B-Spline Discretization of Inextensible Curves
Tuesday, 4.2.25, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Two models for the numerical approximation of the elastic movement of inextensible curves are investigated. The configuration with the least possible elastic energy can be approached by employing a gradient flow of the corresponding energy functional. This gradient flow is then discretized using cubic spline functions. First, we examine a scheme that is based on functions that are once globally differentiable, and then we try to recreate that scheme using the twice globally differentiable B-splines. We show the convergence of the discretizations to the continuous problem and compare the performance of the two discretization schemes.\n
CoHAs of Weighted Projective Lines
Friday, 7.2.25, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
The Cohomological Hall Algebra (CoHA), introduced by Kontsevich--Soibelman, is a cohomological analogue of the Ringel--Hall algebra. While for a symmetric quiver this algebra is free (super) symmetric, there were essentially no other examples that were known explicitly in terms of generators and relations, until Franzen--Reineke computed the CoHA of (regular representations) of the 2-Kronecker quiver. The 2-Kronecker quiver is derived equivalent to the projective line and regular representations correspond to torsion sheaves; thus Franzen--Reineke's algebra can also be seen as the CoHA of torsion sheaves on \(\bmathbb{P}^1\). We extend this computation to CoHAs of torsion sheaves on the so-called weighted projective lines. As a special case, we also obtain the CoHAs of regular representations of all other extended Dynkin quivers (satisfying a certain natural condition on the Euler form).