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Category: books

Mathematics in times of internet

Published November 26, 2017 by lievenlb

A few weeks more of (heavy) teaching ahead, and then I finally hope to start on a project, slumbering for way too long: to write a book for a broader audience.

Prepping for this I try to read most of the popular math-books hitting the market.

The latest two explore how the internet changed the way we discuss, learn and do mathematics. Think Math-Blogs, MathOverflow and Polymath.

‘Gina says’, Adventures in the Blogosphere String War



The ‘string wars’ started with the publication of the books by Peter Woit:

Not even wrong: the failure of string theory and the search for unity in physical law

and Lee Smolin:

The trouble with physics: the rise of string theory, the fall of a science, and what comes next.

In the summer of 2006, Gil Kalai got himself an extra gmail acount, invented the fictitious ‘Gina’ and started commenting (some would argue trolling) on blogs such as Peter Woit’s own Not Even Wring, John Baez and Co.’s the n-Category Cafe and Clifford Johnson’s Asymptotia.

Gil then copy-pasted Gina’s comments, and the replies they provoked, into a leaflet and put it on his own blog in June 2009: “Gina says”, Adventures in the Blogosphere String War.

Back then, it was fun to waste an afternoon re-reading all of this, and I wrote about it here:

Now here’s an idea (June 2009)

Gina says, continued (August 2009)

With only minor editing, and including some drawings by Gil’s daughter, these leaflets have now resurfaced as a book…?!

After more than 10 years I had hoped that Gil would have taken this test-case to say some smart things about the math-blogging scene and its potential to attract more people to mathematics, or whatever.

In 2009 I wrote:

“Having read the first 20 odd pages in full and skimmed the rest, two remarks : (1) it shouldn’t be too difficult to borrow this idea and make a much better book out of it and (2) it raises the question about copyrights on blog-comments…”

Closing the gap: the quest to understand prime numbers



I can hear you sigh, but no, this is not yet another prime number book.

In May 2013, Yitang Zhang startled the mathematical world by proving that there are infinitely many prime pairs, each no more than 70.000.000 apart.

Perhaps a small step towards the twin prime conjecture but it was the first time someone put a bound on this prime gap.

Vicky Neal‘s book tells the story of closing this gap. In less than a year the bound of 70.000.000 was brought down to 246.

If you’ve read all popular prime books, there are a handful of places in the book where you might sigh: ‘oh no, not that story again’, but by far the larger part of the book explains exciting results on prime number progressions, not found anywhere else.

Want to know about sieve methods?

Which results made Tim Gowers or Terry Tao famous?

What is Szemeredi’s theorem or the Hardy-Littlewood circle method?

Ever heard about the Elliot-Halberstam or the Erdos-Turan conjecture? The work by Tao on Erdos discrepancy problem or that of James Maynard (and Tao) on closing the prime gap?

Closing the gap is the book to read about all of this.

But it is much more.

It tells about the origins and successes of the Polymath project, and details the progress made by Polymath8 on closing the gap, it gives an insight into how mathematics is done, what role conferences, talks and research institutes a la Oberwolfach play, and more.

Looking for a gift for that niece of yours interested in maths? Look no further. Closing the gap is a great book!

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Pariah moonshine and math-writing

Published October 20, 2017 by lievenlb

Getting mathematics into Nature (the journal) is next to impossible. Ask David Mumford and John Tate about it.

Last month, John Duncan, Michael Mertens and Ken Ono managed to do just that.

Inevitably, they had to suffer through a photoshoot and give their university’s PR-people some soundbites.

CAPTION

In the simplest terms, an elliptic curve is a doughnut shape with carefully placed points, explain Emory University mathematicians Ken Ono, left, and John Duncan, right. “The whole game in the math of elliptic curves is determining whether the doughnut has sprinkles and, if so, where exactly the sprinkles are placed,” Duncan says.

CAPTION

“Imagine you are holding a doughnut in the dark,” Emory University mathematician Ken Ono says. “You wouldn’t even be able to decide whether it has any sprinkles. But the information in our O’Nan moonshine allows us to ‘see’ our mathematical doughnuts clearly by giving us a wealth of information about the points on elliptic curves.”

(Photos by Stephen Nowland, Emory University. See here and here.)

Some may find this kind of sad, or a bad example of over-popularisation.

I think they do a pretty good job of getting the notion of rational points on elliptic curves across.

That’s what the arithmetic of elliptic curves is all about, finding structure in patterns of sprinkles on special doughnuts. And hey, you can get rich and famous if you’re good at it.

