Today, 18 August 2017, I went to a talk at the University of Queensland on the KPZ equation, a difficult-to-solve differential equation which appears to describe a large number of processes where interfaces grow, like “sticky Tetris”, above.

If a height h grows by random accumulation of deposits and by diffusion, the change in time has three contributions:

1. Slope-dependent “outwards” growth

2. Inbuilt tendencies to diffuse and smooth out

3. Random accretion (represented by below)

So the equation then reads

Afterwards, I searched the web for articles explaining it. This one doesn’t have much maths, but it is brisk and readable and has good graphics.

“The universal laws behind growth patterns, or what Tetris can teach us about coffee stains”

This is shorter:

“Coffee Stains Test Universal Equation”

This is a video of a fairly clear lecture about the maths

And this is an article discussing how mathematical equations come to describe “universal” (or almost universal) laws describing diverse physical processes

“E pluribus unum: from complexity, universality”

All this is a current area of research, and Martin Hairer of Warwick University won a Fields Medal in 2014 for a discovery in the area.

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Topology is the study of the properties of shapes which are unaffected by smoothly stretching or squeezing or twisting the shape (but not breaking or tearing it).

So a topologist is a mathematician who can’t see the difference between a ring doughnut and a coffee cup.

To see the difference between tearing and smoothly stretching or squeezing or twisting, we have to have some idea of *continuity*. The conceptual difference here is similar to the difference between a continuous curve and a discontinuous one. And we need some idea of the edges of shapes, which requires the concept of *convergence*.

We can study “topological equivalence” of shapes in the three-dimensional Euclidean space, , in which we’re thinking of the doughnut and coffee cup. But also in more general mathematical contexts.

In those more general contexts, we need to know what *structure* distinguishes between stretching or squeezing or twisting, and tearing, or, in other words, defines continuity and convergence.

Topologists work with *metric spaces*, in which a *metric*, a concept of distance, is defined. That allows us to see what is close and what is distant, what has moved smoothly and what has “jumped” or torn apart.

When they want more generality, they work with *topological spaces*. We say that *a topology* is defined on a space when we have workable rules to define in it what is:

- an
*open set*(something analogous to the interval containing all numbers greater than 0 and less than 1, written ), which has the property that every point in it has a surrounding neighbourhood (sub-interval), maybe a small one but a neighbourhood, completely contained within the interval) - a
*closed set*(something analogous to the interval containing all numbers greater than or equal to 0 and less than or equal to 1, written ), which has the property that every point which is a limit of a sequence of points in the interval is also within the interval.

is not closed, because the sequence has a limit, 0, outside the interval.

is not open, because every neighbourhood (sub-interval) completing surrounding 1 contains some points > 1 and so outside the interval.

You can then show that a subset A of a metric space X is closed if and only if its complement X−A (meaning: all the points in X which are *not* in A) is open.

A *topology on a space X* is a set of rules for defining “open” subsets for X which fits these conditions:

- (the subset with nothing in it) is open
- X is open
- if you stick together any amount of open subsets, the “total” subset (containing all the points which are in
*any*of those subsets) is open - if you have two open subsets A and B, and C is the subset of points which are in
*both*A and B, then C is open.

At one extreme, rules which say that only and X are open define a topology.

At the other extreme, rules which say that *every* subset of X is open also define a topology.

Usually we want something in between.

For example, if X is an infinite set, the set of rules which says that A is open if A is or X−A is finite (i.e. if A contains all but a finite number of the points in X) is called the Zariski topology on X.

Relative to a topology, we can decide whether a space is *connected* (all in one piece):

A space X is connected if the only subsets of X which are both open and closed are and X.

Then a transformation or mapping f from X to Y is *continuous* if, for every open subset B in Y, the subset of all points in X which f maps into B is also open.

f is a *homeomorphism* (it defines X and Y as shapes which can be stretched, squeezed, or twisted into each other, without tearing) if f is a *continuous* one-to-one mapping, and its inverse (what you get by backtracking it) is also continuous.

For much more, presented clearly and in manageable chunks, see

http://www-history.mcs.st-and.ac.uk/~john/MT4522/index.html, which I’ve used in compiling this summary.

