Arrangements with symmetries

• Applicant: Professor Dr. Michael Cuntz
• Subject Area: Mathematics
• Term: from 2015 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 280581905

Arrangements of hyperplanes play a central role in many areas of mathematics. This proposal has two major goals: to understand the structure of the special simplicial arrangements of hyperplanes, including classifications, and to apply the results to construct a counterexample to the longstanding conjecture of Terao.The classification of a large subclass of the class of simplicial arrangements defined by a certain integrality condition, the so-called crystallographic arrangements, was achieved recently by Heckenberger and the applicant in a series of papers. This was a fundamental result for the theory of Nichols algebras, but meanwhile it is clear that it will have an even greater impact on discrete geometry. The first goal of this proposal is to use similar techniques to prove bounds for the larger class of finite reflection groupoids reducing their classification to a finite problem. An enumeration using further symmetries will provide a classification in dimension three which will be extended to arbitrary dimensions subsequently.The most recent breakthrough towards disproving Terao's conjecture is the counterexample by Hoge and the applicant to the closely related conjecture that free arrangements are recursively free in characteristic zero. The construction of this example is based on a simplicial arrangement. Using the results of the first part, we will produce a database of free arrangements which are not inductively free. The emerging patterns should expose infinite series and eventually allow to classify the set of potential counterexamples completely.Combining representation theory, computational enumerations, and traditional methods of geometry is an approach to these problems which has not been attempted before and is very promising.

• DFG Programme: Research Grants
• Cooperation Partner: Professor Dr. Gerhard Röhrle