Their Nature-paper Pariah moonshine is a must-read for anyone aspiring to write a math-book aiming at a larger audience.

It is an introduction to and a summary of the results they arXived last February O’Nan moonshine and arithmetic.

Update (October 21st)

John Duncan send me this comment via email:

“Strictly speaking the article was published in Nature Communications (https://www.nature.com/ncomms/). We were also rejected by Nature. But Nature forwarded our submission to Nature Communications, and we had a great experience. Specifically, the review period was very fast (compared to most math journals), and the editors offered very good advice.

My understanding is that Nature Communications is interested in publishing more pure mathematics. If someone reading this has a great mathematical story to tell, I (humbly) recommend to them this option. Perhaps the work of Mumford–Tate would be more agreeably received here.

By the way, our Nature Communications article is open access, available at https://www.nature.com/articles/s41467-017-00660-y.”

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how much to spend on (cat)books?

Published December 9, 2016 by lievenlb

My favourite tags on MathOverflow are big-lists, big-picture, soft-question,
reference-request and the like.

Often, answers to such tagged questions contain sound reading advice, style: “road-map to important result/theory X”.

Two more K to go, so let’s spend some more money.

[section_title text=”Category theory”]

One of the problems with my master course on algebraic geometry is that the students are categorical virgins.

They’ve been studying specific categories, functors, natural transformations and more all over their bachelor years, without knowing the terminology.

It then helps to illustrate these concepts with examples. For example that the determinant is a natural transformation, or that $\mathbb{C}[t]$ represents the functor forgetting the ring structure.

The more examples the merrier. I like Riehl’s example that in the category of graphs, the complete graph $K_n$ represents the functor assigning to a graph the set of all its $n$-colourings.

So, I had a look at the MathOverflow question Is Mac Lane still the best place to learn category theory?.

It is always a good idea to support authors offering a free online version of their book.

Abstract and Concrete Categories: The Joy of Cats by J. Adamek,H. Herrlich and G. Strecker. Blurb: “This up-to-date introductory treatment employs the language of category theory to explore the theory of structures. Its unique approach stresses concrete categories, and each categorical notion features several examples that clearly illustrate specific and general cases.”

Free online version : The Joy of Cats

Category Theory for the Sciences by David Spivak. Blurb: “Using databases as an entry to category theory, it begins with sets and functions, then introduces the reader to notions that are fundamental in mathematics: monoids, groups, orders, and graphs — categories in disguise. After explaining the “big three” concepts of category theory — categories, functors, and natural transformations — the book covers other topics, including limits, colimits, functor categories, sheaves, monads, and operads. The book explains category theory by examples and exercises rather than focusing on theorems and proofs. It includes more than 300 exercises, with solutions.”

Free online version: Category theory for scientists

Category Theory in Context by Emily Riehl. Blurb: “Suitable for advanced undergraduates and graduate students in mathematics, the text provides tools for understanding and attacking difficult problems in algebra, number theory, algebraic geometry, and algebraic topology. Drawing upon a broad range of mathematical examples from the categorical perspective, the author illustrates how the concepts and constructions of category theory arise from and illuminate more basic mathematical ideas. ”

Free online version: Category theory in context

Now, for the heavier stuff.

If I want to study Jacob Lurie’s books “Higher Topoi Theory”, “Derived AG”, what prerequisites should I have?

Simplicial Objects in Algebraic Topology by Peter May. Blurb: “Since it was first published in 1967, Simplicial Objects in Algebraic Topology has been the standard reference for the theory of simplicial sets and their relationship to the homotopy theory of topological spaces. ”

Free online version: Simplicial Objects in Algebraic Topology (h/t David Roberts via the comments)

A Concise Course in Algebraic Topology by Peter May. Blurb: “J. Peter May’s approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. ”

Free online version: A Concise Course in Algebraic Topology

Or in Lurie’s words: “To read Higher Topos Theory, you’ll need familiarity with ordinary category theory and with the homotopy theory of simplicial sets (Peter May’s book “Simplicial Objects in Algebraic Topology” is a good place to learn the latter). Other topics (such as classical topos theory) will be helpful for motivation.”

He also has a suggestion for the classic topos theory stuff:

“”Sheaves in Geometry and Logic” by Moerdijk and MacLane is a pretty good read (as is Uncle John, but I’ve never seen topos theory in there).”

I’ve had this book on permanent loan from our library over the past two years, so it’s about time to have my own copy.

Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Mac Lane and Moerdijk. Blurb: “Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various kinds of manifolds. Sheaves also appear in logic as carriers for models of set theory. This text presents topos theory as it has developed from the study of sheaves. Beginning with several examples, it explains the underlying ideas of topology and sheaf theory as well as the general theory of elementary toposes and geometric morphisms and their relation to logic.”

Higher Topos Theory by Jacob Lurie. Blurb: “Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory’s new language. The result is a powerful theory with applications in many areas of mathematics.”

Free online version: Higher topos theory

Although it is unlikely that I can use this left-over money from a grant to pre-order a book, let’s try

Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic ‘bridges’ by Olivia Caramello. Blurb: “According to Grothendieck, the notion of topos is “the bed or deep river where come to be married geometry and algebra, topology and arithmetic, mathematical logic and category theory, the world of the continuous and that of discontinuous or discrete structures”. It is what he had “conceived of most broad to perceive with finesse, by the same language rich of geometric resonances, an “essence” which is common to situations most distant from each other, coming from one region or another of the vast universe of mathematical things”. ”

And, as I also teach a course on the history of mathematics, let’s include:

Tool and Object: A History and Philosophy of Category Theory by Ralph Krömer. Blurb: “This book describes the history of category theory whereby illuminating its symbiotic relationship to algebraic topology, homological algebra, algebraic geometry and mathematical logic and elaboratively develops the connections with the epistemological significance.”

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let’s spend 3K on (math)books

Published December 8, 2016 by lievenlb

Santa gave me 3000 Euros to spend on books. One downside: I have to give him my wish-list before monday. So, I’d better get started. Clearly, any further suggestions you might have will be much appreciated, either in the comments below or more directly via email.

Today I’ll focus on my own interests: algebraic geometry, non-commutative geometry and representation theory. I do own a fair amount of books already which accounts for the obvious omissions in the lists below (such as Hartshorne, Mumford or Eisenbud-Harris in AG, Fulton-Harris in RT or the ‘bibles’ in NCG).

[section_title text=”Algebraic geometry”]

Here, I base myself on (and use quotes from) the excellent answer by Javier Alvarez to the MathOverflow post Best Algebraic Geometry text book? (other than Hartshorne).

In no particular order:

Lectures on Curves, Surfaces and Projective Varieties by Ettore Carletti, Dionisio Gallarati, and Giacomo Monti Bragadin and Mauro C. Beltrametti.
“which starts from the very beginning with a classical geometric style. Very complete (proves Riemann-Roch for curves in an easy language) and concrete in classic constructions needed to understand the reasons about why things are done the way they are in advanced purely algebraic books. There are very few books like this and they should be a must to start learning the subject. (Check out Dolgachev’s review.)”

A Royal Road to Algebraic Geometry by Audun Holme. “This new title is wonderful: it starts by introducing algebraic affine and projective curves and varieties and builds the theory up in the first half of the book as the perfect introduction to Hartshorne’s chapter I. The second half then jumps into a categorical introduction to schemes, bits of cohomology and even glimpses of intersection theory.”

Liu Qing – “Algebraic Geometry and Arithmetic Curves”. “It is a very complete book even introducing some needed commutative algebra and preparing the reader to learn arithmetic geometry like Mordell’s conjecture, Faltings’ or even Fermat-Wiles Theorem.”

Görtz; Wedhorn – Algebraic Geometry I, Schemes with Examples and Exercises. labeled ‘the best on schemes’ by Alvarez. “Tons of stuff on schemes; more complete than Mumford’s Red Book. It does a great job complementing Hartshorne’s treatment of schemes, above all because of the more solvable exercises.”

Kollár – Lectures on Resolution of Singularities. “Great exposition, useful contents and examples on topics one has to deal with sooner or later.”

Kollár; Mori – Birational Geometry of Algebraic Varieties. “Considered as harder to learn from by some students, it has become the standard reference on birational geometry.”

And further, as a follow-up on their previous book on the computational side of AG:

Using Algebraic Geometry by Cox, Little and O’Shea.

