I just had the chance to attend two sets with Vijay Iyer and Wadada Leo Smith tonight! That happened at The Stone, a small music room in NYC owned by John Zorn that features avant-garde jazz music (but not only).

The first set was a plain duet of these two artists. The Stone was packed and we had to sit on the floor. After 10 quite boring minutes during which Vijay played electronics only, he took on the piano and music emerged. Although Vijay had sheets of music prepared, this set sounded very free, especially concerning Wadada Leo Smith's playing—it seems he used all what a trumpet allows to create sound. However, the atmosphere was peaceful. For those who know some of Wadada Leo Smith's music, this was closer to Kulture Jazz than to Ten Freedom Summers which I had discussed on this blog last year

For the second set, came along Reggie Workman at the bass, Nitin Mitta on the tablas, and Patricia Franceschy on vibes. This made the music sound quite differently. The musicians had decided of a few melodic lines and ostinatos, and grooved on that. The tablas gave a wonderful color to the music, similar as the one on Tirtha (with Prasanna on the guitar, and Nitin Mitta on the tablas). The vibes also gave a good touch. It seems that there are nice vibes players in free jazz nowadays; I'm thinking for example of Jason
Adasiewicz who plays in Nicole Mitchelle's Ice Crystals group.

It was my first night in New York City since 2 years. I am happy to have had the opportunity to hear these great artists. Tomorrow night, if the announced snow storm permits, I'll go listen to Ari Hoenig at the Smalls!

# Freedom Math Dance

A blog about math (mainly), computer tricks (sometimes) and jazz music.

## Monday, January 26, 2015

## Wednesday, October 1, 2014

### Book review: Contemplative Practices in Higher Education

*Contemplative Practices in Higher Education*, by Daniel P. Barbezat and Mirabai Bush.

Jossey-Bass, 2014.

[Center for Contemplative Mind in Society] [Library of Congress] [Amazon] [Fnac] [Barnes&Noble]

“Contemplative practices in higher education”? what the f...? Does this means that we should have to have our students meditate instead of practicing mathematics by doing more and more exercises? Again, what the f... ? And is it really appropriate, in our universities (which, in France, are mostly

*laïques et républicaines*) to experiment such practices?

The subtitle of the book under review should perhaps reassure us:

*Powerful methods to transform teaching and learning*. Indeed, as its authors explain to us in the very first lines of its preface, contemplative practices always has a well established place in the intellectual inquiry, a place which goes well beyond their vital role in all the major religions and spiritual traditions. The authors acknowledge many objectives to these practices, pointing out 4 of them whose importance can difficultly be denied:

- Development of attention and focus;
- Deeper understanding of the content of the course;
- Compassion, relation with self; deepening of the moral and spiritual component of education;
- Development of personality, and of creativity.

Daniel Barbezat, a professor in economics at Amherst, explains for example how these methods allowed him to solve the following contradiction: how is it possible that his field (economics) pretends studying the mechanisms of decision that are supposed to lead people to well-being, without every considering the nature of well-being? He proposed to his class various alternatives, of the following kind:

- The class is divided in ten groups of three people; the member of one group receive $1000 each, the other nothing
- Everybody receives $200

David Haskell, who teaches environmental sciences and biology, adapted the reading method of monks (as he says,

*lectio*without too much

*divina*) to have his class study problems of hunger and development. He asked his students to alternate between periods of quiet rest (say one minute) and the reading of one or two sentences of the text (each one reads by turns) and to brief commentaries by the students, etc. Other teachers propose the students to behold some text, or some graphic representation, and then to comment it. Examples are given of the probability distribution of the hydrogen bromide atom, according to its energy levels, or to two charts of industrial production (in absolute vs relative value). The authors claim that such exercises deepen the relation with self, with the studied document, and with other materials of the course.

Mathematics are absent from this book. In a blog post hosted by the American mathematical society, Luke Wolcott evokes this possibility, but acknowledges that he did not go further than personal meditation. In fact, I could not find other explicit examples in various sources, even none in the archives of the Center for contemplative mind in society that the authors of this book lead. However, it seems to me that some practical exercises organised by a teacher such as Adrien Guinemer in his middle/high school classes go in that direction (notably, the study of sections of cubes, cones, cylinder made from plasticine).

