FB 6 Mathematik/Informatik

Institut für Mathematik

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WS 2018/19

14.11.2018 um 17:15 Uhr in 69/125:

Prof. Dr. Franz-Viktor Kuhlmann (Universität Szczecin, Polen)

Resolution of Singularities and the Defect

In 1964 Hironaka proved resolution of singularities for algebraic varieties of arbitrary dimension over fields of characteristic 0. For this result, which has applications in many areas of pure and applied mathematics, he received the Fields Medal. In positive characteristic, resolution has only been proven for dimensions up to 3 by Abhyankar and recently by Cossart and Piltant. The general case has remained open although several working groups of algebraic geometers have attacked it. Following ideas of Zariski, one can also consider a local form of resolution, called local uniformization. Already in 1940 he proved it to hold for algebraic varieties of all dimensions in characteristic 0. But again, the case of positive characteristic has remained open. By its definition, local uniformization is a problem of valuation theoretical nature. In my talk, I will give a quick introduction to valuations and sketch the main idea of local uniformization. In positive characteristic, finite extensions of valued fields can show a nasty phenomenon, the defect. It has been identified as one of the main obstacles to local uniformization. I will present examples of defects and some results, as well as some main open problems. 

21.11.2018 um 17:15 Uhr in 69/125:

Prof. Dr. Rob Eggermont (University of Technology Eindhoven)

Finitely Generated Spaces in High-Dimensional Settings with Symmetry

In a high-dimensional space with the Zariski topology, it is generally difficult to find equations generating a
given subspace. If the space has additional symmetries, the problem becomes easier, and it is sometimes
possible to use equations describing a smaller space in a lower-dimensional setting. As an example, to describe
matrices of rank at most 1 it suffices to know that any 2 by 2 determinant vanishes. This does not depend on
the size of the matrix. In this talk, we give some examples of high-dimensional settings with symmetry, and
describe more formally what we mean by finitely generated spaces in these settings. We will also talk about
some recent results.