We have been asked to share opportunities offered by Jump Trading, one of the TMS’ sponsors. Please find information attached in the pdf here. Jump Trading Quant Research Opportunities
we look forward to seeing all of you there very soon!
Constable: Warren Li
Our final talk – yes, final talk! – of this year is given by Professor Eric Lauga. Details below, hope to see lots of you there!
Monday 4 March, 8:30PM
Elliptical billiards and Poncelet trajectories
Professor Pelham Wilson (DPMMS)
Given an elliptical billiard table, to any ball trajectory which doesn’t cross the line segment joining the two foci, there is an associated smaller confocal ellipse inscribed in the trajectory. A Poncelet trajectory is one which is closed after a finite number of bounces. We’ll see that if there is one such closed trajectory with n segments, then starting from every point on the outer ellipse, there is a similar closed trajectory with n segments and the same inscribed ellipse, and indeed all these trajectories have the same length
Analogous geometric properties hold more generally for any pair of conics in the plane, and in modern terminology the existence of analogous Poncelet polygons is related to the torsion points on an associated elliptic curve.
Monday 25 February, 8:30PM
Addition, multiplication, and why they don’t get along
Dr Julia Wolf (DPMMS)
The sum-product conjecture, put forward by Erdős and Szemerédi in the 1980s, states that the set of all pairwise sums and the set of all pairwise products of a finite subset of the reals cannot simultaneously be close to minimal in size. Despite the simplicity of its statement and a significant amount of research effort devoted to its resolution, the conjecture remains open to this day. In this talk I will explain the motivation for the conjecture as well as some fascinating partial results.
Monday 18 February, 8:30PM
Dr Hamza Fawzi (DAMTP)
A polynomial that is a sum of squares of other polynomials can only take nonnegative values. This trivial observation is surprisingly powerful: many inequalities in mathematics have simple sum-of-squares proofs. I will discuss algorithms that can automatically search for sum-of-squares proofs for polynomial inequalities, and the extent to which they can be considered as “automatic proof machines”.
Monday 11 February, 8:30PM
The Continuum Hypothesis
Professor Imre Leader (DPMMS)
We’ll explore a statement known as the Continuum Hypothesis, which states that there are no `sizes’ of sets between the natural numbers and the reals — or, more precisely, that every uncountable subset of the reals bijects with the reals.