There are at least two methods that I find interesting and that could easily be implemented in our classes:

- Meditation exercises at the beginning of the class — first have everybody focus his attention on its breath during five minutes, and then report it on the subject of the class.
- Introspection techniques to fight failure anxiety — the student is asked to solve an exercise while writing on his sheet everything that comes to his mind, whatever relation it has with the exercise.

The first part of the book proposes a theoretical and practical background that is necessary to appreciate the variety of these methods, as well as some issues that need to be avoided. Three of them seem particularly crucial to me, all of them requiring from the teacher a quite deep personal involvement in these contemplative practices:

- Assign to the contemplative exercises a clear pedagogical goal, whose impact can be evaluated;
- Disjoint the practice of these exercises from the cultural and religious backgrounds in which they were first devised;
- Be able of managing students who would not be at ease, or even would reject, such practices.

So let us try, and see.

## Saturday, May 31, 2014

### The evolution of higher education

After a few months of silence, a short blog post to indicate a few web links that I found interesting, rising concern about the evolution of higher education.

In February 2014, Counterpunch published a series of remarks by Noam Chomsky under the title

More recently (May 2014), the New York Times published an editorial,

In fact, I had been made aware of the problem by a few posts from the blog The Homeless Adjunct, notably this post from 2012 that clearly explains

In February 2014, Counterpunch published a series of remarks by Noam Chomsky under the title

*On Academic Labor*. (I found it first on Alternet, under the alternate title*How America's Great University System Is Getting Destroyed*.)*Fat-Cat Administrators at the Top 25*, where they quote a report from the Institute for Policy Studies indicating that "*student debt and low-wage faculty labor are rising faster at state universities with the highest-paid presidents*."In fact, I had been made aware of the problem by a few posts from the blog The Homeless Adjunct, notably this post from 2012 that clearly explains

*how the American university system was killed in five easy steps*:- Defund the university system;
- Deprofessionalize and impoverish the professors;
- Install a managerial/administrative class who take over governance of the university;
- Move in corporate culture and corporate money;
- Destroy the students.

- Decisive progress towards defunding was made in 2009 by the Sarkozy-Pécresse LRU-law. While the acronym stands for Liberty and Responsability of Universities, this law has been infamously referred to Autonomy of Universities. The French public universities are now allocated a global budget by the State, which they are now supposed to manage as they wish, except that the allocated budget is insufficient, and that they have almost no control of whatever. Many universities are on the edge of defaulting. So what we have under the eyes is nothing but a defunding of the system disguised as a change of allocation model.
- The number of permanent positions is sharply decreasing. Of course, the age-pyramid of the present professors is also a cause for this evolution, since almost all baby-boomers have now retire. But the decrease is not at all the same in all fields—for example, this year, there were many more open positions in applied mathematics than in pure mathematics. Probably, when it comes about cutting positions, the "applied"-color makes it nicer for university boards. Probably too, applied mathematicians have been better at explaining their rôle in society.
- Meanwhile, the administration is getting fatter. To manage the global budget, it has been necessary to hire full-time "managers". And to be able to attract them, it seems that their pay has nothing to do with the usual range among French public servants. At the same time, a new law reorganizes the higher-education system by forcing universities (as well as our innumerous engineering schools) to regroup themselves. This will create enormous beasts that will look like the Lernean Hydra. For example, the Paris-Saclay University regroups 22 higher education schools, among which 2 universities and 10 "grandes écoles"; it will host around 50.000 students and more than 10.000 professors and researchers! No doubt that it will require a heavy bureaucracy to manage this high number of people. And since we're split in many institutions, it will be hard to have the voice of academic freedom be listened to.

## Friday, February 14, 2014

### A map of the universe

If you'd be asked to tell what a

Here theories have esoteric nicknames, such as ACF, ACVF, SCF$_p^n$, or ``universal graph omitting a bowtie'' (an homage to Tom S. ? :-)), and properties have even more esoteric nicknames — NIP, o-minimal, NSOP$_{n+1}$, or superstable. To make it something more than an enjoyable

By the way, this is also a beautiful illustration of the power of HTML5.

