## Plus Blog

September 9, 2010
For the latest episode of the Math/Maths podcast Peter Rowlett interviewed a very tired and exhausted The |
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September 6, 2010
Many things in life are more than the sum of their parts. Whether its the behaviour of crowds of people, flocking birds or shoaling fish, the unpredictable patterns of the weather or the complex structure of the Internet, it's often the interaction between things, rather than the things themselves, that generates complexity. It's a challenge to science, whose traditional approach of taking things apart and looking at the individual bits doesn't work when faced with emergent complexity. But there are mathematical techniques to understand this phenomenon. The Living in a Complex World website, originally launched to accompany an exhibit at the Royal Society Summer Science exhibition, explores complexity in the real world and has some great factsheets looking at the maths used to understand it. It's well worth a look! |
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August 27, 2010
Here are some pictures from the ICM 2010:
Well, maybe not quite ... it's a panel discussion on popularisation of maths. (Thanks to Jaime Carvalho e Silva for both of these photos.)
3000 mathematicians trying to have dinner. 3000 mathematicians trying to have lunch. 3000 mathematicians trying to catch a hotel bus.
The old town of Hyderabad |
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August 20, 2010
What would you think if the nice café latte in your cup suddenly separated itself out into one half containing just milk and the other containing just coffee? Probably that you, or the world, have just gone crazy. There is, perhaps, a theoretical chance that after stirring the coffee all the swirling atoms in your cup just happen to find themselves in the right place for this to occur, but this chance is astronomically small. Cédric Villani, Institut Henri Poincaré The fact that such spontaneous separation never occurs in practice is an illustration of a deep physical law: it says that the In the late 19th century the physicist Ludwig Boltzmann studied this phenomenon, looking at what happens when a gas is released into a room from a bottle. He came up with an equation describing the evolution of this process — the change over time — in terms of how individual atoms collide and other forces acting on the gas. His calculations showed that indeed entropy doesn't decrease. The atoms of gas start out in an ordered state — all sitting in the bottle — and end up in a state of maximum complexity, dispersed throughout the room. Intriguingly, this result meant that there is what physicists call an Villani, however, does understand was entropy is. One question that has until recently remained open was how fast the entropy of a system increases. Cédric Villani received his Fields Medal for developing rigorous mathematical techniques that provide an answer. They show that while entropy never decreases, it sometimes increases faster and sometimes slower. Based on this work, Villani developed a general theory, Villani also used his understanding of entropy to explain a phenomenon that had puzzled physicists for 60 years. Back in the 1940s, the Soviet physicist Lev Davidovich Landau claimed that plasma, a form of matter similar to gas, spreads and reaches its equilibrium state without increasing its entropy. Landau argued that unlike gas, whose approach to equilibrium is driven by particle collisions (which also lead to a loss of order), plasma reaches equilibrium through a decay in its electric field. Landau made some progress on proving his claim, but despite hard work by many physicists over 60 years, it wasn't until Villani's arrival on the scene that Landau was proved right. All this might sound somewhat esoteric, but Villani's deep understanding of entropy has direct consequences for real-life problems. In the 18th century the French mathematician Gaspard Monge started to think about how to transport goods to various places in an efficient way. For example, you might want to distribute a bunch of letters sitting in a post office to the various addresses in a way that minimises transport cost. Villani and his colleague Felix Otto made a crucial connection between this problem and gas diffusion, by drawing the following analogy: loosely speaking, the initial state (all letters in the post office) corresponds to the ordered state of gas sitting in a bottle, while the end state (letters delivered) corresponds to a state where the gas has dispersed. To any configuration of dispersed gas particles you can assign a cost by seeing how far the particles had to travel from the original ordered state. Using this analogy and their understanding of entropy, Villani and Otto made important contributions to optimal transport theory. So next time you have a cup of milky coffee or receive a letter, stop to think that a deep understanding of either, or even better a combination of both, could have earned you a Fields medal. You can find out more about Villani's work in this excellent description on the ICM website, on which this blog post was based. To find out more about the Boltzmann equation and optimal transport problems, read the |
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August 20, 2010
Suppose you throw an equal number of white and black balls into a rectangular box which is, say, 30 balls long, 10 balls wide and is now 5 layers deep in balls. What it the probability that you have a run of touching white balls from one end of the box to the other? Stanislav Smirnov
This question, asked all the way back in 1894 in the first issue of the
Just as the name conjures up the image of water percolating through soil or porous rock,
It turns out that there is a particular p < p, then the liquid will never percolate through the lattice. When the probability of the vertices being open passes this critical point, and _{c}p > p, the system flips behaviour and the liquid can trickle all the way through. (You can read an excellent more technical introduction in Percolation: Slipping through the cracks.)
