In many areas of maths, especially mechanics, drawing clear, neat, big diagrams is useful to organise your information.

Also, in many areas, *imaginative* diagrams, which do not just organise your information, but change the way you see it, are useful.

For example, a problem about a triangle can be transformed into a problem about three points on a circle. A problem about three circles can be transformed into a problem about three right-angled triangles.

]]>Click here for notes from James Jones, a maths teacher in the USA

]]>Convalescing from an operation, I’ve been reading *The Girl Who Played With Fire*, a thriller by Stieg Larsson, who as well as being a novelist and investigative journalist was also a revolutionary socialist.

It’s a good thriller, but also includes what has plausibly been called “the silliest fictional mathematics” ever, the silliest ever version of the myth that mathematics is all about inexplicable, given-by-nature, eccentric flashes of inspiration. As the Polish-Canadian mathematician Izabelle Laba has explained:

The worst thing about the series *[of stories, of which TGWPWF is no.2]* is the mathematical interludes in *The Girl Who Played With Fire*. We’re told that Lisbeth Salander *[the heroine]* is also a puzzle-loving math genius who solves Fermat’s last theorem, or thinks she does, in a passage that *[the Cambridge mathematician]* Tim Gowers singled out for attention some time ago *[as “the silliest piece of fictional mathematics I have ever come across”]*.

Mind you, I’m all for having more novels and movies with strong, resourceful and mathematically talented heroines. I just wish that the math part weren’t so far off the mark…

Salander comes to mathematics by way of puzzles: Rubik’s cube, intelligence tests in magazines, every logical puzzle that she can lay her hands on. She has always been good at solving them, but was not aware of their mathematical side until sometime between the end of TGWTDT and the start of TGWPWF. Mathematics, to her, is “a logical puzzle with endless variations”, a meta-riddle where the goal is to understand the rules for solving numerical or geometric puzzles.

Salander’s primary resource is a book called *Dimensions in Mathematics* by a Dr. L. C. Parnault (Harvard University Press), a 1,200 page book that’s allegedly considered the bible of mathematics. Quite unsurprisingly, neither Dr. Parnault nor the book in question exist in real life, but Larsson tells us that *Dimensions* is a book about “the history of mathematics from the ancient Greeks to modern-day attempts to understand spherical astronomy”. It’s supposed to be pedagogical, entertaining, gorgeously illustrated and full of anecdotes. Salander is fascinated by a theorem on perfect numbers – one can verify it for as many numbers as one wishes, and it never fails! – and then advances through “Archimedes, Newton, Martin Gardner, and a dozen other classical mathematicians”, all the way to Fermat’s last theorem.

Unwilling to look at the “answer key”, she skips the section on Wiles’s proof, and tries to figure it out for herself… *[Andrew Wiles’s proof, with Richard Taylor, of Fermat’s last theorem, completed in 1995, is lengthy and uses 20th-century techniques which require years of postgraduate study to acquire: click here for an idea of it. There is no way that a brilliant amateur could find it in a flash of inspiration, any more than she could design a self-driving truck in a similar flash].*

This is all easy to mock, but unfortunately it seems to be a pretty accurate reflection of what mathematics means to most people… *[Gardner, who died in 2010, was a (brilliant) writer of popular mathematical puzzles, not a working, let alone a “classical”, mathematician]*… With all due respect to Gardner and his work, I have a problem with the image of mathematics as the art of puzzle solving.

Sure, mathematics involves logical arguments and so do mathematical puzzles. An appreciation of that does offer some insight into what we do. Regrettably, it can also lead to the notion that we get paid for playing with Rubik’s cube and solving crossword puzzles and newspaper-style intelligence tests. It’s the equivalent of a blind person touching an elephant’s trunk and concluding that elephants look like snakes.

In case any non-mathematicians are reading this: logical puzzles convey no sense whatsoever of how vast the subject actually is or how much work it takes to learn the craft. They can’t, for the simple reason that they’re created for entertainment. Their target audience can’t be expected to take a calculus class, never mind advanced graduate courses, before they can even understand the statement of the question. This already narrows it down to simple Euclidean geometry, basic combinatorics, possibly some manipulation of numbers, and eliminates most of mathematics as we know it.

You’d never learn for example that analysis, PDE or ergodic theory even exist, let alone how much accumulated knowledge there is in each of these areas. You wouldn’t get any good picture of contemporary geometry or combinatorics, either. The puzzles you’re left with may be tricky and entertaining, but they’re at best peripheral to mainstream mathematics.

The difficulty of math puzzles is usually calibrated so that the readers would have a good shot at solving them within a short time, usually ranging from a few minutes to an hour or two. A really hard puzzle is one that takes more than a few hours. No wonder that Salander was disappointed when she couldn’t solve Fermat’s theorem within a couple of days, or that she would expect a short solution with no background required.

