Laboratory of Geometry

1. prof. dr hab. Tomasz Brzeziński
2. dr Krzysztof Petelczyc
3. dr Mariusz Żynel

Scientific researches in the Department all concern projective geometry and its generalizations, with special emphasis on geometries induced by projective quadrics. Preferable languages to characterize the structures considered are the language of partial linear spaces (this one is suitable, primarily, to develop geometry of spaces of pencils - Grassmann spaces, quadratic Grassmann spaces, Veronese products, Segre products, spine spaces) and the language of weak chain spaces (convenient to develop general Laguerre geometry). Having so many different models at hand one can try to find generalizations, which may be of interest in foundations: weak theories of parallelism (affine partial linear spaces, quasi hyperbolic geometry), or weak theories of orthogonality for example. Other “foundational” question investigated in the Department concerns primitive notions which are sufficient to express considered geometries; in particular cases these are adjacencies, orthogonality (of subspaces), orthoadjacency, or plain incidence (on “less classical” universes). Within this framework two common mathematical problems: the problem of characterization and the problem of classification are investigated, applied to underlying geometrical spaces and their standard derivatives, such as automorphism groups, lattices of subspaces, or characteristic subconfigurations. To this area there belong also problems concerning embeddings. Some attention is also paid to “abstract” configurations on their own right, especially to those which can be represented as set-theoretical analogues of “classical” spaces, such as e.g. combinatorial Grassmannians, combinatorial Veronesians, multi-Veblen configurations.

• Katedra Algebry i Geometrii WMiI UWM, Centrum Nauczania Matematyki i Kształcenia na Odległość PG