pure innocence of a bygone era singing to me a vivid, soul-felt song, healing me with a curious kind of hope.

In the Moonrise Kingdom there reigns disarming honesty, simplicity, sweetness, and a vision of reality that clings to hope.

By the Light of the Moon in Broad Daylight

Sometimes a piece of music resonates so deeply it seems to be singing from inside you. The music is your own — you are confident it is music you would have written, had you been in the habit of writing music.

The movie starts with music — Benjamin Britten’s A Young person’s Guide to the Orchestra being played on a child’s portable record player. Delicate, yet robust — reanimating things past, painting pictures with innocence (and a little bit of sad, jaded reality), Moonrise Kingdom is a tone poem that will stay with you long after the movie is over. The story of two 12 year olds, running away together into the wilder parts of a small island on which the entire story unfolds, is captured with a simplicity and joy that defies words. But listening to The Heroic Weather-Conditions of the Universe again, I am drawn back into the story. Clearly inspired by Britten’s piece and the story unfolded in the movie, the simplicity of Desplat’s song almost without words, captures the tale completely, vividly.

It was at the end of a day with disappointments that I went to see the movie “Moonrise Kingdom”. Letting go, I found the almost-pure innocence of a bygone era singing to me a vivid, soul-felt song, healing me with a curious kind of hope.

In the Moonrise Kingdom there reigns disarming honesty, simplicity, sweetness, and a vision of reality that clings to hope.

Cultures of Disrespect

Reading a mathematical reference today, I came across a not so unusual phrase “It is easy to convince oneself …”. In this particular case, I had to have a quick look at the example they gave to see the “easy” fact for I was thinking along a nonproductive direction and had nothing to reposition my point of view. So, at that point in time, it was not “easy to convince” myself of the fact. And yet, the example was one I could easily have gotten if I had been in a slightly different frame of mind.

I began to think about the numerous phrases that can be found in usage that hide alternating sentiments of superiority and inferiority. We use these patterns often and they encode into our creative environment the limitations implicit in those ideas of comparison and measurement.

When we accept this language as our own, we limit ourselves, often quite severely. There are no fundamental limits or bounds on our creativity if we take into consideration the fact that we are here and not there. That is, if we accept where we are at, we are then free to move anywhere from there. The art with which we move, the creativity that we exhibit, the innovation and originality are never intrinsically limited. Yet so many believe that they are limited, so many have no idea of where they are, that their behavior shouts of limitations and impediments.

And the language we use either reinforces this or helps bring us to the freedom that the creative will needs to really soar.

Our culture is often a culture of disrespect. Because we do not dwell in the atmosphere of respect that characterizes quietness and stillness, our response to this culture is to begin desperately trading in a currency of disrespect.

Quietness sets us free.


fear is a scream, 
frozen into a prison.


light roars, silence 
slays fear

deeper, plunging
downward into silence and light


the deep, encompassing 
me in boundlessness

an infinite music fills
the void 


silence, failing
to contain the light, sings to me

intensity invades me with stillness,
setting me free


light sings to me

Anarchy as Optimal Versatility

In the very teachings that Christians claim to base their religion on we find clear revelations of the non-institutional structure of the flow of life and innovation:

“Ye are the salt of the earth”

with the accompanying admonition to distribute and mix throughout the world.

Shallow readings of this can be viewed as admonitions to send out missionaries and to evangelize boldly. But a deeper reading will connect with the anti-institutional, anti-organizational and even anarchist nature of the most innovative streams of inspiration and life. Freedom is ever at odds with the propagation of organizations and the rise of institutions.

Mass movements very quickly gain an organizational, institutional structure that begins immediately to destroy the pure creative fiber that is at the foundation of whatever is good in the initial inspiration. There rarely is anyone bold enough, wise enough, to remind the inspired, who are in the process of being carried away with the euphoria of revelation, that “Ye are the salt of the earth”. Freedom is quickly challenged and slowly (sometimes quickly) falls prey. Group dynamics begin to dictate individual behavior and constrain what is and what is not acceptable. At this point the inspiration has been hijacked and the demonic nature of the institution begins to hold sway.

This is not to say that all forms of individual behavior are harmonious with life and innovation. Indeed there are sociopaths and psychopaths that would, if permitted, exploit any situation or collective or group. But it is often the tyranny of the majority, expressed in the form of an organization, that exerts its destructive will on the individual, limiting the free action of the individual and the unfettered creation of living diversity.

It is one of the apparent mysteries that inspiration and degradation share such close quarters, that the euphoria of inspiration can so quickly turn into evil. Part of the unraveling of this mystery surely lies in the fact that inspiration gives power and power very easily corrupts — and in groups, humans do things that they would hesitate to do individually. For in quietness we see most clearly.

But there are organizations that emerge spontaneously and are purely cooperative, transitory phenomena, not violating or leading to the loss of freedom. This kind of grass roots behavior is highly fluid. In its purest form it leads to the accomplishment of some immediate goal at which time the collective dissipates into its creative, living pieces, gathering new energy and diversity, becoming better prepared for the next emergent goal.

This, though seen faintly, through a glass darkly, keeps our hope alive.

Brilliance and Renaissance

Today, I was moved and inspired by the documentary “The Philosopher Kings”. It reminded me of a deeper level of awareness, of life and brilliance that awaits our quietness and attention. Life as ideas, as art, as the true university — life as immersion in those bright beams of illumination awaiting the attentive.

Brilliance breaks through to the ready and attentive, here and there, now and then. Illumination floods through the embrace of even one insight. Brilliance is often described as a feeling, because it is nothing more or less than full immersion into the stream of life. At a fundamental level it has nothing to do with the recognition of others or the acclaim of an adoring audience.

Illumination waits everywhere, at all times, for anyone who will see.

Quietness opens the eyes.

Brilliance is freely available, yet many avoid the quietness that would make them aware of brilliance. In quietness, the life and depth in everything becomes visible. In the mundane, uninspired lives so many believe they have, inspired joy is close at hand, just below the surface, ever ready to illuminate. The smallest steps in the direction of life begin to transform, to open and heal.

Solitude and connection, point and counterpoint, brings a growing awareness of light and creative power. The world becomes a deeply informative study, the invitation to illumination, to brilliance that encompasses and moves to something larger.

Then we teach.

Teaching, we have come full circle, but not to the place we began. Having embraced life and illumination, this is simply the place that we consciously take on the mentoring of others. Dwelling in the place without limits, we find others attracted to the life flowing over and outward.

Others join us.

That is renaissance.

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