While in linear algebra linear equations are studied in many variables, in a classical introductory lecture of algebra mostly polynomial equations in one variable are considered. Algebraic geometry links these two topics by studying polynomial equations in many variables. In general, even any commutative rings are linked to the geometry. How does this linkage work? In the case of polynomial equations, their sets of zeros represent a geometric object.
Among other things, the research of algebraic geometry is concerned with classifying the obtained curves, surfaces and higher-dimensional varieties with respect to suitable invariants. It is particularly interesting to examine the algebraic properties of, for example, the set of all curves with a fixed genus (moduli space).
Even at school, you learn that you can not divide by zero.
But it is in the nature of mathematics to question the familiar:
So what happens when a dividend approaches to zero more and more?
In the most diverse situations, this leads to so-called singularities, that is, isolated places where objects have particular behavior or structure. Singularities of equation systems usually have strong effects on the (geometry of) corresponding solution sets. The theory of singularities makes use of a variety of methods, i.a. from topology, algebra and analysis.
Tropical geometry is a new and active area of mathematics that seeks to reduce complex algebraic or geometric problems to combinatorial ones. Many constructions in algebraic geometry have in this way a combinatorial equivalent in tropical geometry. If this new situation is then easier to understand, one can try to translate the tropical results back into algebraic geometry. In addition, tropical geometry has many relationships with other fields of research, such as algebraic statistics and even bioinformatics.
Groups describe the symmetries of mathematical or real objects; they occur in mathematics and the natural sciences. The nature of their occurrence is described by the mathematical concept of a representation. In order to understand these symmetries, it is therefore necessary to know the possible representations of the groups in question.
The research interests of AGAG in the field of group and representation theory include, among others, the modular representation theory of finite groups, local-global conjectures of Alperin, Brauer and McKay, linear algebraic groups, finite groups of Lie-type, and character theory.
Number theory is the investigation of number fields: Starting from a polynomial, we consider (symbolic) properties of the zeros. Number theory is a classic area of pure mathematics that today has far-reaching applications in cryptography and messaging: Both most modern encryption techniques and error-correcting codes are based in number theory.
Classical mathematics often deals with the existence of interesting objects - often without even having any beginnings to find these objects. For example, it is well known that there are infinitely many primes. On the other hand, checking this for a (very) large number is much more difficult. The AGAG develops constructive methods that support the theoretical answers. This also allows for "experimental" action to gather ideas for evidence.