What applications does abstract algebra and algebraic geometry have in computer science and programming? I love math and programming. Abstract Algebra and algebraic geometry seems very pleasant to me.
But I also would like to improve my skills as a programmer, but I would love to do so in the fiels that very related to 'real' math (combine my two passions).
We also have a very strong algebraic school where I live, so it would be great to learn from those professors. But, as I've mentioned already, I would love to apply this to elaborate programming. 
So what fiels can I chose? 
 A: Cryptography is an area with lots of abstract algebra. 
A little bit shows up in functional programming. (Get an impression here)
You might want to have a look at the GAP computer algebra project.
A: Haskell programming language (one of the most widely-used functional languages today) almost entirely builds up on category theoretic stuff.
Algebraic topology is used in high-dimensional data analysis (for example, persistent homology).
3D rendering is full of geometry and linear algebra. Probably abstract algebra or algebraic geometry can be usefull there, too.
A: The theory of Error-Correcting Codes addresses the problem of recovering the intended information when a digital message gets corrupted in transit. A good code is a collection of words made from some alphabet such that if the recipient of a string (of alphabet letters) discovers that the string is not a word (in which case it is a corruption of a word), they can identify the word it was intended to be with little or no ambiguity.
A Goppa Code is such a code which is constructed using algebraic geometry. Roughly, one fixes a curve over a finite field $\mathbb{F}_q$ together with a divisor on the curve and a set of $n$ rational points. The code is then the image (in $\mathbb{F}_q^n$) of the associated evaluation map on the Riemann-Roch space of the divisor.
