31.01.2022 um 14:00 Uhr Meetingroom
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.
17.01.2022 um 14:00 Uhr Meetingroom
Galen Dorpalen-Barry (Ruhr-Universität Bochum)
Varchenko-Gel’fand Ring of a Cone of a Hyperplane Arrangement
The coefficients of the characteristic polynomial of an arrangement in a real vector space have many interpretations. An interesting one is provided by the Varchenko-Gel’fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise multiplication. Varchenko and Gel’fand gave a simple presentation for this ring, along with a filtration whose associated graded ring has its Hilbert function given by the coefficients of the characteristic polynomial. In this talk, I will generalize Gel'fand and Varchenko's results to cones defined by intersections of halfspaces of some of the hyperplanes.
13.12.2021 um 14:00 Uhr in 69/117
Alessio D'Alí (Universität Osnabrück)
Symmetric edge polytopes and their gamma-vectors
Symmetric edge polytopes are lattice polytopes depending on the data of a finite simple graph. Such objects have been studied both for their intrinsic combinatorial properties and their connections to other areas of mathematics (like finite metric space theory) and physics: in particular, they arise naturally in the study of the Kuramoto synchronization model, which describes how oscillators can influence each other. It has been recently conjectured by Ohsugi and Tsuchiya that every symmetric edge polytope has a nonnegative gamma-vector. We present some evidence supporting this claim, both from a deterministic point of view (proving that the second coefficient of the gamma-vector is always nonnegative) and from a probabilistic one (studying what happens when we use the Erdős-Rényi random model to generate the graph). This is joint work with D. Köhne, M. Juhnke-Kubitzke and L. Venturello.
06.12.2021 um 14:00 Uhr in 69/117
Daniel Köhne (Universität Osnabrück)
Laplacian Polytopes of Oriented Simplicial Complexes
The Laplacian simplex of a finite simple graph is the convex hull of the columns of its Laplacian matrix. Since graphs can be seen as 1-dimensional simplicial complexes, we extend this approach to finite oriented simplicial complexes. Focusing on the boundary of (d+1)-dimensional simplices, we will investigate serveral properties for the associated Laplacian polytopes as dimension, simpliciality and facet description. Moreover we show in this case that they admit a regular unimodular triangulation and study the h^*-vector.
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.