In real-life mathematics, we don’t have a Ceiling Cat to set up problems for us and control their level of difficulty. Advisors can sometimes do that for their graduate students, to a very limited extent, but mostly we’re left stumbling in the dark, not knowing whether there even is a solution or whether we’re asking the right question in the first place. Learning to navigate this is possibly the hardest part of becoming an independent researcher.

Then there’s *Dimensions of Mathematics*. The very idea that mathematics should have a “bible” looks like a continued misunderstanding of the nature and scope of the subject. However, Larsson’s description is more reminiscent of any number of popular math books, except for the length.

If I had to suggest a real-life book for Larsson to use instead, it might be a collection of national or international Math Olympiad problems with solutions. It would not be a bible of anything, but it should present a challenge to someone like Salander at about the right level.

Olympiad problems only require normal high school background, which Salander should be able to catch up on. (Did I mention that she had dropped out of school?) They can be very hard, immeasurably more so than logical puzzles in popular magazines, but not necessarily out of reach for an extremely smart newcomer to mathematics who’s willing to put in the time and effort.

*[Or the Stanford Mathematics Problem Book, of which there is a copy in the CoLA Maths Library. The ingenuity and imagination and persistence required for these problems is something which every research mathematician has to learn, as well as learning the more advanced techniques developed by other mathematicians’ collective ingenuity and imagination over the centuries. It is also useful in many other areas of thought.*

*And sometimes a mathematician does – through knowing a lot of possible ways to approach problems, through being imaginative and persistent about trying them out, and through being quick and precise in checking them – make a simple-once-you-see-it breakthrough which produces a quicker, easier proof of something which previously required complicated and abstruse reasoning – John Conway’s proof of Morley’s Miracle, for example.]*

Click here for pdf. These notes on some beginning ideas of combinatorics cover only:

the Pigeonhole Principle

basic rules about combinations and permutations

a few features of Pascal’s Triangle

the Inclusion-Exclusion principle

]]>No-one pays attention to alchemy any more. Horoscopes are still published, but few people take astrology seriously. However, “numerology” and other types of fake maths have proved hardier than fake chemistry or fake astronomy. Their strongest hold seems to be where you might think people would be exceptionally hard-headed: in high finance.

It is common for young children to be fascinated by numbers. No wonder: numbers are among their first encounters with abstract thought. No wonder also that as small children we attach emotions to numbers, and those emotions can stick: lots of people have “favourite numbers”.

The Pythagoreans of the 5th century BC gave us the Pythagoras Theorem and others maths of value but were also number-mystics. By about 300 BC, when Euclid wrote the first ever maths book, the real maths had been sifted out from the number mysticism.

Few people today would take seriously the Pythagorean ideas that odd numbers are male, even numbers are female. Five represents justice. Ten is a sacred number.

Number-mysticism still around, but it has about the same status as astrology: see here and here. People can be amused and intrigued about it, but are pretty much aware that it is not real maths.

The idea that 4 (in China) or 13 (in the West) are “unlucky numbers” is widespread enough to make some tall buildings skip those numbers when numbering floors (see below for a lift in Shanghai). But few people take it seriously: I know of no-one on the 13th floor of the block of flats where I live expressing worries about it.

Yet there still seem to be thousands of stock-market or foreign-exchange traders seriously basing real-life strategies, with a lot of money at stake, on mathematical superstitions. One such superstition is that if stock-market prices, or currency-exchange rates, fall to 61.8% of a previous high, or, when on a down-trend, rise 38.2%, then that “retracement” (against-the-trend movement) is about to reverse, so you should buy (or sell).

This superstition is called “Fibonacci retracements”.

If you take the Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

in which each number is the sum of the previous two, then the ratio of successive numbers approaches ½(1+√5), called φ for short, as you go further along the sequence.

That’s true. And 61.8% is about φ−1 (which is equal to 1/φ); 38.2% is about (φ−1)^{2}.

There really is a lot of interesting maths connected with the Fibonacci sequence. There are also *some* connections between the Fibonacci sequence and biology, based on mathematical properties of φ (see here: more in this short video, and at the end of Keith Devlin’s longer video, below).

One high school where I worked in Brisbane did an experiment for a year by giving over the whole of Year 9 maths entirely to investigations of the Fibonacci sequence.

The experiment didn’t work well, but it was plausible enough for a serious maths department to try. And in any case the Year 9 students learned something useful quickly enough: that the Fibonacci sequence and φ (called the “golden ratio”) attract vast varieties of superstitions.

Here is an account by Keith Devlin of the real maths and some of the superstitions around the Fibonacci sequence.

Click here for more references.

What do stock-market prices, or currency-exchange rates, have to do with Fibonacci numbers, or the golden ratio? You can easily find very straight-faced recommendations for using “Fibonacci retracements”

but they don’t include any mathematical argument connecting stock-market prices, or currency-exchange rates, to the real maths of the Fibonacci sequence or the “golden ratio”.

Click here for a rational discussion of “Fibonacci retracements” by a more scientifically-minded financial trader, Adam Grimes; and here is an article from The Economist magazine, 21/9/2006.

