Probability, Trees and Algorithms

Just got back from a very nice week at Oberwolfach, organized by Luc Devroye and Ralph Neininger. I took notes on the talks on my tablet, using GoodNotes and a Jot Touch 4. People seemed to like them so I’m putting them up here for anyone who’s interested. The main downside of the app is that it creates huge PDFs: the above file is 50MB. This is mostly due to my inclusion of pictures of some slides, but the program creates rather large files even if you don’t include pictures.

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Nobel Prize in Chemistry

Today the 2014 Nobel Prize in Chemistry was awarded, to Eric Betzig, Stefan Hell, and William Moerner. They were awarded their prize in honour of their advances to microscope technology. In brief, the breakthrough was an invention of methods which allow for optical microscopy on scales that are smaller than the wavelength of visible light (the so-called diffraction limit).

Below are pictures of the medal presented to Robert Robinson after he won the Nobel Prize for Chemistry in 1947. (The citation is “for his investigations on plant products of biological importance, especially the alkaloids”. Apparently his achievements included synthesizing a cocaine precursor called tropinone, and discovering the molecular structures of morphine and penicillan.) I found this medal in a rather run-down presentation box on the mantlepiece in the Magdalen Fellows coffee room. (I returned it to the mantle after photographing it.)

1947 Nobel Prize in Chemistry

1947 Nobel Prize in Chemistry


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Magdalen college

As of October 1, for the fall term I am a visiting fellow at Magdalen College, Oxford. I’m quite pleased to be here, in particular because of the great crop of mathematicians that the college claims as their own (of these, the ones I know personally are Julien BerestyckiAlison Etheridge, Ben Green, and James Maynard; another who I hope to meet is Don Knuth), and because of the lovely accommodation they’ve offered me, overlooking Oxford’s botanical gardens. I thought I’d record some of my observations about the college while they’re still fresh. Continue reading

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Pinching a penny from the past

My grandmother, Margaret Berry (née Wynne), was born on November 4, 1915, in Bolton, Lancashire. She grew up in an orphanage and was working by the age of 12. One of the first places she worked was TM Heskeths and Sons, a cotton mill. When Margaret started working at the mill, the chairman was likely George Hesketh, one of the sons, who was previously the Mayor of Bolton, as well as the High Sheriff of Lancashire. George Hesketh’s cousin (quite possibly first cousin, though I haven’t managed to pin this down) was William Hesketh Lever, First Viscount Leverhulme, who also had previously been Mayor as well as High Sheriff. I doubt my grandmother cared much about all this, though, while living in an orphanage and working in a cotton mill. Continue reading

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Random infinite squarings of rectangles

Nicholas Leavitt and I have recently uploaded our paper, Random infinite squarings of rectangles, to arXiv. The paper relies on a beautiful result proved by four Cambridge undergraduates (Brooks, Smith, Stone and Tutte) in 1940, which connects squarings of rectangles with the theory of electrical networks. The history of the problem, and of their paper, is very nicely detailed at the Squaring the Square website.

Left: A random squaring of a rectangle with 6345 squares. Right: A magnification of a region within the squaring. The region on the right is contained within the red pixel on the left.

Left: A random squaring of a rectangle with 6345 squares. Right: A magnification of a region within the squaring. The region on the right is contained within the red pixel on the left.

The question Brooks, Smith, Stone and Tutte (hereafter BSST) set out to answer was this: can the unit square be exactly tiled by sub-squares, no two having equal area? Such a tiling is called a perfect squaring of a square. (More generally, a collection {(S_i,i \in I)} of closed squares in {\mathbb{R}^2} with pairwise disjoint interiors, whose union is a closed rectangle, gives a squaring of a rectangle; it is perfect if the squares of {(S_i,i \in I)} all have different sizes.) BSST showed that every finite squaring of a rectangle may be built by a construction which I now describe, and which is illustrated in Figure 2. Continue reading

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Growing random 3-connected maps, or Comment s’enfuir de l’Hexagone

I have just uploaded a paper to the arXiv which describes a growth procedure for random planar maps. There has been substantial recent interest in random planar maps which have the Brownian map as their conjectural scaling limit. For a number of classes of random maps, such convergence has now been rigorously established. Random quadrangulations were the first model for which convergence to the Brownian map was proved (by Miermont and, independently, Le Gall, and to date all other proofs of convergence to the Brownian map rely on the known convergence of random quadrangulations. The most recent ensemble for which this convergence was proved is that of random maps with a fixed number of edges; this latter result was made possible due to the discovery by Ambjorn and Budd of a beautiful new family of bijections between labelled quadrangulations and maps. The depths of the Ambjorn–Budd bijection are far from having been plumbed, but that is not the point of the current post.

Continue reading

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Brownian objects

What have I missed?

Posted in Math.PR, Mathematics, Non-technical | Tagged , , | 3 Comments