[section_title text=”Non-commutative geometry”]

ncgbookspng

Noncommutative Geometry and Particle Physics by Walter van Suijlekom. Blurb: “This book provides an introduction to noncommutative geometry and presents a number of its recent applications to particle physics. It is intended for graduate students in mathematics/theoretical physics who are new to the field of noncommutative geometry, as well as for researchers in mathematics/theoretical physics with an interest in the physical applications of noncommutative geometry. In the first part, we introduce the main concepts and techniques by studying finite noncommutative spaces, providing a “light” approach to noncommutative geometry. We then proceed with the general framework by defining and analyzing noncommutative spin manifolds and deriving some main results on them, such as the local index formula. In the second part, we show how noncommutative spin manifolds naturally give rise to gauge theories, applying this principle to specific examples. We subsequently geometrically derive abelian and non-abelian Yang-Mills gauge theories, and eventually the full Standard Model of particle physics, and conclude by explaining how noncommutative geometry might indicate how to proceed beyond the Standard Model.”

An Invitation To Noncommutative Geometry by Matilde Marcolli. Blurb: “This is the first existing volume that collects lectures on this important and fast developing subject in mathematics. The lectures are given by leading experts in the field and the range of topics is kept as broad as possible by including both the algebraic and the differential aspects of noncommutative geometry as well as recent applications to theoretical physics and number theory.”

Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory. Blurb: “This collection of expository articles grew out of the workshop “Number Theory and Physics” held in March 2009 at The Erwin Schrödinger International Institute for Mathematical Physics, Vienna. The common theme of the articles is the influence of ideas from noncommutative geometry (NCG) on subjects ranging from number theory to Lie algebras, index theory, and mathematical physics. Matilde Marcolli’s article gives a survey of relevant aspects of NCG in number theory, building on an introduction to motives for beginners by Jorge Plazas and Sujatha Ramdorai.”

Feynman Motives by Matilde Marcolli. Blurb: “This book presents recent and ongoing research work aimed at understanding the mysterious relation between the computations of Feynman integrals in perturbative quantum field theory and the theory of motives of algebraic varieties and their periods. One of the main questions in the field is understanding when the residues of Feynman integrals in perturbative quantum field theory evaluate to periods of mixed Tate motives.” But then, check out Matilde’s recent FaceBook status-update.

[section_title text=”Representation theory”]

repthybookspng

An Introduction to the Langlands Program by J. Bernstein (editor). Blurb: “This book presents a broad, user-friendly introduction to the Langlands program, that is, the theory of automorphic forms and its connection with the theory of L-functions and other fields of mathematics. Each of the twelve chapters focuses on a particular topic devoted to special cases of the program. The book is suitable for graduate students and researchers.”

Representation Theory of Finite Groups: An Introductory Approach by Benjamin Steinberg.

Representation Theory of Finite Monoids by Benjamin Steinberg. Blurb: “This first text on the subject provides a comprehensive introduction to the representation theory of finite monoids. Carefully worked examples and exercises provide the bells and whistles for graduate accessibility, bringing a broad range of advanced readers to the forefront of research in the area. Highlights of the text include applications to probability theory, symbolic dynamics, and automata theory. Comfort with module theory, a familiarity with ordinary group representation theory, and the basics of Wedderburn theory, are prerequisites for advanced graduate level study.”

How am I doing? 914 dollars…

Way to go, same exercise tomorrow. Again, suggestions/warnings welcome!

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human-, computer- and fairy-chess

Published December 1, 2016 by lievenlb

It was fun following the second game last night in real time. Carlsen got a winning endgame with two bishops against a rook, but blundered with 62. Bg4?? (winning was Kf7), resulting in stalemate.

There was this hilarious message around move 60:

“The computer has just announced that white mates in 31 moves. Of course, the only two people in the building who don’t benefit from that knowledge are behind the pieces.”

[section_title text=”Alice’s game from ‘Through the Looking-Glass'”]

The position below comes from the preface of Lewis Carroll’s Through the Looking-Glass



The old notation for files is used:

a = QR (queen’s side rook)
b = QKt (queen’s side knight)
c = QB (queen’s side bishop)
d = Q (queen)
e = K (king)
f = KB (king’s side bishop)
g = KKt (king’s side knight)
h = KR (king’s side rook)

Further, the row-number depends on whose playing (they both count starting from their own side). Here’s an animated version of the game:



And a very strange game it is.

White makes consecutive moves, which is allowed in some versions of fairy chess.

And, as the late Martin Gardner explains in his book The Annotated Alice:

“The most serious violation of chess rules occurs near the end of the
problem, when the White King is placed in check by the Red Queen without
either side taking account of the fact. “Hardly a move has a sane purpose,
from the point of view of chess,” writes Mr. Madan. It is true that both sides
play an exceedingly careless game, but what else could one expect from the
mad creatures behind the mirror? At two points the White Queen passes up
a chance to checkmate and on another occasion she flees from the Red
Knight when she could have captured him. Both oversights, however, are in
keeping with her absent-mindedness.”