Practical traders, who believe themselves to be quite exempt from any intellectual influences, are usually slaves of some defunct mathematician. That is what Keynes might have said had he considered the faith placed by some investors in the work of Leonardo of Pisa, a 12th and 13th century number-cruncher.

Better known as Fibonacci, Leonardo produced the sequence formed by adding consecutive components of a series — 1, 1, 2, 3, 5, 8 and so on. Numbers in this series crop up frequently in nature and the relationship between components tends towards 1.618, a figure known as the golden ratio in architecture and design.

If it works for plants (and appears in “The Da Vinci Code”), why shouldn’t it work for financial markets? Some traders believe that markets will change trend when they reach, say, 61.8% of the previous high, or are 61.8% above their low.

Believers in Fibonacci numbers are part of a school known as technical analysis, or chartism, which believes the future movement of asset prices can be divined from past data. Some chartists follow patterns such as “head and shoulders” and “double tops”; others focus on moving averages; a third group believes markets move in pre-determined waves. The Fibonacci fans fall into this last set…

A new study * by Professor Roy Batchelor and Richard Ramyar of the Cass Business School, finds no evidence that Fibonacci numbers work in American stockmarkets. The academics looked at the Dow Jones Industrial Average over the period 1914-2002 and found no indication that trends reverse at the 61.8% level, or indeed at any predictable milestone.

This research may well fall on stony ground. Experience has taught Buttonwood that chartists defend their territory with an almost religious zeal. But their arguments are often anecdotal: “If technical analysis doesn’t work, how come so-and-so is a multi-millionaire?”. This “survivorship bias” ignores the many traders whose losses from using charts drive them out of the market.

Furthermore, the recommendations of technical analysts can be so hedged about with qualifications that they can validate almost any market outcome. As Professor Batchelor writes: “The root of the problem is the failure of technical analysts to specify their trading rules and report trading results in a scientifically acceptable way. Too often, rules are so vague and complex as to make replication impossible.”

Fibonacci numbers at least have the virtue of creating a testable proposition; one that they appear to fail. But chartists will not be completely discouraged. A review of the academic literature ** finds that, of 92 modern studies of technical analysis, 58 produced positive results (although the researchers say some of these studies may be flawed and that the best results occurred before the early 1990s).

If the efficient market theory is correct, technical analysis should not work at all; the prevailing market price should reflect all information, including past price movements. However, academic fashion has moved in favour of behavioural finance, which suggests that investors may not be completely rational and that their psychological biases could cause prices to deviate from their “correct” level.

Chartism probably holds most sway in the foreign-exchange market. Although currency markets are liquid and transparent, many participants (such as central banks) are not “profit-maximising”. So it is possible that currency prices are not completely efficient. Furthermore, some technical predictions may be self-fulfilling; if everyone believes that the dollar will rebound at ¥100, they will buy as it approaches that level.

Technical analysts also make the perfectly fair argument that those who analyse markets on the basis of fundamentals (such as economic statistics or corporate profits) are no more successful. Nevertheless, Buttonwood urges extreme caution in relying on their claims.

All that talk of long waves is distinctly mystical and seems to take the deterministic view of history that human activity is subject to some pre-ordained pattern. Chartists fall prey to their own behavioural flaw, finding “confirmation” of patterns everywhere, as if they were reading clouds in their coffee futures.

Besides, technical analysis tends to increase trading activity, creating extra costs. Hedge funds may be able to rise above these costs; small investors will not. As illusionists often proclaim, don’t try this at home.

* “Magic numbers in the Dow: http://www.cass.city.ac.uk/magicnumbers

** “The Profitability of Technical Analysis: a review”, by Cheol-Ho Park and Scott H Irwin, University of Illinois: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=603481

The simple explanation for “Fibonacci retracements” or other ideas of “chartism” still being popular is that, if we really don’t know how stock-market prices or currency exchange-rates will move – and we don’t – then a more-or-less random rule will produce gains better than the market average half of the time, and worse than the market average half of the time.

The half who do better than the market average are much more likely to strut around boasting about it than those who do worse. And those who do better are more likely to boast about the ingenious rule they’ve discovered; those who do worse are more likely to bemoan their bad luck, or blame themselves for choosing the wrong superstitious rule or using it ineptly, or just retire from high finance.

There’s a further twist to this. According to Batchelor and Raymar’s research, the “Fibonacci” stuff just doesn’t work. But it is possible for some of the high-finance fake maths to “work” after a fashion, if only enough people think it will work.

To take a simplified example, suppose everyone becomes convinced, with the certainty of religious dogma, that the price of any share is bound to stagnate or fall once it reaches the unlucky value of $13. Then nobody will buy shares for over $13, and the prophecy will be self-fulfilling.

After a while some people will notice that some $13 (or $12.99) shares are yielding much more in dividends than others. Since all religious prescriptions have some let-out clauses, the less strait-laced will find ways to sell and buy those high-dividend shares at over $13.

