The notion of homology arose over a century ago as a tool to distinguish geometric objects: e.g. a doughnut and a ball are certainly quite different geometric objects - the most obvious difference beeing the existence of a 'hole' in the former. Homology theory is a way to detect this hole - and conversely, also the absence of a hole - in a mathematically rigorous manner. Moreover, it yields a precise notion of 'dimension', allowing for example, the distinction of a line and a plane.
The key idea is to associate an algebraic structure to a geometric object in a 'natural' way, i.e. if we perform some 'admissable' geometric operation - such as stretching or shrinking - then the algebraic structure should stay the same. This translates our original geometric problem - to distinguish geometric objects - into an algebraic one, which allows for computations and is often easier to solve, e.g. in the case of the line and the plane, we end up with the different numbers 1 and 2. Homology theory was very successful: its offspring, the field of Homological algebra, permeates many areas of pure mathematics today.
We want to use the power of homological techniques: the aim of this research project is to build new relations between representation theory
and geometry - in particular, of singular spaces. The objects of our study are complicated structures, a deeper understanding of which will have many applications in these and related fields. Let us briefly explain these two areas in elementary terms.
1. Representation theory.
Symmetry is a central idea in mathematics, which often leads to simplifications of arguments and calculations. The collection of all transformations preserving the symmetry of a space, satisfies certain axioms turning it into a group. Conversely, given such a group, we can often elucidate its structure, by realising it as collection of symmetries on a space - such a realisation is called a representation of the group. Representation theory is the study of representations of groups and more general algebraic structures.
2. Geometry & Singularity theory.
Polynomials belong to the simplest mathematical objects. Although, the study of (common) zero sets of several polynomials dates back to antiquity, it remains challenging today. These vanishing sets are called 'varieties'. A typical point on a typical variety will be nice: it will locally resemble affine space, just like smooth curves locally look like lines from a topological viewpoint. Singularities are places where this nice correspondence breaks down. They are abundant in mathematics, physics and almost any field in which either mathematics or physics is applied.
3. How they are connected.
Given any variety (possibly with singularities), we can associate an object from homological algebra to it ('the derived category of coherent sheaves'). This object does not allow us to reconstruct the original structure completely, some information is lost in the transformation process. This, however, is a good thing: some of the information we lose is superfluous anyway and by reducing to more essential quantities, our life simplifies. Moreover, this allows us to see certain symmetries, that were hidden before, more clearly.
Now we look at objects from representation theory: given an algebra, we can perform the same process and study its derived category. Often this derived category coincides with that of the variety, revealing the existence of an underlying structure that both objects share. This coincidence and related constructions form a bridge between two different areas of mathematics, which can be exploited in both ways to increase our understanding of (singular) varieties as well as algebras.
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