As those notes point out, topology has developed in different directions:

1. Differential topology studies surfaces, solutions of differential equations, etc.

2. Algebraic topology: the study of algebraic (in the university sense: groups, rings, etc.) invariants of topological spaces.

3. Combinatorial or geometric topology

4. General or point-set topology is the basic theory underlying the above.

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A good introduction by Jack Huizenga, from quora.com

At a very bare-bones level, algebraic geometry is technically the study of solutions of systems of polynomial equations. As it is actually studied and practiced, however, it attempts to answer qualitative and geometric questions about such solution sets.

In high school algebra, you might do things like try to determine the precise solutions to a system of equations such as

Here this is the intersection of a line and a circle, and consists of two points. It is possible to determine precisely the coordinates of those two points, and there isn’t a whole lot of “geometry” going on.

But what if we consider a system of the form

where p,q are two polynomials in three variables? We’d expect that each equation describes a surface in 3-space, and that their intersection is a curve.

What does it mean to “solve” a system of this form? Perhaps it means to fully describe the solution set in some kind of explicit way. For instance, one could try to parameterize the curve of intersection. That is, maybe you can write down some function

such that

for all t, and likewise with the other equation.

For example, maybe we’re considering the system

Then the solution set can be parameterized by

Parameterizing the solution set could be viewed as the algebraic way of describing a zero set of a system of polynomials. We learn a lot (in some sense everything) about the individual solutions, but the structure of the set of solutions as a whole is not really illuminated.

On the other hand, in algebraic geometry one usually doesn’t care so much about explicit equations. In the last example, you might just say the intersection is a circle and be done with it. Typically the types of questions you ask in algebraic geometry are more qualitative questions about the geometric structure of the solutions. Of course, the theory is capable of asking precise questions about, for instance, the coordinates of points in the solution set, but it usually doesn’t have too much to say. The interesting things you can learn from the theory involve deeper information about the whole collection of solutions, instead of individual solutions.

**The cubic surface**. For a less trivial example, a cubic surface is a surface in 3-space defined by a polynomial equation of degree 3. One such example is the Fermat cubic surface

*The Fermat cubic surface.*

If we are given two different cubic surfaces, each given by their own polynomial of degree 3, it is not so clear from an algebraic perspective what intrinsic properties the two surfaces have in common. They are just given by some equation, and it is hard to directly make analogies between them besides the fact that they have somewhat similar equations.

In algebraic geometry, you would study the cubic surface not necessarily by looking for instance at the coordinates of its points, but by trying to answer questions such as

What kinds of curves are on this surface?

Are there other ways of describing this surface in terms of some fundamental geometric operations?

Is it possible to parameterize this surface by a plane? (Without necessarily caring about what the parameterization is in terms of explicit formulas.)

Let me try and describe a bit about each of these three questions in the case of a cubic surface. Let’s assume our surface is smooth, which is a technical condition ensuring the surface has no singularities. In other words, if you zoom in really close at a point of the surface, it looks basically like a plane.

**Curves on a cubic surface**. There are obviously many different curves which are contained in a cubic surface. For instance, if you intersect the surface with a plane, you’ll get a cubic curve in that plane. Most of the time, this looks something like this:

*The typical plane section of a cubic surface.*

If you were to intersect your cubic with some higher degree surface, such as a surface defined by an equation of degree 2, you’d get some complicated space curves.

It’s also possible to choose a plane which intersects the surface in such a way that the intersection looks like this:

*Picture shamelessly copied from earlier.*

In particular, the surface actually contains some lines. It is a fact that any smooth cubic surface has 27 lines. (In the earlier picture of the Fermat cubic, several of the lines are relatively easy to see. Some of the lines may be “complex” or “at infinity” and not visible in the finite real picture.) The lines on a cubic surface intersect in a beautiful pattern called the double-six configuration.

