Pariah moonshine and math-writing

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.”

how much to spend on (cat)books?

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.”

let’s spend 3K on (math)books

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!

human-, computer- and fairy-chess

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. ”