The net effect of the widespread superstition is to clog up the market and mess up what is supposed to be the market’s great merit, its information-transmitting property. If a share costs $13, you don’t know whether that means that enough traders are rigidly loyal to the “unlucky $13” rule, or whether a balance of informed opinion has estimated its probable future yields to correspond to that price.

The other effect of the superstitious rules is to increase trading volumes (because they lead to people buying and selling for superstitious as well as more-or-less realistic reasons), and thus to increase the incomes of brokers and the like.

Yet the financial traders responsible for this nonsense are the people who reckon themselves “Masters of the Universe”; who actually do exercise huge economic power in our capitalist economy; and who sometimes rake in huge salaries and bonuses for their work.

Better if economic life were regulated using democracy guided by real science and real maths.

]]>Yesterday, 22 August, at the University of Queensland library, I came across a (fairly old) book which I didn’t know about before, but is very good and very accessible for school students with a will to inquire into the more interesting areas of maths beyond the syllabus.

It’s Ivan Niven’s “The mathematics of choice – how to count without counting”, which is about the maths of counting and tabulating possibilities, usually called “combinatorics”

Combinatorics involves relatively little in the way of formulas, theorems, special notations, complicated chains of calculation, and such. It is mostly about developing systematic and smart ways of mathematical thinking, which are often applicable way beyond combinatorics itself.

Click here for a pdf of the book.

An example of a combinatorics problem (from Niven, p.3):

You work in a building located seven blocks east and eight blocks north of your home. (This is a US city, constructed on a grid pattern, not London!)

How many different routes can you take from home to work, walking only 15 blocks?

(The answer is 6435).

]]>On Monday 20 August my daughter Daisy (not a mathematician) and I went to a commemoration event for Maryam Mirzakhani organised by the School of Mathematics and Physics at the University of Queensland.

In 2014 Mirzakhani became the first woman ever to win a Fields Medal (the maths equivalent of a Nobel Prize). In 2017 she died of cancer.

Cecilia González Tokman gave the event an idea of Mirzakhani’s areas of research. Cheryl Praeger, herself one of the first women ever to serve on the executive committee of the International Mathematical Union, and the second woman ever to become a professor of mathematics in Australia, talked about Mirzakhani’s work fitted in with the battle for equality for women in mathematics.

Azam Asanjarani (now in New Zealand) and another Iranian woman mathematician (now in Australia) talked about their memories of Maryam Mirzakhani as a teenager in Iran, where she did high school and university before going to Harvard University in 1999. Mirzakhani worked in the USA, at Harvard, then Princeton, then Stanford University, until her death.

In middle school Mirzakhani was more interested in reading novels and becoming a writer than in maths. In one round of tests, she got 20/20 in every subject, except maths, where she got 16/20. Annoyed, she declared in future that she “wouldn’t even try” in maths.

But she did. Gradually she realised that there are stranger riches, greater beauties, in maths than the routine school syllabus suggests. On one account, she was first inspired by learning about Gauss’s solution to the problem of adding all the whole numbers from 1 to 100.

Solution: Write those numbers in a column. In another column, next to it, write those numbers in reverse order. Then each row – 100 + 1, 99 + 2, 98 + 3, etc. – adds up to 101. There are 100 rows. So the total of the whole numbers from 1 to 100 is ½ (101 × 100), or 5050.

She became one of the first Iranian girls to go to the International Mathematical Olympiad, and did well there. But even up until shortly before she went to university she would say: “I love maths, but I’m afraid I’m not good enough at it for university”.

She gained confidence, and had her first research paper published even before she finished her first university maths course. It was about “graph theory”, that is, the maths of networks studied as consisting of vertices connected by edges, without regard to the length or shape of edges.

Click here to read the research paper.

It was about “complete tripartite graphs”, that is, graphs whose vertices can be divided into three groups so that every vertex in every group is connected to every vertex in every *other* group but not to any vertex in its own group. When, it asked, can those “complete tripartite graphs” be made up by putting together “5-cycles”, that is, sets of five vertices connected one after another in a loop.

*Below: a complete tripartite graph where the 3 groups are (1) the single vertex on the left (2) the two vertices in the middle (3) the four vertices on the right.*

That problem belongs to “discrete mathematics”, the maths of whole numbers and of objects and patterns which can be encoded in whole numbers rather than by the uncountable “real” number line which includes √2 and π and such, and which we need to do calculus. In fact, according to her CV, Mirzakhani with her friend Roya Beheshti (who came to the USA to do maths postgrad work at the same time as her, and also stayed) published a book or pamphlet on “Elementary Number Theory” before leaving Iran. “Number theory” means theory to do with the counting numbers 1, 2, 3, …

Mirzakhani stood out, mathematically, for her ability to link different areas of maths. Her Fields Medal was for work on Riemann Surfaces. Riemann Surfaces came out of work on “complex analysis”, i.e. calculus of functions w = f(z) where w and z are both *complex* numbers. Click here for a summary of how Riemann Surfaces come from complex analysis.