In fact, the whole game reflects the book’s story (Alice is the white pawn travelling to the other side of the board), with book-pages associated to the positions listed on the left. Martin Gardner on this:

“Considering the staggering difficulties involved in dovetailing a chess
game with an amusing nonsense fantasy, Carroll does a remarkable job. At
no time, for example, does Alice exchange words with a piece that is not
then on a square alongside her own. Queens bustle about doing things while
their husbands remain relatively fixed and impotent, just as in actual chess
games. The White Knight’s eccentricities fit admirably the eccentric way in
which Knights move; even the tendency of the Knights to fall off their
horses, on one side or the other, suggests the knight’s move, which is two
squares in one direction followed by one square to the right or left. In order
to assist the reader in integrating the chess moves with the story, each move
will be noted in the text at the precise point where it occurs.”

The starting position is in itself an easy chess-problem: white mates in 3, as explained by Gardner:

” It is amusing to note that it is the Red Queen who persuades Alice to advance along her file to the eighth square. The Queen is protecting herself with this advice, for white has at the outset an easy, though inelegant, checkmate in three moves.
The White Knight first checks at KKt.3. If the Red King moves to either Q6
or Q5, white can mate with the Queen at QB3. The only alternative is for
the Red King to move to K4. The White Queen then checks on QB5,
forcing the Red King to K3. The Queen then mates on Q6. This calls, of
course, for an alertness of mind not possessed by either the Knight or
Queen. ”

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NaNoWriMo (3)

Published November 30, 2016 by lievenlb

In 2001, Eugenia Cheng gave an interesting after-dinner talk Mathematics and Lego: the untold story. In it she compared math research to fooling around with lego. A quote:

“Lego: the universal toy. Enjoyed by people of all ages all over the place. The idea is simple and brilliant. Start with some basic blocks that can be joined together. Add creativity, imagination and a bit of ingenuity. Build anything.

Mathematics is exactly the same. We start with some basic building blocks and ways of joining them together. And then we use creativity, and, yes, imagination and certainly ingenuity, and try to build anything.”

She then goes on to explain category theory, higher dimensional topology, and the process of generalisation in mathematics, whole the time using lego as an analogy. But, she doesn’t get into the mathematics of lego, perhaps because the talk was aimed at students and researchers of all levels and all disciplines.

There are plenty of sites promoting lego in the teaching of elementary mathematics, here’s just one link-list-page: “27 Fantastic LEGO Math Learning Activities for All Ages”. I’m afraid ‘all ages’ here means: under 10…

lego-math-teaching-children-alycia-zimmerman-fb__700-png

Can one do better?

Everyone knows how to play with lego, which shapes you can build, and which shapes are simply impossible.

Can one tap into this subconscious geometric understanding to explain more advanced ideas such as symmetry, topological spaces, sheaves, categories, perhaps even topos theory… ?

Let’s continue our

[section_title text=”imaginary iterview”]

Question: What will be the opening scene of your book?

Alice posts a question on Lego-stackexchenge. She wants help to get hold of all imaginary lego shapes, including shapes impossible to construct in three-dimensional space, such as gluing two shapes over some internal common sub-shape, or Escher like constructions, and so on.

escherlego

Question: And does she get help?

At first she only gets snide remarks, style: “brush off your French and wade through SGA4”.

Then, she’s advised to buy a large notebook and jot down whatever she can tell about shapes that one can construct.

If you think about this, you’ll soon figure out that you can only add new bricks along the upper or lower bricks of the shape. You may call these the boundary of the shape, and soon you’ll be doing topology, and forming coproducts.

These ‘legal’ lego shapes form what some of us would call a category, with a morphism from $A$ to $B$ for each different way one can embed shape $A$ into $B$.

Of course, one shouldn’t use this terminology, but rather speak of different instruction-manuals to get $B$ out of $A$ (the morphisms), stapling two sets of instructions together (the compositions), and the empty instruction-sheet (the identity morphism).

Question: But can one get to the essence of categorical results in this way?

Take Yoneda’s lemma. In the case of lego shapes it says that you know a shape once you know all morphisms into it from whatever shape.

For any coloured brick you’re given the number of ways this brick sits in that shape, so you know all the shape’s bricks. Then you may try for combination of two bricks, and so on. It sure looks like you’re going to be able to reconstruct the shape from all this info, but this quickly get rather messy.

