Termcard for Michaelmas Term 2014

Week 1, Tuesday 14th October – An unqualified success in promoting Mathematics!

Johnny Ball Johnny was writing comedy for TV in the late 60’s. When asked what subject he might write his own series on, he said, “Maths” and produced Think of a Number – which won BAFTA in its first year – and subsequently 20 TV series on Maths and Science. Amazingly Johnny had left school aged 16 with just 2 O’levels in Maths (of course – 100%) and Geography. But he was clearly gifted and made Recreational Maths his hobby while in the RAF. Since then he has been conveying his love of all things mathematical to audiences young and old, receiving 7 honorary degrees including Honorary Fellow of the Royal College of Teaching and of the Actuaries Livery Company. He has run Maths Exhibition stands for HM Dept of Education and written 5 educational stage musicals including Tales of Maths and Legends which headed the UK’s Maths Year 2000 efforts and averaged 3500 bums on seats per day in major provincial theatres. Also in 2000 his Mind Zone Live show in the Dome SOLD OUT for the entire year. Earlier this year HM Government brought Chinese Maths teachers to the UK. At precisely the same time he heard that his Think of a Number maths book had sold 50,000 copies in Beijing alone? Johnny will talk about the Maths he loves, virtually without “doing any Maths” so those pro or anti maths should all enjoy his talk equally.

Week 2, Tuesday 21st October – Decision Problems for Linear Recurrence Sequences

Joel Ouaknine Linear recurrence sequences (LRS), such as the Fibonacci numbers, permeate vast areas of mathematics and computer science. In this talk, we consider three natural decision problems for LRS, namely the Skolem Problem (does a given LRS have a zero?), the Positivity Problem (are all terms of a given LRS positive?), and the Ultimate Positivity Problem (are all but finitely many terms of a given LRS positive?). Such problems (and assorted variants) have applications in a wide array of scientific areas, such as theoretical biology (analysis of L-systems, population dynamics), economics (stability of supply-and-demand equilibria in cyclical markets, multiplier-accelerator models), software verification (termination of linear programs), probabilistic model checking (reachability and approximation in Markov chains, stochastic logics), quantum computing (threshold problems for quantum automata), discrete linear dynamical systems (reachability and invariance problems), as well as combinatorics, statistical physics, formal languages, etc. Perhaps surprisingly, the study of decision problems for LRS involves advanced techniques from a variety of mathematical fields, including analytic and algebraic number theory, Diophantine geometry, and real algebraic geometry.

Week 3, Tuesday 28th October – Invariants Puzzle Drive

None Join us for a competitive evening of puzzle solving. Compete in teams to complete a variety of entertaining problems set by our puzzle master. Prizes will be awarded to the winners and other successful teams and there will be free snacks. First prize this year is a huge £200, courtesy of Oxford Asset Management. This event is free!

Week 4, Tuesday 4th November – Points and lines

Ben Green Suppose you have n points in the plane, not all on a line. A famous result called the Sylvester-Gallai theorem states that there must be at least one “ordinary line”, by which we mean a line through precisely two points of the set. I will discuss this theorem and then turn to some more recent developments connected with the question of how many ordinary lines there must be. Perhaps unexpectedly, this involves considerations about elliptic curves as well as some results from additive number theory. The talk should be accessible to undergraduates (you won’t need to know what an elliptic curve is).

Week 4, Thursday 6th November – It’s a talk of two halves: A crash course in football and tennis forecasting

ATASS Sports – Dr Tim Paulden Forecasting the outcomes of sports events is vital to the sports betting industry, sports clubs and – like me – those with fantasy football teams. Dr Tim Paulden will be presenting models to predict football and tennis. The talk will provide an insight into the work that ATASS Sports carry out and after the talk there will be an opportunity to talk to Dr Tim Paulden and Rich Hill about career opportunities with firm.

Week 5, Tuesday 11th November – Gravity and the Arrow of Time

Julian Barbour Since the time of Boltzmann, one of the great mysteries in physics has been the apparent mismatch between the time-reversal symmetry of the laws of nature and the irrerversible growth of entropy expressed by the second law of thermodynamics. It is widely believed that special initial conditions must be imposed on any time-symmetric law if its solutions are to exhibit behaviour of any kind that defines an ‘arrow of time’. In a recent paper in Physical Review Letters (arXiv:1409.0917), collaborators and I have shown this is not so. My talk will be based on this paper, in which we study the simplest non- trivial time-symmetric gravitational law that can be used to model a dynamically closed universe. Because of specific properties of this law, its typical solutions all divide at a uniquely defined point into two halves. In each a well-defined measure of shape complexity fluctuates but grows irreversibly between rising bounds from that point. Each solution can be viewed as having a single past and two distinct futures emerging from it. Any internal observer must be in one half of the solution and from observations within it will find an arrow of time despite the underlying time-symmetry of the law. We are still far from a complete explanation of the second law, but this seems to be a promising first step.

Week 6, Tuesday 18th November – Why I am not a Platonist: quantum information, life and the universe

Vlatko Vedral I will start by introducing the concept of information due to Shannon and will then argue that it actually only starts to make sense within the framework of quantum physics. Here, the key will be the intrinsic randomness existing within all elementary quantum phenomena and which allows for information to be created ex nihilo (out of no prior information). This counterintuitive feature will be seen to be linked with the bizarre quantum effects such as being in two places at the same time as well as the famous “spooky action at the distance” that Einstein alleged quantum physics to permit. The latter is based on the phenomenon of quantum entanglement which all the experiments so far have firmly confirmed. Technologically, entanglement allows us to teleport as well as to perform quantum computations that outperform their classical counterparts. Recent experiments also suggest that living systems might be using entanglement to improve processes such as photosynthesis and magneto-reception. Finally, I will speculate that we should think of quantum information in entangled states also as being behind notions such as time and temperature. This may allow us to understand how the observable universe could have started with low (or no) entropy, but has then adiabatically evolved into states with high entropy such as the current one (which seems to contradict the basic laws of thermodynamics).

Week 7, Tuesday 25th November – Early computers, and how I met my wife – Number theory on the EDSAC 55 years ago

Bryan Birch I shall reminisce about the early days of computers, and work I did with Peter Swinnerton-Dyer on Wround about 1958-61.

Week 8, Tuesday 2nd December – Profinite group theory

Dan Segal A suitably coherent family of finite groups can be ‘stuck together’ to make a compact (infinite) group, by forming an ‘inverse limit’. The resulting object is called a profinite group. For example, the family (Z/p^nZ) (n>0) corresponds to the ring of p-adic integers Z_p. All sorts of interesting questions about infinite groups, or about infinite families of finite groups, can be approached by studying the associated profinite group. I will give various illustrations.

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