Mirzakhani, however, who seems to have had an uncanny ability to visualise these Riemann Surfaces (2-dimensional surfaces sitting in four-dimensional space), and get an idea of them by sketches, would use them to solve problems of counting possibilities (combinatorics) in discrete maths.

To students at Stanford, she described herself as a “slow mathematician”. Hardly. But what really set her out was not quickness in calculation, or instant insight, but, as Curtis McMullen, her Ph D supervisor at Harvard, put it, “relentless questions” and “fearless ambition”.

McMullen has recounted that she would come to his office and pose questions that were “like science fiction stories… vivid scenes… some unexplored corner of the mathematical universe… strange structures… beguiling patterns…”. “Is it right?” she would ask. “As if I might know the answer”, comments McMullen.

“I believe that many students don’t give maths a real chance”, Mirzakhani later commented. After completing one of her research projects, she said to a co-worker: “If we knew things would be so complicated, I think we would have given up”. Then she added: “I don’t know; actually, I don’t know. I don’t give up easily.”

Click here to watch a video of Mirzakhani lecturing on “dynamics of moduli surfaces of curves”. (A “Riemann surface” is also a “complex curve”). She is lecturing for researchers in a specialised field of maths, so do not expect to understand the detail. You can, however, get a taste of what sort of ideas Mirzakhani was working with, her enthusiasm, her humour.

One set of problems she mentions in the lecture are billiard-table problems. If you shoot one ball at another on a square billiard table, for example, can you protect the target ball by having only a *finite* number of guard balls around it?

One guard ball will protect the target ball from a direct shot. One will protect it from a shot bouncing directly off one wall. But then there are an infinite number of paths with more and more bounces which the shot can take.

It turns out, however, that you *can* protect the target ball with only a finite number of guard balls.

The proof (click here) uses the idea that you can map the bouncing paths as straight lines if you draw copies of the square adjacent to it, and rewrite the bouncing path as going straight into the neighbouring copies.

The fundamental idea here is equating a square (actually, in this case, a 2×2 square, the original square plus three adjacent squares which allow us to model the first bounces as continuations along straight lines) with a torus (a ring donut) by gluing together the opposite pairs of edges of the square. An ordinary flat space is equated with a Riemann surface.

Click here for a video of “gluing a torus”.

This is the short video produced to mark Mirzakhani’s Fields Medal: click here.

One of her other notable achievements is to have become the first woman, since the 1979 “Islamic revolution”, to appear in the Iranian press without a headscarf. Apparently the Supreme Leader, Ayatollah Khamenei, was so taken with Iranian national pride when Mirzakhani won her Fields Medal that he allowed a picture of her. Like other Iranian women emigre mathematicians, she didn’t wear a headscarf after she left Iran and no longer was compelled to.

The Iranian women mathematicians who knew Mirzakhani recounted that when she and Roya Beheshti went to the International Mathematical Olympiad, they had to work in full, all-swathing, loose-fitting religious dress. Their enthusiasm and dynamism could be detected only by the quick movements of their trainers beneath their robes.

Look at the video of Mirzakhani lecturing, and you can see she couldn’t do as she did if swathed in robes on the pretext that God would be angered by seeing her expound mathematics more vividly and directly.

]]>The idea of complex numbers, as points in a number *plane* with rules for adding and multiplying them (etc.), was first formulated clearly early in the 19th century.

Complex numbers had been used “on the side” in maths for hundreds of years before that. For example, they turn out to be necessary in the calculation of roots of cubic equations *even if* all the roots of the equation are real.

But formulating the idea clearly set off a whole new wave of investigation. For example, how does calculus work if we have functions

where z and w range over the whole complex number plane, rather than just along the real number line?

It’s difficult to tell, because if we try to draw a graph, then it has to be in *four dimensions* (two for the real and imaginary components of z, two for the real and imaginary components of w).

It seems it should still be true that if for example

then . And it is.

Calculus using complex numbers (usually called complex analysis) is of course more difficult than calculus using real numbers. Also, it turns out, it is necessary for answering even some questions which seem to be only about calculus with real numbers.

When we do calculus with real numbers, at school we pretty much take it for granted that any function we deal with will have a derivative. The question is how to find it, not whether it exists.

In fact, plenty of functions do not have derivatives. The modulus function

has no derivative (no ) when x=0.

at 0 would have to be the limit (the number approached more and more closely) by the average speed of change in y as x ranges over smaller and smaller neighbourhoods of 0.

But in every neighbourhood of 0, however tiny, there is an average speed of change in y relative to x of +1 as x ranges bigger than 0, and an average speed of change in y relative to x of −1 as x ranges bigger than 0. Choose the neighbourhood (however small) to be mostly x>0, and the average speed will be near +1. Choose the neighbourhood (however small) to be mostly x<0, and the average speed will be −1.

There is no one set value that the average speed approaches, and so no when x=0.

The Weierstrass function, pictured below, is continuous everywhere, but has no *anywhere*.

With complex functions of complex variables, oddly, the separation between differentiable functions and non-differentiable functions is more drastic.

