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Project
A study of the geometric structure of near-vector spaces and generalized total colorings of planar graphs (OZRSAB2)
We will study the geometry of near-vector spaces and generalized total colorings of planar graphs. For this we plan to collaborate with world experts in the field.
Near-vector spaces were introduced in [Andr74] as a generalization of vector spaces with the idea to use scalars from a near-field instead of from a (skew) field. These non-linear structures did not get much attention back then but recently came back into fashion, due to a new interest in non-linear structures and applications in for example cryptography. Much of the geometry behind these nearvector spaces is unknown and our goal is to analyze this aspect in detail. We will seek to compare near-vector spaces with the classical vector spaces and to understand the striking differences like, for example, single vectors which generate the whole space. We will also look at derived geometric structures like affine and projective spaces as well as substructures thereof.
The second topic of research is concerned with graph theory. Here we will work on conjectures and open problems concerning colorings of the vertex and edge sets of graphs such that the graphs induced by the vertices (resp. edges) of a certain color always belong to a certain prescribed family of graphs. More precisely we will be interested in the minimum number of colors needed to color the edges and vertices of a planar graph such that the induced graphs are always forests.
Near-vector spaces were introduced in [Andr74] as a generalization of vector spaces with the idea to use scalars from a near-field instead of from a (skew) field. These non-linear structures did not get much attention back then but recently came back into fashion, due to a new interest in non-linear structures and applications in for example cryptography. Much of the geometry behind these nearvector spaces is unknown and our goal is to analyze this aspect in detail. We will seek to compare near-vector spaces with the classical vector spaces and to understand the striking differences like, for example, single vectors which generate the whole space. We will also look at derived geometric structures like affine and projective spaces as well as substructures thereof.
The second topic of research is concerned with graph theory. Here we will work on conjectures and open problems concerning colorings of the vertex and edge sets of graphs such that the graphs induced by the vertices (resp. edges) of a certain color always belong to a certain prescribed family of graphs. More precisely we will be interested in the minimum number of colors needed to color the edges and vertices of a planar graph such that the induced graphs are always forests.
Date:1 Oct 2019 → 30 Sep 2020
Keywords:geometry
Disciplines:Geometry