*map of the universe*looks like, I'm pretty sure you'd imagine something on a dark background, with many dots representing planets, and shaded areas corresponding to galaxies. That map of the universe, drawn by Gabriel Conant, a graduate student at Berkeley, is of more or less like that. Except that dots are mathematical theories, and galaxies correspond to some stability properties defined in model theory.Here theories have esoteric nicknames, such as ACF, ACVF, SCF$_p^n$, or ``universal graph omitting a bowtie'' (an homage to Tom S. ? :-)), and properties have even more esoteric nicknames — NIP, o-minimal, NSOP$_{n+1}$, or superstable. To make it something more than an enjoyable

*invitation au voyage*, Gabriel indicated important specific examples, with their definitions and references.By the way, this is also a beautiful illustration of the power of HTML5.

## Saturday, February 1, 2014

### Wadada Leo Smith: Ten Freedom Summers

Last Saturday (January 25th), I attended a concert by trumpetist Wadada Leo Smith in Vitry/Seine, within the Sons d'hiver festival, which serves as a pretext for this blog entry.

I had been introduced to Wadada Leo Smith's music thanks to France Musique program

In

For

Then, although some parts of the concert seemed to be improvised, it all looked as if they played the music as it is written on a score. This was the more surprising for the drummer who, most always in jazz music, is left to imagining by himself how he should bring his playing to the music being created. (When drummers have scores, that's rarely drum scores, but more often that of the bass player, or simply the main theme with the chords changes.)

Even Pheeroan akLaff was obviously playing the drums as written on the score, but the compositions gave him a quite interesting role in the development of the music. Wadada Leo Smith had written long solos for the drums which began or ended the pieces. In fact, since the group that night had no bass player, but a cellist who played with the bow — anyway, Lindberg mostly plays with the bow on the CD too — the other musicians are not given the explicit harmonic/rhythmic pattern that a “walking bass” can impose on the music, so there's probably no point for the drummer to play a definite swing rhythm, which akLaff did not do.

And within Wadada Leo Smith's mostly meditative music, that was akLaff's playing — sometimes forceful, or with traces of military marches — that reminded us that Freedom is a fight.

An everyday-fight.

The first part featured the Anti Pop Consortium's machine-player HPrizm, accompanied by improvisers David Virelles (piano), Steve Lehman (saxophone) and Wadada Leo Smith (trumpet), as well by Emanuuel Pidre (visuals). I found this part a bit bland. HPrizm's music lacked inspiration, rhythm, and although the improvisers are remarkable musicians, it was probably difficult for them to build on a lame material. Steve Lehman saved most of it, I think, because his playing is very lyrical, and quite dense, so that he could made the music.

The second part was Wadada Leo Smith's

The second part was Wadada Leo Smith's

*Ten Freedom Summers —*well, only a part of it, although they played for almost two hours.*Ten Freedom Summers*is the title of a monumental series of compositions by Wadada Leo Smith: 19 pieces, lasting for 4 hours and a half, depicting those moments of American history where African american people fought for Freedom. The first piece, “Dred Scott, 1857” recalls the story of Dred Scott, a slave who filed a suit at the Supreme Court to be able to buy his freedom, and lost, when the Supreme court ruled (1857) that people of African origin, whether slave or free, were not citizens of the United States —anyway, Scott had been freed the very same year by his new owner. The second piece is about the Montgomery bus boycott initiated by Rosa Parks in 1944. Two pieces are also devoted to the US presidents John F. Kennedy and Lyndon Johnson, celebrating the New Frontier, and the Civil Rights act of 1964.I had been introduced to Wadada Leo Smith's music thanks to France Musique program

*Open Jazz*, when Alex Dutilh aired the piece “Kulture of Jazz”, from the*Kulture Jazz*CD. Most of the pieces of that disc are evocations of jazz through some prominent figures of jazz (Louis Armstrong, Billie Holiday, John Coltrane, Albert Ayler), African literature (Ayl Kwel Armah), or his personal life (Sarah Brown-Smith-Wallace). The “K” in the title, which reminds me of the Klan, already emphasized the fact that jazz is an African-american music of emancipation. That is the music that black people played, but didn't have the right to listen to.In

