31.01.2022 um 14:00 Uhr in 69/117
Lukas Kühne (Universität Bielefeld)
Matroids and Algebra
A matroid is a combinatorial object based on an abstraction of linear independence in vector spaces and forests in graphs.
I will discuss how matroid theory interacts with algebra via the so-called von Staudt constructions. These are combinatorial gadgets to encode polynomials in matroids.
A main application is concerned with generalized matroid representations over division rings, matrix rings and probability space representations together with their relation to group theory.
10.01.2022 um 14:00 Uhr in 69/117
Galen Dorpalen-Barry (Ruhr-Universität Bochum)
13.12.2021 um 14:00 Uhr in 69/117
Alessio D'Alí (Universität Osnabrück)
06.12.2021 um 14:00 Uhr in 69/117
Daniel Köhne (Universität Osnabrück)
29.11.2021 um 14:00 Uhr in 69/117
Robin Suxdorf (Leibniz-Universität Hannover)
Perfection of Rings and Schemes
A ring of characteristic p is called perfect if its Frobenius homomorphism is an isomorphism. Since non-perfect rings can cause many problems it is of interest to turn non-perfect rings into perfect rings. We construct the so-called perfect closure of a ring, which is the solution to a certain universal problem. Further, we study the properties of the perfect closure with a special interest on ideals.
Similarly, a scheme of characteristic p is called perfect if its absolute Frobenius morphism is an isomorphism and there is a functor, the inverse perfection functor, which turns non-perfect schemes into perfect schemes. We construct the inverse perfection by first extending the perfect closure functor to the category of sheaves and then applying this construction to schemes.
15.11.2021 um 14:00 Uhr in 69/117
Lorenzo Venturello (KTH Stockholm)
01.11.2021 um 14:15 Uhr in 93/E09
Mandala von Westenholz (Universität Osnabrück)
Covariance matrices of length power functionals of random geometric graphs -- an asymptotic analysis
Asymptotic properties of a vector of length power functionals of random geometric graphs, which arise as the 1-skeleton of considered random simplicial complexes, are investigated. More precisely, its asymptotic covariance matrix is studied as the intensity of the underlying homogeneous Poisson point process increases. This includes a consideration of matrix properties like rank, definiteness, determinant, eigenspaces or decompositions of interest. For the formulation of the results a case distinction is necessary. Indeed, in the three possible regimes the respective covariance matrix is of quite different nature which leads to different statements.