But then, someone tells you the key argument in Yoneda’s proof: you only have to look for the shape to which the identity morphism is assigned. Bingo!

Question: Wasn’t your Alice interested in the ‘illegal’ or imaginary shapes?

Once you get to Yoneda, the rest follows routinely. You define presheaves on this category, figure out that you get a whole bunch of undesirable things, bring in Grothendieck topologies to be the policing agency weeding out that mess, and keep only the sheaves, which are exactly the desired imaginary shapes.

Question: Your book’s title is ‘Primes and other imaginary shapes’. How do you get from Lego shapes to prime numbers?

By the standard Gödelian trick: assign a prime number to each primitive coloured brick, and to a shape the product of the brick-primes.

That number is a sort of code of the shape. Shapes sharing the same code are made up from the same set of bricks.

Take the set of all strictly positive natural numbers partially ordered by divisibility, then this code is a functor from Lego shapes to numbers. If we extend this to imaginary shapes, we’ll rapidly end up at Connes’ arithmetic site, supernatural numbers, adeles and the recent realisation that the set of all prime numbers does have a geometric shape, but one with infinitely many dimensions.

primenumbers

Not sure yet how to include all of this, but hey, early days.

Question: So, shall we continue this interview at a later date?

No way, I’d better start writing.

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Ulysses and LaTeX

Published November 29, 2016 by lievenlb

If you’re a mathematician chances are that your text-editor of choice will be TeXShop, the perfect environment for writing papers. Even when writing a massive textbook, most of us stick to this or a similar LaTeX-frontend. The order of chapters in such a book is usually self-evident, and it is enough to use one TeX-file per chapter.

If you’re a blogger, chances are you spend a lot of writing time within the WordPress-editor. If you have a math-blog, there’s no longer the issue of including TeX-output images in some laborious way, thanks to MathJax. Even for a longer series of blog-posts there’s no problem staying within the WordPress-environment.

However, if you’re reckless enough to want to write a novel, or a math-book for a larger audience, you may need different equipment.

You will have to be able to follow story-lines, to follow your main characters throughout the plot, get word counts on scenes and chapters, jot down ideas and results from research, but most of all: you will have to be able to remain focussed just on your writing, as far away as possible from all bells and whistles and thrills of internet and preview-on-the-go editors.

In short, you may consider moving all of your writing to Ulysses.

I’ve been an early adopter from the days their iPad-app was called Daedalus, which I found cute, being a pathetic Joyce-fan. However, the app’s iCloud syncing sucked, but it is now replaced by the Ulysses.app which works like magic, syncing every keystroke between iPad, iPhone and whatever Mac you use as your workhorse.

But, what if you want to write about math and are unwilling to ban all LaTeX-formulas from your text.

Well, I’ve tried everything, including the approach below (in a faulty way), and figured it was impossible due to the fact that Ulysses is a MarkDown editor in which underscores are entirely different from indices.

Fortunately, yesterday Eline Steffens posted “Writing Mathematical Equations in Ulysses” showing me what I did wrong.

If you want MathJax to parse your text you need to include the standard code in your header. What I missed was that you have to include it as a ‘Raw source block’ (under ‘Markup’ in Ulysses).

Further, I forgot to prepend dollar-signs with a tilde, which works as an escape character in Ulysses so that all underscores are safe within the LaTeX-boundaries.

But now it works like a charm.

Ulysses is able to export your text in a variety of ways. You can preview it as HTML, including all rendered LaTex, and you can export directly either to Medium (on which I should begin to cross-post stuff asap) or your own WordPress-site.

In fact, I wrote this in Ulysses, then clicked the export-icon, choose ‘Publishing’ and NeverEndingBooks, and bingo I was able to post it as a draft. I can even fill in categories and tags, even add the featured image appearing at the top of this post, check everything in WordPress-admin and hit: “Publish”.

I guess I’ll be doing all my non-paper writing from now on entirely in Ulysses.

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NaNoWriMo (2)

Published November 28, 2016 by lievenlb

Two more days to go in the NaNoWriMo 2016 challenge. Alas, it was clear from the outset that I would fail, bad.

I didn’t have a sound battle plan. Hell, I didn’t even have a a clue which book to write…

But then, I may treat myself to a SloWriMo over the Christmas break.

For I’ve used this month to sketch the vaguest possible outlines of an imaginary book.

ulysses2

[section_title text=”An imaginary interview”]

Question: What is the title of your book?

I don’t know for sure, but my working title is Primes and other imaginary shapes.

Question: What will the cover-illustration look like?

