Reduction techniques are pervasive in mathematics. For example, in
commutative algebra, one uses localisations at prime ideals to work over
local rings, as many important ring-theoretic or homological properties
are known to hold globally if and only if they hold locally. Another
strategy which is widely employed in algebra, geometry, and topology is
the decomposition of a category into smaller parts that are still big
enough to allow for reconstruction of the whole category. This leads to
the notion of a torsion pair and its variants, e.g. t-structures, or
recollements. In representation theory, one decomposes finite
dimensional modules into indecomposable summands; the indecomposable
modules over a given ring can often be classified and yield information
on the whole module category. Classification results in many areas of
modern mathematics typically provide complete lists of objects up to
some equivalence relation, that is, after performing some
identifications which simplify the structure but preserve the relevant
information. Also labellings in combinatorics, e.g. vertex or edge
colourings of graphs, or conservative extensions of logical theories
reduce complexity of data and structure enough as to facilitate insight,
proof and computation, while keeping the balance between useful
simplification and meaningful distinction.
Our project is aimed at theory and practice of such reduction
techniques, as ranging from axiomatic foundations including
computer-based formalisation, to mathematical applications. The topics
are cross- sectorial over three main areas of modern mathematics:
algebra (ring theory, representation theory, homological algebra);
mathematical logic (proof theory and constructive mathematics, model
theory); and combinatorics and graph theory (colourings, matchings and
flows). This allows us to exploit methodical synergies and to address
questions at the interface between the different sectors.