*Kulture Jazz*, Wadada Leo Smith is the only credited musician. He mostly plays the trumpet, an instrument that truly belongs to jazz music (although it is slightly less heared these days), but he also sings, plays percussions, as well as koto, a rarely heard instrument in this context!For

*Ten Freedom Summers,*he combines a jazz quartet (trumpet, bass, drums and piano) and the Southwest Chamber Ensemble, a 9-musician strings combo. The combo that was initially announced for the concert was Smith's Golden quartet (with Anthony Davis, piano; John Lindberg, bass; Pheeroan akLaff, drums), except that the bassist had broken his wrist and could not play. He was thus replaced by Ashley Waters (from the Southwest Chamber Ensemble) on cello. Consequently, but that's probably one of the miracles allowed for by improvised music, the concert sounded pretty much like the recorded music.
Anyway, both

First, the music sounds different. For example, there is no rhythm section in *Kulture Jazz*and*Ten Freedom Summers*are very different from other jazz pieces devoted to the civil rights movements that I know (such as Max Roach's*Freedom Now. We Insist!*whose “Triptych: Prayer/Protest/Peace” is one of the rare jazz pieces that made me cry, or from Charles Mingus's “Fables of Faubus”).*Kulture Jazz*, and almost nothing as such in*Ten Freedom Summers.*And the pieces are definitely not built on the classical form (rhythmic/harmonic) we're now used to, either from listening to classical music, or from blues, or from the modal pieces played by Miles Davis, John Coltrane and others in the 50s-60s. Maybe not unlike latter pieces by Coltrane (Love Supreme, or Interstellar Space), Wadada Leo Smith's music is an abstract meditation about the place of an African-american musician in History.Then, although some parts of the concert seemed to be improvised, it all looked as if they played the music as it is written on a score. This was the more surprising for the drummer who, most always in jazz music, is left to imagining by himself how he should bring his playing to the music being created. (When drummers have scores, that's rarely drum scores, but more often that of the bass player, or simply the main theme with the chords changes.)

Even Pheeroan akLaff was obviously playing the drums as written on the score, but the compositions gave him a quite interesting role in the development of the music. Wadada Leo Smith had written long solos for the drums which began or ended the pieces. In fact, since the group that night had no bass player, but a cellist who played with the bow — anyway, Lindberg mostly plays with the bow on the CD too — the other musicians are not given the explicit harmonic/rhythmic pattern that a “walking bass” can impose on the music, so there's probably no point for the drummer to play a definite swing rhythm, which akLaff did not do.

And within Wadada Leo Smith's mostly meditative music, that was akLaff's playing — sometimes forceful, or with traces of military marches — that reminded us that Freedom is a fight.

An everyday-fight.

## Thursday, January 9, 2014

### Radon measures form a sheaf for a natural Grothendieck topology on topological spaces

First post of the year, so let me wish all of you a happy new year!

Almost two years ago, Antoine Ducros and I released a preprint about differential forms and currents on Berkovich spaces. We then embarked in revising it thoroughly; unfortunately, we had to correct a lot of inaccuracies, some of them a bit daunting. We made a lot of progress and we now have a much clearer picture in mind. Fortunately, all of the main ideas remain the same.

A funny thing emerged, which I want to explain in this blog.

One of our mottos was to define

Also, Grothendieck topologies play such an important rôle in analytic geometry over non-archimedean fields; this is obvious for classical rigid spaces, but they are also important in Berkovich geometry, in particular if you want to care about possibly non-good spaces for which points may not have a neighborhood isomorphic to an affinoid space. So it was natural to sheafify the presheaf of tropical forms for the G-topology, giving rise to a G-sheaf of G-forms.

Now, every differential form of maximal degree $\omega$ on a Berkovich space $X$ gives rise to a measure on the topological space underlying $X$. Our proof of this is a bit complicated, and was made more complicated by the fact that we first tried to define the integral $\int_X \omega$, and then defined $\int_X f\omega$ for every smooth function $f$, and then got $\int_X f\omega$ for every continuous function with compact support $f$ by approximation, using a version of the Stone-Weierstrass theorem in our context.

In the new approach, we directly concentrate on the measure that we want to construct. For G-forms, this requires to glue measures defined locally for the G-topology. As it comes out (we finished to write down the required lemmas today), this is quite nice.