_{c}
Physicists are interested in this area as it is one of the simplest models to have a
But the systems physicists are interested aren't neatly space lattices in regular shapes. Instead physicists are interested in what happens at the smallest possible scale when the spacing in the mesh of the lattice becomes finer and finer, which they call taking the
In 2001 physicists and mathematicians alike breathed a sigh of relief when Smirnov proved that the crossing probability existed for the scaling limit for a two-dimensional triangular lattice, and that it was equal to the value calculated by Cardy's formula. Keston, himself a pioneer in the percolation theory, said that Smirnov's work would make statistical physicists very happy as it confirmed their assumptions and put the area on a solid mathematical foundation. And it is hoped that the novel techniques Smirnov used to prove this and related results will allow him and others to extend these results to any two-dimensional lattice (square, hexagonal, and so on), proving that Cardy's conjecture is At the end of Shmirnov's own presentation of his work at the ICM, he was asked if his work could be extended to three-dimensional lattices. Smirnov held up his hands, very little is known about how these models work in three-dimensions. Perhaps Smirnov, or a future Fields medallist, will take us there. |
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August 19, 2010
Elon Lindenstrauss got the Fields Medal for developing tools in the area of dynamical systems and using them to crack hard problems in number theory. Elon Lindenstrauss, Princeton University As the name suggests, number theory studies the basic properties of numbers. The whole numbers 1, 2, 3, etc are probably the first thing that spring to mind when you think about numbers. Close to follow are the In turns out that you can approximate an irrational number, call it , by a fraction to any degree of accuracy. If you give me a really small number , then no matter how small is, I can find you a fraction that's within of . But some approximations are better than others. The fraction 2147865/68341 is a tiny bit closer to than 22/7, but it's also much more horrible to write down because it has such a large denominator (and as a result a very large numerator). So what's the ideal relationship between the accuracy of approximation and the denominator of a fraction? In the 19th century the German mathematician Johan Dirichilet came up with a notion of this ideal relationship. He decided that an approximation of an irrational number should be no further from than . In other words, if the denominator is large, (so that is even larger and therefore very small), then the fraction should make up for this by being close enough (within ) of . Dirichilet proved, and the proof wasn't very hard, that given any irrational number , you can always find infinitely many fractions which satisfy this criterion. So there's a "nice" approximation, in Dirichilet's sense, for any level of accuracy. It turns out that something similar is true for pairs of irrational numbers and . There are infinitely many fractions and which are nice In the twentieth century the mathematician John Littlewood decided that we should be able to do even better than this. Given any two irrational numbers and and an that's as small as you like, there should be fractions and so that Number theory is littered with statements that look like they should be easy to prove but turn out to be incredibly hard. In these cases you have to look for clever tools to help you find a solution. In his work Elon Lindenstrauss did just that, using tools from dynamical systems theory. As an example of a dynamical system, think of the 2D plane in which every point is defined by its co-ordinates, a pair of numbers . Now take any such point and shift it by a certain distance to the right and up by another distance . This rule gives you a dynamical system. You can apply it again and again and see what happens to the trajectories of various points. In the case of the plane, nothing very interesting happens, as trajectories just move further and further away from the centre of the plane, given by the coordinates . If, however, if you look at the surface of a doughnut, things get more interesting. You can make such a surface by taking a square from the plane, turning it into a cylinder by gluing together the left and right edges, and then bending it around and gluing together the circles on either end of the cylinder. In this way, the doughnut's surface inherits the coordinates defined on the original square. Things now become more interesting as you shift points around as before, using the numbers and . Trajectories can travel round and round and visit the same patch of doughnut lots of times. It turns out that if your two numbers and are irrational, then the dynamical system is what's called Using a more complicated dynamical system, Lindenstrauss and his colleagues made massive progress towards a proof of the elusive Littlewood conjecture. They showed that if there are any pairs of numbers that can't be approximated in the nice way stipulated by the conjecture, then they make up only a negligible portion of the plane in which they live. There are pairs for which the conjecture isn't yet proven, in fact there's infinitely many of them, but as Lindenstrauss showed, collectively they are nothing more than drops in the ocean of the 2D plane. It's this progress on Littlewood's conjecture that forms part of the body of work for which Lindenstrauss is being honoured. You can find out more about his work in this excellent description on the ICM website. |