Martin's Blog

Rendering OpenStreetMap with Tilemill

Posted by Martin Orr on Tuesday, 20 May 2014 at 19:30

I have decided to play around with creating some maps using the freely available data from OpenStreetMap. I am mainly interested in creating maps of London bus routes but first I had to learn how to use the Tilemill rendering software and how to load OpenStreetMap data into Tilemill. In this post I will describe how I rendered my first map, a very simple line map of London.

My first map: Line map of London

no comments Tags osm, tilemill Read more...

Universal vector extensions of abelian varieties

Posted by Martin Orr on Friday, 09 May 2014 at 18:25

In the last post we showed that extensions of an abelian variety A by the additive group \mathbb{G}_a are classified by a vector space \operatorname{Ext}^1(A, \mathbb{G}_a) which is canonically isomorphic to T_0(A^\vee). In this post I will show that there is a so-called universal vector extension of A, that is, a vector extension E_0 of A such that every vector extension of A can be obtained in a unique way as a pushout of E_0. The vector group part of E_0 is \operatorname{Ext}^1(A, \mathbb{G}_a)^\vee, and an ingredient which we require from last time is that this is finite-dimensional.

Even if you are not interested in vector extensions of abelian varieties for their own sake (and I am not), they still have interesting properties. For example, I will show that they provide examples of algebraic varieties over the complex numbers which are non-isomorphic as algebraic varieties but which are isomorphic as complex manifolds.

2 comments Tags abelian-varieties, alg-geom, maths Read more...

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