At the moment I would settle for something like this:

Question: Does your book have an epigraph?

That’s an easy one. Whenever this works out, I’ll use for the opening quote:

[quote name=”David Spivak in ‘Presheaf, the cobbler'”]God willing, I will get through SGA 4 and Lurie’s book on Higher Topos Theory.
[/quote]

Question: Any particular reason?

Sure. That’s my ambition for this book, but perhaps I’ll save Lurie’s stuff for the sequel.

Question: As you know, Emily Riehl has a textbook out: Category Theory in Context. Here’s a recent tweet of hers:

Whence the question: does your book have a protagonist?

Well, I hope someone gave Emily the obvious reply: Yoneda! As you know, category theory is a whole bunch of definitions, resulting in one hell of a lemma.

But to your question, yes there’ll be a main character and her name is Alice.

I know, i know, an outrageous cliché, but at least I can guarantee there’ll be no surprise appearances of Bob.

These days, Alices don’t fall in rabbit holes, or crawl through looking-glasses. They just go online and encounter weird and wondrous creatures. I need her to be old enough to set up a Facebook and other social accounts.

My mental image of Alice is that of the archetypical STEM-girl

In her younger years she was a lot like Lewis Carroll’s Alice. In ten years time she’ll be a copy-cat Alice Butler, the heroine of Scarlett Thomas’ novel PopCo.

Question: What will be the opening scene of your book?

Alice will post a question on Lego-stackexchenge, and yes, to my surprise such a site really exists…

(to be continued, perhaps)

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NaNoWriMo (1)

Published November 5, 2016 by lievenlb

Some weeks ago I did register to be a participant of NaNoWriMo 2016. It’s a belated new-year’s resolution.

When PS (pseudonymous sister), always eager to fill a 10 second silence at family dinners, asked

(PS) And Lieven, what are your resolutions for 2016?

she didn’t really expect an answer (for decades my generic reply has been: “I’m not into that kinda nonsense”)

(Me) I want to write a bit …

stunned silence

(PS) … Oh … good … you mean for work, more papers perhaps?

(Me) Not really, I hope to write a book for a larger audience.

(PS) Really? … Ok … fine … (appropriate silence) … Now, POB (pseudonymous other brother), what are your plans for 2016?

nanowrimo_2016_webbanner_participant

If you don’t know what NaNoWriMo is all about: the idea is to write a “book” (more like a ‘shitty first draft’ of half a book) consisting of 50.000 words in November.

We’re 5 days into the challenge, and I haven’t written a single word…

Part of the problem is that I’m in the French mountains, and believe me, there always more urgent or fun things to do here than to find a place of my own and start writing.

A more fundamental problem is that I cannot choose between possible book-projects.

Here’s one I will definitely not pursue:

“The Grothendieck heist”

“A group of hackers uses a weapon of Math destruction to convince Parisian police that a terrorist attack is imminent in the 6th arrondissement. By a cunning strategy they are then able to enter the police station and get to the white building behind it to obtain some of Grothendieck’s writings.

A few weeks later three lengthy articles hit the arXiv, claiming to contain a proof of the Riemann hypothesis, by partially dismantling topos theory.

Bi-annual conferences are organised around the globe aimed at understanding this weird new theory, etc. etc. (you get the general idea).

The papers are believed to have resulted from the Grothendieck heist. But then, similar raids are carried out in Princeton and in Cambridge UK and a sinister plan emerges… “

Funny as it may be to (ab)use a story to comment on the current state of affairs in mathematics, I’m not known to be the world’s most entertaining story teller, so I’d better leave the subject of math-thrillers to others.

Here’s another book-idea:

“The Bourbaki travel guide”

The idea is to hunt down places in Paris and in the rest of France which were important to Bourbaki, from his birth in 1934 until his death in 1968.

This includes institutions (IHP, ENS, …), residences, cafes they frequented, venues of Bourbaki meetings, references in La Tribu notices, etc.

This should lead to some nice Parisian walks (in and around the fifth arrondissement) and a couple of car-journeys through la France profonde.

Of course, also some of the posts I wrote on possible solutions of the riddles contained in Bourbaki’s wedding announcement and the avis de deces will be included.

Here the advantage is that I have already a good part of the raw material. Of course it still has to be followed up by in-situ research, unless I want to turn it into a ‘virtual math traveler ’s guide’ so that anyone can check out the places on G-maps rather than having to go to France.

I’m still undecided about this project. Is there a potential readership for this? Is it worth the effort? Can’t it wait until I retire and will spend even more time in France?