If a function is differentiable everywhere in an open disk around a complex number q in ℂ, then it has *all* derivatives at q for all n, and it is equal to its Taylor series:

The function is then called analytic, or holomorphic (the two terms mean the same) at q. Just as a lot of calculus with real variables is about differentiable functions, a lot of complex analysis is about analytic or holomorphic functions.

It is not easy for a complex function to be analytic or holomorphic. For example the conjugate function

is not differentiable. (The average speed of increase of w relative to z around z=a is +1 if you move parallel to the real axis, but -1 if you move parallel to the imaginary axis).

If f(z) is analytic (holomorphic) at z=0, then the Maclaurin series (i.e. Taylor series based at z=0) for

“works” from z=0 out to the nearest “singularity”, i.e. point when does *not* exist.

A bit more work shows that that the region in which the Maclaurin series “works” is a disk around z=0. (Makes sense, if you think about it: the Maclaurin series not working probably means the z^{n} terms becoming too big for big n).

It follows that the range of convergence for Maclaurin series considered only for *real* variables is always symmetrical around the origin (an interval [-b, b] or, for super well-behaved functions, the whole real line).

The Maclaurin series stops working for both real and complex variables once we get far enough out to hit a singularity. So for example the Maclaurin series for stops working outside the range −1 < x < 1. The function doesn't blow up or misbehave anywhere around x=−1 or x=1. But it does blow up at x=±i, and so its disk of convergence has radius 1.

As Nets Katz says in his Caltech lecture notes: “The complex numbers overshadow the elementary calculus of one variable, silently pulling its strings”. (More here).

The *only* analytic and invertible functions mapping the unit disc in the z-plane to the unit disc in the w-plane are “Möbius transformations”, transformations of the form

for some given a, b, c, d.

Möbius transformations map circles to circles (so long as we interpret a line as a special circle with infinite radius and centre at infinity). We can see what’s happening with Mobius transformations by drawing pictures side by side of circles (or lines) in the w-plane and their images in the z-plane (one example below: for more click here).

For all other complex functions of complex variables, this method of drawing z-shapes and seeing what w-shape they map into does not work well.

The w-shapes may be very complicated. They may well involve overlaps: the function w = z^{2} transforms the circle |z| = 1 into the circle |w| = 2 *travelled twice*, but how do you show the twice-travelling?

There is another problem, and this is where *Riemann surfaces* begin to kick in.

Consider

If z is real, we get round the problem of every z (≠ 0) having *two* square roots by having the convention that √z means the positive square roots. Then we can draw the graph and do calculus on it.

For complex z, though, it’s more complicated. Write z in the form

, choosing θ to be between −π and π, and so defining w as the square root which has argument between −½π and ½π, i.e. the square root with positive real part.

Follow z from the value z=4 (say) anticlockwise as θ increases round to 4e^{iπ}, i.e. − 4: w moves from 2 through 2e^{iπ/4} to 2e^{iπ/2}, i.e. 2i.

Follow z from the value z=4 *clockwise*, and w moves from 2 through 2e^{−iπ/4} to 2e^{−iπ/2}, i.e. −2i.

If we make a consistent choice for which square root w to choose, and then follow it through round a circle, it becomes inconsistent!

This is the best we can do to picture in three dimensions what’s happening. It plots only the real part of w.

But the w-surface does not really cross itself along the half-line where z is real and negative. It just seems to do that because we’re squashing four dimensions down to three.

These pictures (below) could show spiralling (one-dimensional) curves – which don’t cross themselves – in three dimensions. Only when they’re flattened to two dimensions do the curves seem to cross themselves. It’s the same sort of thing when we try to “flatten” the surface formed by all the values of w and z where w^{2}=z – a two-dimensional surface sitting in four-dimensional space – down to a two-dimensional surface which we can see in three dimensions.

This video gives some idea of what’s happening in four dimensions.

The w-surface is a two-dimensional surface, but a two-dimensional surface sitting in four-dimensional space. To get an idea of how this works, think about the (one-dimensional) w-curve, sitting in three dimensions, which is the w-image of the (one-dimensional) curve |z|=1. It is a one-dimensional w-curve which “flies over” itself, as of course it easily can in three dimensions.

To get a two-dimensional picture of this, cut the w-curve at the flyover point and glue the bits back again with a flip to avoid the flyover. We can then get a “flat” version of the w-curve.

which, with a bit of stretching or nudging, is equivalent to a circle. By analogous cutting-and-gluing, the whole w-surface can be flattened to a copy of the complex plane.

This cutting, gluing, stretching etc. obviously changes the detail of the w-surface. But some important properties of the general shape of the surface remain unchanged, and much of the study of Riemann surfaces is to do with those proporties of their general shape.

The pictures below (from Matt Kerr) show what happens with .

