IMath Webinar Series: Dr. Bernhard Klaassen and Jesus Paolo Joven
You’re all invited to attend the 4th seminar of our webinar series on Monday, May 8, 2023 at 2:00 PM. Our speakers are Dr. Bernhard Klaassen (from Fraunhofer Institute for Algorithms and Scientific Computing, Germany, hosted by the Discrete Geometry and Combinatorics Research Group) and Mr. Jesus Paolo Joven (from the Matrix Analysis and Linear Algebra Research Group).
Dr. Bernhard Klaassen (Fraunhofer Institute for Algorithms and Scientific Computing, Germany, hosted by the Discrete Geometry and Combinatorics Group)
Title: Recent attempts to solve the problem of aperiodic monotiles
Abstract: After Penrose’s famous discovery of aperiodic two-tile sets in the 1970s, it was an open problem until March 2023 whether an aperiodic monotile exists without any matching rule. In other words: Is there a shape (a topological disc) that can tile the plane without gaps or overlaps, but only in nonperiodic ways? A first important step towards this goal was made by Socolar and Taylor in 2010, however, as it holds also for most of the following attempts, some matching rules had to be obeyed to guarantee the nonperiodicity of any possible tiling. In the last three years, some more alternative approaches appeared – including a very new spectacular solution from 2023 – that will be discussed and compared in this talk.
Jesus Paolo Joven (Matrix Analysis and Linear Algebra Research Group)
Title: Generators of the Symplectic Group
Abstract: A \(2n\)-by-\(2n\) matrix \(S\) over a field \(\mathbb{F}\) is said to be symplectic if \(S^\top \Omega_{2n} S = \Omega_{2n}\), where \(\Omega_{2n}\) is defined by \[\Omega_{2n} = \left[ \begin{array}{cc} 0 & I_n \\ -I_n & 0 \end{array}\right] ,\] and \(I_n\) is the \(n\)-by-\(n\) identity matrix. The set of all symplectic matrices, which is a group under matrix multiplication, is called the symplectic group. We consider known generators and look for alternative generating sets of the symplectic group. In particular, we show that the real and complex symplectic groups are generated by symplectic skew-involutions (\(S^2=-I_{2n}\)).