4th April 2016 at 4:15 p.m. in room 69/125:
Mateusz Michalek (Freie Universität Berlin)
From topology to algebraic geometry and back again
Secant varieties are known to play an important role in complexity theory, representation theory and algebraic geometry, relating to ranks of tensors. In my talk I would like to present applications of secants in topology through k-regular embeddings. An embedding of a variety in an affine space is called k-regular if any k points are mapped to linearly independent points. Numeric conditions for the existence of such maps are an object of intensive studies of algebraic topologists dating back to the problem posed by Borsuk in the fifties. Recent new results were obtained by Pavle Blagojevic, Wolfgang Lueck and Guenter Ziegler. Our results relate k-regular maps to punctual versions of secant varieties. This allows us to prove existence of such maps in special cases. The main new ingredient is providing relations to the geometry of the punctual Hilbert scheme and its Gorenstein locus. The talk is based on two joint works: with Jarosław Buczynski, Tadeusz Januszkiewicz and Joachim Jelisiejew and with Christopher Miller: arXiv:1511.05707 and arXiv:1512.00609.
19th April 2016 at 4:15 p.m. in room 69/125:
Holger Brenner (Osnabrück University)
The Symmetric Signature
We prove that for invariant rings of a small finite group G the differential symmetric signature is 1/|G|. This is based on joint work with Alessio Caminata.
17th May 2016 at 4:15 p.m. in room 69/125:
Richard Sieg (Osnabrück University)
Representations of Hilbert Series in Normaliz
One of the major computation goals in Normaliz is the Hilbert Series of an affine monoid, that is, the generating function for counting elements in each degree. We will discuss the current calculation and representation of this series in Normaliz. Next we will present a new form which is compact and allows for a combinatorial interpretation. This form is attained by calculating the degrees of a homogeneous system of parameters for the monoid algebra and we will present and discuss an algorithm to compute them.