Posted by Martin Orr on Monday, 04 October 2010 at 12:37
In this post I will define the Mumford-Tate group of an abelian variety.
This is a 
Posted by Martin Orr on Friday, 24 September 2010 at 08:48
I spend most of my time thinking about the Hodge structures attached to abelian varieties, so I decided that I should explain what these Hodge structures are. A Hodge structure is a type of algebraic structure found on the (co)homology of complex projective varieties.
Here I will discuss only the special case of the first homology of abelian varieties. This is the simplest case, but is nonetheless very important. In particular, the Hodge structures on other homology and cohomology groups for abelian varieties can be calculated from that of the first homology. Also Hodge structures on the first (but not higher) cohomology of non-abelian varieties can be reduced to the case of abelian varieties by passing to something called the Albanese variety, generalising the Jacobian of curves.
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.
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.
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.