This, from Constantin Teleman’s notes, shows how cutting, gluing, and stretching can make a two-dimensional shape *which can be modelled in three dimensions* from the Riemann surface for:

Surfaces which have the same sort of properties as these Riemann surfaces derived from fairly-well-behaved functions w=f(z) are called “abstract” Riemann surfaces; so the basic “moral” definition of a Riemann surface (as Teleman puts it) is that it is a two-dimensional manifold which is similar in structure to ordinary (Euclidean) two-dimensional space in a neighbourhood around every point and “with a ‘good’ notion of complex-analytic functions”.

Those functions are thought of as mapping from the surface to a complex plane, so in the case of the Riemann surface for w=√z we reconceptualise the function as mapping from points on the 2-dimensional surface (sitting in 4-dimensional space) defined by w^{2}=z to points in the w-plane.

An important example of an abstract Riemann surface is the “Riemann sphere”, which means: a sphere conceptualised as a model of the complex plane plus a single point at infinity.

It’s a higher-dimensional equivalent of the idea of a circular (rather than linear) “number line”.

Suppose instead of listing the numbers along a line, we list them round a circle, starting from 0 at the bottom and marking the positions round the circle in the way shown, so that the numbers are closer and closer together as we get nearer the top, and in theory we can get numbers as high as we like marked on the circle.

This “circular number line” will then be not exactly a full circle, but a “punctured” circle, with one point missing, at the top.

The full circle will be a model, not just of the usual number line, but of the number line plus a single infinity. (Not separate +∞ and −∞, but a single infinity).

The Riemann sphere is the equivalent for complex numbers.

]]>Short videos made with each of the four winners of Fields Medals (“the Nobel prize of maths”) have been released.

The video-makers have done a good job of making the videos accessible to non-specialists, yet without dumbing down.

I found the video with Caucher Birkar – an Iranian Kurd by origin, who came to Britain as a refugee, and is now a professor at Cambridge – particularly vivid. The name Birkar goes by is not his birth name, but one he chose when coming to Britain which means “migrant mathematician” in Kurdish. He expresses hope that his achievement may bring some pride and joy to the 40 million Kurdish people, still denied their right to self-determination in a state of their own.

The video with Peter Scholze is also excellent.

Most people with maths degrees – probably, I’d guess, most people with maths PhDs outside the particular areas of maths studied by the medallists – will not understand the technicalities of the medallists’ research. The videos are still worth watching.

Two of the medals this year have been awarded for work in algebraic geometry – roughly speaking, the study of the geometrical shape of the sets of solutions of algebraic equations. Students at Warwick and Cambridge universities, for example, can study algebraic geometry as a third-year option, but at many universities it is not even an option at undergraduate level. The field is quite old, but was revolutionised by new concepts in the 1950s and 60s (thanks mostly to French mathematicians), and has been very active ever since.

One medal has been awarded for work on partial differential equations – equations connecting rates of change of variables, rather than just the variables themselves, central to mathematical physics – and one for work on number theory (the theory of the counting numbers, 1, 2, 3… – again, a very old field, but one which has revitalised in recent decades).

]]>The development of calculus in the 19th century did not only help calculations useful in physical science. It provided solutions to paradoxes that had puzzled philosophers for thousands of years.

Zeno’s arrow paradox, for example, dates back to ancient Greece. Bertrand Russell discussed it in his book, “The Principles of Mathematics” (1903).

The background is that Russell had studied maths at Cambridge in 1890-3. The syllabus at Cambridge then was old-fashioned and heavily tilted towards mechanics and the maths used in physics. Russell had become interested in philosophical problems, like Zeno’s arrow and other paradoxes. He had friends who were philosophers, like G E Moore and Alfred North Whitehead.

In 1895 he visited Berlin. He went to study economics, and one of the results of the visit was a (sympathetic) book he published in 1896 about the German socialist movement. But he also discovered that mathematical ideas about calculus which he’d never heard of in Cambridge, but were well-known in Germany, could solve a lot of the philosophical problems he’d puzzled over. (Arthur Cayley, professor of pure maths at Cambridge, was a front-rank researcher of world repute, and would have known all those ideas which Russell discovered. However, Cayley was a mild-mannered person: he hadn’t been able to change the syllabus, and by the time Russell got to Cambridge Cayley was old and semi-retired).

Philosophers in Germany, as distinct from mathematicians, were not very interested in those problems. Russell made the connection. From then to 1913 Russell focused on the overlap between maths and philosophy. He moved on to other pursuits during and after World War One.

Russell cites the paradox in the following form:

“If everything is in rest or in motion in a space equal to itself, and if what moves is always in the instant, the arrow in its flight is immovable”.

Another way of stating it is: “If the instant is indivisible, the arrow cannot move, for if it did the instant would immediately be divided. But time is made up of instants. As the arrow cannot move in any one instant, it cannot move in any time”.

One form in which I have come across it is in economics. In a simplified capitalist economy, at any given instant there is only a fixed quantity of effective demand, constituted by what capitalists have to pay each other at that moment for raw materials of production and replacement or repair of fixed capital, plus what they have to pay workers, plus what they pay each other for their own consumer goods and services. That quantity is what it is. Therefore there can be no growth of aggregate purchasing power.

