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# Plus Magazine Monthly Column – May-June 2018 – Hawking, Billiards and Randomness

By on 02/07/2018

Plus http://plus.maths.org is a free online magazine about mathematics aimed at a general audience. It is part of the Millennium Mathematics Project, based at the University of Cambridge and our aim is to open a door onto the world of maths for everyone. We run articles, videos and podcasts on all aspect of mathematics, from pure maths and theoretical physics to mathematical aspects of art, medicine, cosmology, sport and more. Plus has a news section, covering news from the world of maths as well as the maths behind the mainstream news, reviews of books, plays and films, as well as puzzles for you to sharpen your wits.

Ask a 12-year-old child to explain the Big Bang or a black hole and chances are you’ll get a reasonably correct answer. Ask a physicist about the same concepts and he or she will tell you they are accepted components of our theory of the Universe. Only a few decades ago neither of these two statements would have been true. The fact that they are now is to no small extent due to a single physicist: Stephen Hawking.

Hawking passed away on March 14, 2018. The aftermath was a sad and emotional time at the Centre for Mathematical Sciences (CMS) in Cambridge, where Hawking formerly held the Lucasian Chair of Mathematics and established the Centre for Theoretical Cosmology. Plus is based at the CMS, we compiled a brief overview of his work, and also collected memories from those who knew him well.

Hawking is remembered fondly by friends and colleagues for his wit, humour and enjoyment of life. He was always keen to attend, and give, parties and was well known for joining the dance floor. Ben Allanach, Professor of Theoretical Physics at the University of Cambridge, remembers a disco, where “us (then young) postdocs were dancing, and Stephen had the bike lights on his wheelchair fixed to ‘flash’ and came and danced with us,” Tim Pedley, Professor of Fluid Mechanics at Cambridge, agrees with this impression of a gregarious and lively character. “Stephen really enjoyed being the centre of attention, and had a wonderful way of doing that.”

Gary Gibbons, also Professor of Theoretical Physics at Cambridge, offered the most succinct description of Hawking’s character. “[I remember] an extremely lively, amusing, interesting and overwhelmingly sharp and intelligent person,” he told us. “He enjoyed life and was absolutely determined not to let his physical condition prevent his enjoyment, both of science and of life in general.”

What struck us about Hawking’s scientific record is its cohesion. The Big Bang and black holes don’t at first appear connected, but both are examples of singularities. Hawking showed, with Roger Penrose and George Ellis, that singularities are an inevitable consequence of the general theory of relativity.  The famous singularity theorem provided the mathematical underpinning for experimental evidence that the Universe did indeed spring into existence from a very hot and very dense beginning. It also showed that, according to general relativity, black holes must exist. Hawking’s work on black holes established our current understanding of these gravitational monstrosities.

In a twist of the theory that seems almost fanciful, Hawking and Gibbons also realised that the whole of the Universe could be regarded — in some sense — as a black hole turned inside out. Using this idea they were able to make precise the inflationary theory of the Universe, first proposed by Alan Guth, which describes the first moments of the Universe and explains how the large-scale structures we see today, planets, stars and galaxies, came into being.

Using approaches that had proved successful both in describing the radiation emanating from black holes and in inflationary physics, Hawking also worked on his own approaches to quantum gravity: the elusive theory of everything that combines quantum physics and general relativity. It is here that the greatest challenges for future generations lie.

Hawking’s work was mathematical in nature, but the spectacular discovery of gravitational waves in 2015 opened up the possibility to test some of his results experimentally.  With more observational evidence due to arrive both from gravitational waves and from further analyses of the cosmic microwave background, Hawking leaves us in exciting times and his successors with important ideas to work on. To find out more about Stephen Hawking and his work, see this collection of articles on Plus magazine.

Image by NASA

Arithmetic billiards

From cosmology and the beginning of the Universe let’s move on to something more elementary: the least common multiple and greatest common divisor of two natural numbers. A recent article by Antonella Perucca explains how you can work out these numbers on a billiard table.