Here’s yet another idea:

“Mr. Yoneda takes the Tokyo-subway”

This is just a working title, others are “the shape of prime numbers”, or “schemes for hipsters”, or “toposes for fashionistas”, or …

This should be a work-out of the sga4hipsters meme. Is it really possible to explain schemes, stacks, Grothendieck topologies and toposes to a ‘general’ public?

At the moment I’m re-reading Eugenia Cheng’s “Cakes, custard and category theory”. As much as I admire her fluent writing style it is difficult for me to believe that someone who didn’t knew the basics of it before would get an adequate understanding of category theory after reading it.

It is often frustrating how few of mathematics there is in most popular maths books. Can’t one do better? Or is it just inherent in the format? Can one write a Cheng-style book replacing the recipes by more mathematics?

The main problem here is to find good ‘real-life’ analogies for standard mathematical concepts such as topological spaces, categories, sheaves etc.

The tentative working title is based on a trial text I wrote trying to explain Yoneda’s lemma by taking a subway-network as an example of a category. I’m thinking along similar lines to explain topological spaces via urban-wide wifi-networks, and so on.

But al this is just the beginning. I’ll consider this a success only if I can get as far as explaining the analogy between prime numbers and knots via etale fundamental groups…

If doable, I have no doubt it will be time well invested. My main problem here is finding an appropriate ‘voice’.

At first I wanted to go along with the hipster-gimmick and even learned some the appropriate lingo (you know, deck, fin, liquid etc.) but I don’t think it will work for me, and besides it would restrict the potential readers.

Then, I thought of writing it as a series of children’s stories. It might be fun to try to explain SGA4 to a (as yet virtual) grandchild. A bit like David Spivak’s short but funny text “Presheaf the cobbler”.

Once again, all suggestions or advice are welcome, either as a comment here or privately over email.

Perhaps, I’ll keep you informed while stumbling along NaNoWriMo.

At least I wrote 1000 words today…

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Oulipo’s use of the Tohoku paper

Published July 17, 2014 by lievenlb

Many identify the ‘Tohoku Mathematical Journal’ with just one paper published in it, affectionately called the Tohoku paper: “Sur quelques points d’algèbre homologique” by Alexander Grothendieck.

In this paper, Grothendieck reshaped homological algebra for Abelian categories, extending the setting of Cartan-Eilenberg (their book and the paper both appeared in 1957). While working on the Tohoku paper in Kansas, Grothendieck did not have access to the manuscript of the 1956 book of Cartan-Eilenberg, about which he heard from his correspondence with Serre.

Concerning the title, an interesting suggestion was made by Mathieu Bélanger in his thesis “Grothendieck et les topos: rupture et continuité dans les modes d’analyse du concept d’espace topologique”, (footnote 18 on page 164):

“There is a striking resemblance between the title of the Grothendieck’s article “Sur quelques points d’algèbre homologique”, and that of Fréchet‘s thesis “Sur quelques points d’analyse fonctionelle”. Why? Grothendieck remains silent about it. Perhaps he saw a methodological similarity between the introduction, by Fréchet, of abstract spaces in order to develop the foundations of functional calculus and that of the Abelian categories he needed to clarify the homological theory. Compared with categories of sets, groups, topological spaces, etc. that were used until then, Abelian categories are in effect abstract categories.”

But, what does this have to do with the literary group OuLiPo (ouvroir de littérature potentielle, ‘workshop of potential literature’)?

Oulipo was founded in 1960 by Raymond Queneau and François Le Lionnais. Other notable members have included novelists Georges Perec and Italo Calvino, poets Oskar Pastior, Jean Lescure and poet/mathematician Jacques Roubaud.

Several members of Oulipo were either active mathematicians or at least had an interest in mathematics. Sometimes, Oulipo is said to be the literary answer to Bourbaki. The group explored new ways to create literature, often with methods coming from mathematics or programming.

One such method is described in “Chimères” by Le Lionnais:

One takes a source text A. One ’empties’ it, that is, one deletes all nouns, adjectives and verbs, but marks where they were in the text. In this way we have ‘prepared’ the text.

Next we take three target texts and make lists of words from them, K the list of nouns of the first, L the list of adjectives of the second and M the list of verbs of the third. Finally, we fill the empty spaces in the source text by words from the target lists, in the order that they appeared in the target texts.

In the example Le Lionnais gives, the liste M is the list of all verbs appearing in the Tohoku paper.


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