*Russell comments on the paradox:*

After two thousand years of continual refutation, these sophisms [Zeno’s arrow paradox, and others] were reinstated, and made the foundation of a mathematical renaissance, by a German professor, who probably never dreamed of any connection between himself and Zeno… [Karl] Weierstrass…

*He continues:*

For the present, I wish to divest the [statement of the paradox] of all reference to change. We shall then find that it is a very important and very widely applicable platitude, namely: “Every possible value of a variable is a constant”. If x be a variable which can take all values from 0 to 1, all the values it can take are definite numbers, such as ½ or ⅓, which are all absolute constants.

*He explains what a “variable” is.*

A variable is a fundamental concept of logic, as of daily life. Though it is always connected with some class, it is not the class, nor a particular member of the class, nor yet the whole class, but any member of the class. On the other hand, it is not the concept “any member of the class”, but it is that (or those) which this concept denotes.

*Then what motion is:*

Motion consists in the fact that, by the occupation of a place at a time, a correlation is established between places and times; when different times, throughout any period however short, are correlated with different places, there is motion; when different times, throughout some period however short, are all correlated with the same place, there is rest.

*And then what speed at an instant means. In other words, what we usually think of as a thing being in movement at a single point of time, though Russell avoids that usage. It is indeed the case that we cannot define speed at an exact point of time without knowing about the object’s position at *other instants* in some *span* of time, however small, which includes that instant.*

If f(x) be a function which is finite and continuous at the point x, then it may happen that the fraction

has a definite limit as h approaches to zero. If this does happen, the limit is denoted by f′(x), and is called the derivative or differential of f(x) in the point x.

*If f(t) describes the positions of an object at time t, for different values of t, then f'(x) is the speed of the object at time x.*

*Russell explains what the word “limit” means here. The idea behind this definition of limit was developed by the early 19th century mathematician Cauchy, and then refined by Weierstrass. I’ve reworded Russell’s account slightly to conform more to modern usage.*

To say that the function f(t) has a derivative d at t=x means that the limit of

as h approaches zero is d. In precise terms: given any number ε however small, we can find another number δ so that for any h with |h| < δ

differs from d by less than ε.

If the limit in question does not exist, then f(x) has no derivative at the point x. If f(x) be not continuous at this point, the limit does not exist; if f(x) be continuous, the limit may or may not exist.

The only point which it is important to notice at present is, that there is no implication of the infinitesimal in this definition… It is the doctrine of limits that underlies the Calculus.

Carl Boyer comments (The History of the Calculus, p.25) that: “The paradox of the flying arrow involves directly the concept of the derivative and is answered immediately in terms of this… Mathematical analysis has shown that the concept of an infinite class is not self-contradictory, and that the difficulties here… are those of conceiving intuitively the nature of the continuum and of infinite aggregates”.

By “continuum”, Boyer means the number line containing all real numbers, fractions, numbers like √2, and numbers like π as well as whole numbers.

Another idea here which is difficulty for everyday thought to get its head round, but which is conceptualised neatly by Weierstrass’s formulation of calculus, is a function:

- having a property
*at a point* - it being possible to change the function at any other point, no matter how close, without changing the property at the first-named point
- but that property being impossible to ascertain from a single “snapshot” of the function
*at the point* - and instead depending on what the function does in some neighbourhoods,
*it doesn’t matter how small*, of the point.

An example here is continuity. Roughly speaking, we think of a function f being continuous over a span of values of x if it has no “gaps” and can be drawn for that span of x without taking the pencil off the paper. Precisely speaking, we can define what it means for a function to be continuous at a single point.

Consider the “popcorn function”.

if x is a rational number expressed in lowest terms as p/q

if x is irrational

It is continuous at all irrational values of x, and not continuous at any rational values. Although the function is continuous at π, say, we can’t “draw” it smoothly for even the tiniest range around π, because that range would include rational values of x where it is discontinuous. And we could change P for any number of other values of x no matter how near to π – for example, change P(22/7) and P(335/113), etc., to zero or one or whatever – without making any difference to whether P is continuous at π.

However, we can’t see that P is continuous at that single value π, just by looking at the function at that single point. All we can see by looking at that single point is that P(π) = 0. We need to look at how P behaves in some span around the point… only it does not matter how small that span is.

Russell makes a big thing of Weierstrass’s argument including no reference to “infinitesimals”. It is true that for two hundred years before Weierstrass, mathematicians had presented arguments in terms of “infinitesimals”, numbers which were somehow infinitely small and yet not zero, with an uneasy feeling that the arguments seemed to work but was hardly watertight. It is also true that Weierstrass’s way of making the arguments watertight is still the standard way today.

From 1960, however, an alternative approach was developed by Abraham Robinson which gives a precise definition of “infinitesimals” and works with them. It is called Non-Standard Analysis. A description of the approach, and an argument for its value, has been written by Joel Tropp.

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