Since Berkovich spaces are locally compact, we may restrict ourselves to classical measure theory on locally compact spaces. However, we may not make any metrizability assumption, nor any countability assumption, since the most basic Berkovich spaces lack those properties. Assume that the ground non-archimedean field $k$ is the field $\mathbf C((t))$ of Laurent series over the field $\mathbf C$ of complex numbers. Then the projective line $\mathrm P^1$ over $k$ is not metrizable, and the complement of its ``Gauss point'' $\gamma$ has uncountably many connected components (in bijection with the projective line over $\mathbf C$). Similarly, the complement of the Gauss point in the projective plane $\mathrm P^2$ over $k$ is connected, but is not countable at infinity, hence not paracompact.

As always, there are two points of view on measure theory: Borel measures (countably additive set functions on the $\sigma$-algebra of Borel sets) and Radon measures (linear forms on the vector space of continuous compactly supported functions). By the theorem of Riesz, they are basically equivalent: locally finite, compact inner regular Borel measures are in canonical bijection with Radon measures. Unfortunately, basic litterature is not very nice on that topic; for example, Rudin's book constructs an outer regular Borel measure which may not be inner regular, while for us, the behavior on compact sets is really the relevant one.

Secondly, we need to glue Radon measures defined on the members of a G-cover of our Berkovich space $X$. This is possible because Radon measures on a locally compact topological space naturally form a sheaf for a natural

Let $X$ be a locally compact topological space and let us consider the category of locally compact subspaces, with injections as morphisms. Radon measures can be restricted to a locally compact subspace, hence form a presheaf on that category.

Let us decree that a family $(A_i)_{i\in I}$ of locally compact subspaces of a locally compact subspace $U$ is a B-cover (B is for Borel) if for every point $x\in U$, there exists a finite subset $J$ of $I$ such that $x\in A_i$ for every $i\in J$ and such that $\bigcup_{i\in J}A_i$ is a neighborhood of $x$. B-covers form a G-topology on the category of locally compact subsets, for which

This said, the proof (once written down carefully) is not a big surprise, nor specially difficult, but I found it nice to get a natural instance of sheaf for a Grothendieck topology within classical analysis.

Almost two years ago, Antoine Ducros and I released a preprint about differential forms and currents on Berkovich spaces. We then embarked in revising it thoroughly; unfortunately, we had to correct a lot of inaccuracies, some of them a bit daunting. We made a lot of progress and we now have a much clearer picture in mind. Fortunately, all of the main ideas remain the same.

A funny thing emerged, which I want to explain in this blog.

One of our mottos was to define

*sheaves*of differential forms, or of currents. Those differential forms were defined in two steps : by definition, they are locally given by tropical geometry, so we defined a presheaf of tropical forms, and passed at once to the associated sheaf. What we observed recently is that it is worth spending some time to study the*presheaf*of tropical forms.Also, Grothendieck topologies play such an important rôle in analytic geometry over non-archimedean fields; this is obvious for classical rigid spaces, but they are also important in Berkovich geometry, in particular if you want to care about possibly non-good spaces for which points may not have a neighborhood isomorphic to an affinoid space. So it was natural to sheafify the presheaf of tropical forms for the G-topology, giving rise to a G-sheaf of G-forms.

Now, every differential form of maximal degree $\omega$ on a Berkovich space $X$ gives rise to a measure on the topological space underlying $X$. Our proof of this is a bit complicated, and was made more complicated by the fact that we first tried to define the integral $\int_X \omega$, and then defined $\int_X f\omega$ for every smooth function $f$, and then got $\int_X f\omega$ for every continuous function with compact support $f$ by approximation, using a version of the Stone-Weierstrass theorem in our context.

In the new approach, we directly concentrate on the measure that we want to construct. For G-forms, this requires to glue measures defined locally for the G-topology. As it comes out (we finished to write down the required lemmas today), this is quite nice.

Since Berkovich spaces are locally compact, we may restrict ourselves to classical measure theory on locally compact spaces. However, we may not make any metrizability assumption, nor any countability assumption, since the most basic Berkovich spaces lack those properties. Assume that the ground non-archimedean field $k$ is the field $\mathbf C((t))$ of Laurent series over the field $\mathbf C$ of complex numbers. Then the projective line $\mathrm P^1$ over $k$ is not metrizable, and the complement of its ``Gauss point'' $\gamma$ has uncountably many connected components (in bijection with the projective line over $\mathbf C$). Similarly, the complement of the Gauss point in the projective plane $\mathrm P^2$ over $k$ is connected, but is not countable at infinity, hence not paracompact.