*The double-six configuration of lines on a cubic surface. (Credit: Wikipedia)*

**Another description of the cubic surface**. Consider the plane. There is an operation called blowing-up, which takes a point in the plane and replaces it by a line. This is perhaps best illustrated by the prototypical picture:

*Schematic diagram of the blow-up of a plane.*

Lines which pass through the point with different slopes are “stretched out” into a spiral staircase. Algebraic geometry is not so much the study of things cut out by specific polynomial equations as it is the study of things which can be described by polynomial equations, and this wonky staircase-thing can in fact be described by polynomial equations.

(Aside: if you are asked by security at an airport what your job is while trying to get on a flight, don’t say “I blow up planes.” Even if it’s true.)

It turns out, if you start from the plane and blow up six different points then the thing you end up with “is” a cubic surface. I put “is” in quotes here because the blowup here is not *embedded* as a surface in space; what this means is that there exists some polynomial map which embeds the blowup in space as a cubic surface.

Conversely, given any smooth cubic surface, there is a collection of six points in the plane such that the surface is the blowup of the plane at those six points. The six lines introduced in the blowup are six of the 27 lines on the cubic surface.

**Parameterizing the cubic surface by a plane**. In fact, it is impossible to parameterize the cubic surface by a plane in the strictest sense: there is no way to make the points of the cubic surface correspond to the points of the plane in a natural one-to-one fashion. But the alternate description of the cubic surface in the previous subsection gives a natural way of parameterizing almost all of the points by the plane. In fact, the blowup of the plane at six points is basically just the plane with six points replaced by lines. Then away from these six lines, the cubic surface is parameterized by the plane with the six points thrown out. We say that the cubic surface is birational to the plane, and it is rational.

**Closing remarks**. As it is actually studied, algebraic geometry has a lot more to do with the geometric pictures I’ve presented here than it does the solving of polynomial equations from high school type algebra. To study higher-dimensional zero sets of systems of equations it is much more profitable to ask qualitative geometric questions than it is to ask about explicit equations and parameterizations.

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Above: Maryam Mirzakhani, the first-ever female Fields Medal winner, died on 14 July 2017. Click here for Stanford University’s obituary.

A friend in Australia wrote to me: “A female mathematician who was awarded the Fields Medal died recently [and she was of] Iranian descent. I have noticed even in Australia several highly skilled Iranian engineers etc. Despite all the problems in Iran, why does it produce such experts in these fields?”

Here are my thoughts.

In general, prosperity and civil liberties are the best conditions for good scientific work.

However, there are other factors.

The Nazi regime in Germany virtually destroyed German mathematics within a matter of months, although Germany had been the centre of world mathematics for over a century before 1933. But not all repressive regimes have the same effect.

In Germany, many of the leading mathematicians were Jewish, or left-wing or at least outspokenly liberal, or both, partly I guess because they had grown up in a world of relatively wide civil liberties.

Max Zorn, famous as the originator of Zorn’s Lemma, was not Jewish, but in later life (in exile in the USA, like most German mathematicians) had a raspy voice because of a throat injury received in street-fighting against the Nazis.

Maybe in part as a reaction to anti-German chauvinism from the victors of World War I (the Allied powers refused sponsorship to any international mathematical conferences involving German mathematicians for some years after 1918, and that blockade was broken only by mathematicians deciding to hold their international conferences without official money), the Nazis wanted to promote “German mathematics” as opposed to other nationalities’ mathematics.

Such chauvinism is fatal in mathematics. The milder form in which British mathematicians insisted on shunning Leibniz’s formulations of the calculus in favour of Newton’s, which continued until broken by a student revolt at Cambridge in the second decade of the 19th century, made British mathematics a backwater for a century and a half even while other sciences were flourishing.

Not all repressive regimes promote such chauvinism in such abstract areas of thought.

Mathematics did well in the Stalinist USSR, and under the fascistic regimes in Poland and Hungary between World War 1 and World War 2. Most of the Hungarian mathematicians did their best work in exile – as did Maryam Mirzakhani, who moved to the USA when she was 22 – but the Poles and Russians did it in their home country.

Some factors involved here:

1. If there is relative freedom in mathematics – because Stalin, or Pilsudski, or Horthy, or Ayatollah Khamenei, however authoritarian in other realms, do not get the idea of laying down the law in maths – then keen researchers may choose to go for maths, while in a freer country they might have gone for another field.

