Posted by Martin Orr on
Monday, 10 May 2010 at 21:41
Singular points in a curve are places where curve fails to be smooth: intuitively, multiple points of the curve pile up on top of each other. In this post I will describe a simple invariant of curve singularities, the multiplicity, which essentially counts how many points are piled up there. In the simplest case of an ordinary multiple point, I describe how to use the previous post's algorithm to compute a power series for each branch of the curve near the singularity.
Tags
alg-geom, maths
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Posted by Martin Orr on
Friday, 30 April 2010 at 20:29
I had a week at home in Belfast last week. Since I resolved a couple of years ago not to fly for routine journeys, it is quite a long journey between Belfast and Paris. In the past I have always done this with a night's stop in England, taking most of two days. This time I broke my journey Paris to Belfast in Leeds for the Future Sounds of Swing weekend. This was almost the only lindy classes I have gone to since last June. It was a great weekend.
On my way back from Belfast to Paris, I took the sleeper train from Edinburgh to London. This allowed me to get from Belfast to Paris in 24 hours (with a bit more time required for travel at either end). It takes about 7 hours leaving at 11.40. The train during the day, which makes several stops, takes less than 5 hours. I suppose it goes slower to give a smoother ride and so that you can get a decent night's sleep. In fact you can stay in your cabin for another hour after it arrives. It cost me £35 with a rail card.
The room is very small with not much more than a pair of bunk beds and a sink. When I saw it I worried whether there would be space for my suitcase, but there was a shelf for it. I was sharing with one other man. It was very comfortable and the movement was barely noticeable. You get brought tea or coffee and a biscuit when the train arrives (in a disposable cup, I suppose so you can take it away if you want).
When I got back to Bures, as soon as I looked across the valley I could see that something had changed. The woods which cover all the upper part of the valley side had gone from brown to green. The leaves have come out on all the deciduous trees on the campus, which was only just starting when I left. This makes everything greener but less open.
Tags
lindy, paris, train, travel
Posted by Martin Orr on
Monday, 05 April 2010 at 22:34
An algebraic function is a function which we obtain by solving a polynomial in two variables 
I shall give an algorithm for expressing an algebraic function (in such a nicely behaved region) as a power series, thereby proving that a power series solution to the original polynomial exists. The generalisation of this result to a complete discrete valuation ring is Hensel's lemma, and is particularly important to number theorists in the case of p-adic integers (which were invented by Hensel). In this post I will focus on the case of algebraic functions, as it is easier to apply geometric intuition.
Tags
alg-geom, maths, number-theory
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Posted by Martin Orr on
Saturday, 20 March 2010 at 22:45
In the past six weeks I have taken four exams: two on the courses from the first semester (October-January) and two on the courses I have taken January-February.
The reason the latter came so soon after the former is because they are in different institutions
- the first two were here in Orsay, the second two in other institutions in Paris with different timetables.
These exams were hard, and very different from exams in Cambridge.
(Note: this post discusses pure maths exams only.)
Tags
exams, m2, paris, tripos
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Posted by Martin Orr on
Wednesday, 03 March 2010 at 10:08
I am finally ready to finish my series on algebraic tori, by talking about their representations.
I shall show that these representations can be classified by a grading on the vector space of the representation, after extending scalars to the separable closure.
I will describe this classification explicitly in a simple case.
Tags
alg-geom, alg-groups, maths, representations, tori
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