Given two natural numbers a and b, start with a rectangular billiard table whose sides have lengths a and b and which doesn’t have any holes that can swallow up balls. Now shoot a billiard ball from one of the corners at a 45 degree angle to both sides. The ball will start moving, bouncing off the sides as it goes. Eventually it will hit a corner. The least common multiple of a  and b is the length of the path from the start of the ball to it hitting a corner divided by $\sqrt{2}$. The greatest common divisor of a and b is the length of the path from its start to its first point of intersection, divided by $\sqrt{2}$.

If you divide the table up into little squares of side length 1, then The least common multiple of a  and b is the number of little squares crossed by the path. The greatest common divisor of a and b is the number of little square the crossed by the portion of the path between its start and its first point of intersection.

Perucca’s article delivers a proof of this result, assuming an idealised ball of no mass which doesn’t experience any friction. The proof uses a clever trick: whenever the ball hits a side of the table, reflect the table in this side. Doing this repeatedly turns the path of the ball into the diagonal of a square whose length is easily calculated.

The maths of randomness

This August the International Congress of Mathematicians will take place in Rio de Janeiro. At the Congress four deserving mathematicians will be awarded the Fields Medal, one of the highest accolades in mathematics. We’re very excited to be attending the ICM and to report from it.

In the meantime we are very pleased to publish a series of articles by one of the Fields Medallist from the last ICM: Martin Hairer, Professor of Pure Mathematics at University College London. Hairer was awarded the Fields Medal in 2014 for his work on equations designed to describe the boundaries at which two substances meet. An example comes from a piece of paper, set alight at one end. As the fire eats its way through the sheet, ash and still intact paper meet at an ever-changing, irregular  boundary, which is difficult to model mathematically. Hairer’s approach at describing this boundary hinged on getting to grips with the element of chance that is involved in the burning process.

Hairer’s articles on Plus explore the mathematics of randomness at a more elementary level. In them Hairer points out that while randomness and chance are hard to define in a philosophical sense, the mathematical theory of probability is well defined and free from controversy. He explores the role of symmetry in probability theory: if a coin is perfectly symmetric, so that no side of it has any reason to come up more often when the coin is flipped than the other, then we can say that each side has a 50% chance of coming up.

In the articles Hairer also examines the interesting concept of universality: even if you can’t describe the microscopic behaviour of a system, you can often pin down its large-scale behaviour. In physics, this is the case for liquids, for example: even though the behaviour of individual molecules is essentially random and will look different for different liquids at different moments in time, on a macroscopic scale all liquids are similar and defined by like viscosity and pressure.

Universality can also occur for systems made up of people. While attributes such as body height or political opinion may vary in all sorts of ways over individuals in a population (the molecules), the averages of such attributes (large-scale properties) always vary over samples from the population in the same kind of way: the averages of these measures are distributed according to the normal distribution. This result, called the central limit theorem, is what enables us to make inferences about a whole population from smaller samples.

Hairer also leaves us a warning in the form of a paradox which illustrates the subtlety of probability theory. Suppose you have two envelopes with money in them, one containing twice as much money as the other. You don’t know which envelope contains which amount. You are asked to pick an envelope and you’ll get to keep the money in it. However, after you have picked one, you get the chance to change your mind.

To decide whether to change your mind or not, you make the following calculation: if the envelope you have picked contains the amount x then the other contains the amount x/2 with probability ½ and the amount 2x with probability 1/2. The expected amount contained in the other envelope is therefore

$$(x/2+2x)/2 = 5x/4.$$

Since this is more than x, you should switch. But what if you get the chance to change your mind again? By the same argument you should switch again, and then again, and then again. You will keep switching forever and never get any money at all.

What is wrong with this thinking? To find out see Hairer’s series of articles on Plus. To find out more about the work that gained Hairer the Fields Medal, see this article.