2. The Hungarian mathematician Paul Erdös suggested that Hungary might have done well in maths because – although it had well-established universities and good connections with a large diaspora, as Iran has – it was a relatively poor country (poorer then than Iran is now, and with a much smaller population, around 9 million then while Iran has 80 million now). Mathematics is a cheap science. Keen researchers who might have chosen another field in a country with greater resources maybe went for maths instead.

3. Contrary to popular image, mathematics is the most social of sciences. It is very difficult and rare to make progress in maths without daily conversation and argument with other mathematicians. Thus talented mathematicians tend to appear in clusters. Where you have a few, they attract more. Where they are scarce, they remain scarce.

Much (not all) of the extraordinary flowering of Polish mathematics between World War 1 and World War 2 was down to a group of mathematicians who met in one particular café in Lviv (now part of Ukraine). The leading role of Germany in world mathematics for a century and a half before 1933 was in large part (not totally) due to work done at Göttingen, a small university in a small town in central Germany. (Mid-19th century, its “Mathematics-Physics Seminars”) still had only about 15 students).

Thus there is an element of “randomness” in where clusters of mathematical advance appear. They are not spread evenly in exact proportion to advantageous general circumstances.

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**Most helpful to other students**

Onesimus, Mohaned, Tegan

**Best questions and objections in class**

Ashley, Genie

**Best diagrams**

Rashi, Michelle

**Hardest working**

Umut, Bolaji

**Neatest work**

Brenda

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The New York World Building (above) was New York’s first skyscraper, in 1890. Lots more were built in the next few decades. Why weren’t skyscrapers built before? Builders had been able to build taller before, and lifts had been built since the 1850s.

Answer: the spread of the telephone. Why? Genie Louis de Canonville produced a good mathematical explanation, with help from the rest of the Year 12 Further Maths class.

This is a “Fermi problem”, meaning that to do the calculation you can’t just go from figures you’re given, but have to make good estimates of some numbers involved.

A skyscraper office building would have at least 100 people working on each floor. (The Empire State Building has an average of 210 per floor).

These are office workers, so their work has to involve getting in items of information (messages) and sending out messages. Today that is done over the internet. Before the invention and spread of the telephone, it had to be done by physical messages carried by messengers. A messenger can carry more than one message, but a busy office could not afford to wait for urgent messages, and some “messages” (those that would be big email attachments today) would be bulky.

The New York building had 20 floors (much fewer than later skyscrapers), but that makes 2000 workers. Suppose each worker generates 100 messages in or out per day. That’s 200,000 messenger-journeys. Since the messages would not be evenly spread over the day, you’d have to allow for say 40,000 messenger journeys in a peak hour.

A lift might do the trip from ground floor to an average higher storey in one minute, and take 10 people, so that’s 600 messenger-journeys per hour.

So you’d need 700 lifts to make the building work. Too many.

I don’t know how many lifts the World Building had. The Empire State Building, built in 1932 and long the tallest office skyscraper in the world, has about 70 lifts for 102 storeys.

Lifts can be built faster and bigger now, so the Shard has 38 lifts. Still, the Shard would be unviable if all the messages into and out of it had to be carried by messengers.

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At the end of the 13th century Pope Boniface VIII wanted to commission some paintings for St. Peter’s Cathedral, in Rome, and sent a courier round Italy to find the best painter.

One artist in Florence, named Giotto, just took a sheet of paper and painted a perfect circle on it, freehand.

The Pope gave Giotto the commission. Over six hundred years later, Giotto’s skill is still famous.

Since you’re not Giotto, use another method.

- Have, with your maths kit, a 2p coin (diameter 25.9 mm). That will do for many diagrams.
- Where a diagram calls for a larger circle, or two circles, one larger than the other, it’s useful to get a larger disc. I’ve got some 50mm and 75mm plastic discs, which you can keep with your maths kit, and should help.
- For a larger or more detailed diagram, use compasses.

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In the next school year we will be asking all A level maths and further maths students to buy the new Casio Classwiz calculator. The new syllabuses require functions available only on this and other similar calculators. It costs about £30 on Amazon, but the school should be able to get a bulk buy at around £20. You can, as of now anyway, get a Classwiz calculator yourself, in advance, at a similar price on eBay.