As always, there are two points of view on measure theory: Borel measures (countably additive set functions on the $\sigma$-algebra of Borel sets) and Radon measures (linear forms on the vector space of continuous compactly supported functions). By the theorem of Riesz, they are basically equivalent: locally finite, compact inner regular Borel measures are in canonical bijection with Radon measures. Unfortunately, basic litterature is not very nice on that topic; for example, Rudin's book constructs an outer regular Borel measure which may not be inner regular, while for us, the behavior on compact sets is really the relevant one.

Secondly, we need to glue Radon measures defined on the members of a G-cover of our Berkovich space $X$. This is possible because Radon measures on a locally compact topological space naturally form a sheaf for a natural

*Grothendieck topology*!Let $X$ be a locally compact topological space and let us consider the category of locally compact subspaces, with injections as morphisms. Radon measures can be restricted to a locally compact subspace, hence form a presheaf on that category.

Let us decree that a family $(A_i)_{i\in I}$ of locally compact subspaces of a locally compact subspace $U$ is a B-cover (B is for Borel) if for every point $x\in U$, there exists a finite subset $J$ of $I$ such that $x\in A_i$ for every $i\in J$ and such that $\bigcup_{i\in J}A_i$ is a neighborhood of $x$. B-covers form a G-topology on the category of locally compact subsets, for which

*Radon measures form a sheaf*! In other words, given Radon measures $\mu_i$ on members $A_i$ of a B-cover of $X$ such that the restrictions to $A_i\cap A_j$ of $\mu_i$ and $\mu_j$ coincide, for all $i,j$, then there exists a unique Radon measure on $X$ whose restriction to $A_i$ equals $\mu_i$, for every $i$.This said, the proof (once written down carefully) is not a big surprise, nor specially difficult, but I found it nice to get a natural instance of sheaf for a Grothendieck topology within classical analysis.

## Sunday, December 8, 2013

### Homotopy type theory on Images des mathématiques

This post will be a short advertisement to a longer general audience text about homotopy type theory that I published on the website Images des mathématiques.

In this text, I try to convey my excitement at the reading of the book published by the participants of last year's IAS program, under direction of Steve Awoodey, Thierry Coquand and Vladimir Voevodsky. As I write there (this is the title of this article), this remarkable work is at the crossroads of foundations of mathematics, topology and computer science. Indeed, the new foundational setup for mathematics provided by type theory may not only replace set theory; it is also at the heart of the systems for computer proof checking, and gave birth to a new kind of ``synthetic homotopy theory'' which is totally freed of the general topology framework.

Also remarkable is the way this book was produced: written collaboratively, using technology well known in open source software's development, then published under a Creative commons's license, and printed on demand.

This is not the only general audience paper on this subject, probably not the last one neither. Here are links to those I know of:

In this text, I try to convey my excitement at the reading of the book published by the participants of last year's IAS program, under direction of Steve Awoodey, Thierry Coquand and Vladimir Voevodsky. As I write there (this is the title of this article), this remarkable work is at the crossroads of foundations of mathematics, topology and computer science. Indeed, the new foundational setup for mathematics provided by type theory may not only replace set theory; it is also at the heart of the systems for computer proof checking, and gave birth to a new kind of ``synthetic homotopy theory'' which is totally freed of the general topology framework.

Also remarkable is the way this book was produced: written collaboratively, using technology well known in open source software's development, then published under a Creative commons's license, and printed on demand.

This is not the only general audience paper on this subject, probably not the last one neither. Here are links to those I know of:

- Voevodsky's mathematical revolution, by Julie Rehmeyer (Scientific American)
- Penser types plutôt qu'ensembles, by Philippe Pajot (Science & Vie, October 2013)
- Voevodsky’s Univalence Axiom in Homotopy Type Theory, by Steve Awodey, Álvaro Pelayo, and Michael A. Warren (Notices of the AMS, September 2013)

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