This post is aimed at secondary-school students pitched roughly at the level of the British Mathematical Olympiad. It is ostensibly about a certain class of number theory problems, but the main underlying mathematical principle is broader than this. The post … Continue reading →
The first round of the British Mathematical Olympiad was marked in early December. Belatedly, here are some thoughts of the problems. These aren’t supposed to be official solutions, and some of them are not in fact solutions at all. Students … Continue reading →
AnalysisMeasure TheoryUniversity teachingborel measurefundamentals of probabilityintegrable functionlebesgue integralmeasurable functionMonotone convergence theoremmonotone limitsigma-finitesimple function
I’m currently lecturing the course Fundamentals of Probability at KCL, where we cover some of the measure theory required to set up probability with a higher level of formality than students have seen in their introductory courses. By this point, … Continue reading →
I’m currently lecturing my first course at King’s, which builds measure theory from the ground up to the construction of the Lebesgue integral, along with some more probabilistic topics. In this second week, we have been discussing various matters related … Continue reading →
Probability TheoryUniversity teaching1A Probabilityalmost sure convergencebirthday problembranching processCambridgecovariancediscrete harmonic extensionexamples sheetexploration processextension problemsharder problemsmultitype branching processPart 1APolya's urnrandom permutationRandom walksize-biasedsnakes and ladderstotal population size
This year, I was lecturing the first year probability course in Cambridge. To supplement the usual excellent problem sets I inherited from James Norris and many previous lecturers, I prepared extension problems for enthusiastic students. A handful of the extension … Continue reading →
The second and final round of this year’s British Mathematical Olympiad took place on Thursday. Here are some thoughts on the problems. I wasn’t involved in choosing the problems, although I did write Q4. I’ll say a bit more about … Continue reading →
British Maths OlympiadEuclidean geometryOlympiad Mathematicsbinary expansionBMOBMO1British Mathematical Olympiadgenerating function
The first round of the British Mathematical Olympiad was sat on Thursday by roughly 2000 pupils in the UK, and a significant number overseas on Friday. For obvious reasons, much of the past 18 months has been dominated by logistical … Continue reading →
For a Galton-Watson tree, can one obtain upper bounds in probability on the height of the tree, uniformly across all offspring distributions with mean $latex \mu$? Continue reading →
[Ho63] addresses the alternative model where the increments of a random walk are chosen uniformly without replacement from a particular set. The potted summary is that the sum of random increments chosen without replacement has the same mean, but is more concentrated that the corresponding sum of random increments chosen with replacement. This means that any of the concentration results proved in the earlier sections of [Ho63] for the latter situation apply equally to the setting without replacement. Continue reading →
British Maths OlympiadEuclidean geometryOlympiad MathematicsBMO1BMO1 2019British Mathematical Olympiadrecurrence relationtwin prime conjecture
The first round of the British Mathematical Olympiad was sat yesterday. The paper can be found here, and video solutions here. Copyright for the questions is held by BMOS. They are reproduced here with permission. I hope any students who … Continue reading →
