In Algebra, one deals with the solutions of polynomial equations in one variable X, such as X2 - 2 = 0 or X5 + 4·X + 2 = 0. The set of all solutions of such polynomial equations with rational numbers as coefficients forms one field Qalg. This means that you can add, subtract, multiply, and divide the numbers in it. One of the main problems of Number Theory is to better understand the field Qalg. On the other hand, if polynomial equations (or systems of polynomial equations) are considered in several variables, their sets of solutions have a geometric structure (see the picture below). Also, e.g. the real solution set of X2 + Y2 - 1 = 0 is exactly the circle line. In addition, these sets of solutions also have a rich algebraic structure that results from considering polynomial equations. The branch of mathematics that investigates these sets of solutions is called Algebraic Geometry. A more detailed description can be found here.
"Zitrus" by Herwig Hauser is the real solution set of X2 + Z2 + Y3·(Y−1)3 = 0.
(Own work by User: Andreasmatt. Licensed under Creative Commons Attribution-Share Alike 3.0 via Wikimedia Commons https://commons.wikimedia.org/wiki/File:IMAGINARY_Zitrus_Herwig_Hauser.jpg)
Other algebraic areas that are the focus in Münster are Representation Theory, Group Theory and the Theory of Automorphic Forms. Of course, algebra is not an isolated individual science, but is closely related to other areas of mathematics (such as differential geometry, topology, logic, and operator algebras).