If you already have a Casio 991ES, previously the most advanced Casio scientific calculator, then as of now (and this may change as more students switch to the Classwiz) you can get a reasonable price by selling it on eBay.

Review of the Classwiz by Abdal Mohammad (thanks to Abdal for this)

- The display is much better. If you want to convert a complex number to polar form, for example, you get the answer clearly on the screen in surd and fraction-of-π form if appropriate, rather than having to scroll across to see the r and then the θ, and having to use the Alpha>X and Alpha>Y key combinations to get them in surd and fraction-of-π form.
- It solves cubics and quartics neatly, including giving complex roots.
- It solves polynomial inequalities neatly (useful for FP2).
- It does matrix calculations more neatly and clearly than the 991ES.
- It find cross-product of vectors, which the 991ES doesn’t, and does many other vector calculations.
- It evaluates sums of series between given numerical bounds.
- The OPTN key provides a menu of further options that are relevant to a particular mode. It gives access to hyperbolic functions, to symbols for different angle measures, and the various engineering symbols.
- Twelve calculator modes are available by tapping on the MENU button. Each mode has an image that better visualises what each mode works to do and what it is.
- Calculations in statistics (e.g. the normal distribution) are smoother and more intuitive.
- It can handle simultaneous equations with up to four unknowns, and deal with quadratics, cubics, quartics, and even quintics, stating the turning point for each.
- It has a QR code generation which you can scan with your iPhone, which makes it a cheap alternative to a graphics calculator if you want to visualise graphs.

Additional comment from me about the Classwiz and statistics:

The new spec for ordinary A level maths says:

“Use of a calculator to find individual or cumulative binomial probabilities”

“Students will be expected to use their calculator to find

probabilities connected with the normal distribution.”

It is not really clear what this means, because the Edexcel formula book still has a table of binomial probabilities (and a few, but only a few, values for the normal distribution); but that’s what the spec says.

On the Classwiz, finding those probabilities is intuitive and easy.

On the 991es, you can find the normal cdf from a z-value. But you have to calculate the z, whereas on the Classwiz you can input x, μ, σ, and get your answer straight from those.

On the 991es, you can calculate binomial probabilities, but only via the nCr function. As far as I can see, you can calculate the binomial cdf only by adding up.

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Sessions at the City of London Academy Southwark, open to Sixth Form students and teachers from other schools. Each session combines a talk from a visiting presenter with interactive work. To book, contact the London South East Maths Hub.

**Topology and Big Data**

Jacek Brodzki (Southampton University) and Derek Hill (UCL) will introduce us to new methods being used to analyse the much larger datasets produced by new information technologies. Traditional statistical methods lose grip, and topology, the branch of maths that studies the properties of shapes which remain when they are bent, twisted, or stretched, gives us keys to the “shape” of the data. Monday 18 September, 9:50-12:10.

**Building a Theorem**

Joe Watkins (Kent University) will introduce us to the exact and careful procedures which have enabled mathematicians to defy prejudice, tradition, and established authority, and prove the most startling results beyond doubt. He’ll follow up with a presentation on the relations between juggling and music. Wednesday 18 October, 9:50-12:10.

**Escher and Coxeter**

Sarah Hart (Birkbeck, University of London) will give us a glance into where geometry meets art. Maurits Escher, one of the most off-beat artists of the 20th century, had an ongoing conversation with Donald Coxeter, the outstanding geometer of the 20th century. Above: Escher’s Circle Limit I. Wednesday 15 November, 16:45-17:45

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Click here to download the task sheet.

Please write your work on A4 lined paper, and hand it in at the start of the September term. These problems aim to:

- help bed in your knowledge of the method of differences
- help you retain the little about Maclaurin series we had time to do at the end of the summer term
- practise the sort of mathematical thinking which will be especially useful to you if you decide to try the extra exams, MAT or STEP, or if you just want to be a better mathematician generally

Because we had so little time on Maclaurin series, there are some reminder notes about it at the end of the task sheet.

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