10.04.2018 um 16:15 Uhr in 69/125:
Arun Kumar (Universität Osnabrück)
We will construct the Grassmannian and the Quaternionic Grassmannian Varieties and see how are they are related to Algebraic and Hermitian K-theory respectively.
24.04.2018 um 16:15 Uhr in 69/125:
Holger Brenner (Universität Osnabrück)
Asymptotic properties of differential operators on a singularity
For a local algebra R over a field, we study the decomposition of the module of principal parts. A free summand of the nth module of principal parts is essentially the same as a differential operator E of order ≤ n with the property that the differential equation E(f) =1 has a solution. The asymptotic behavior of the seize of the free part gives a measure for the singularity represented by R. We compute this invariant for invariant rings, monoid rings, determinantal rings and compare it with the F-signature, which is an invariant in positive characteristic defined by looking at the asymptotic decomposition of the Frobenius. This is joint work with Jack Jeffries and Luis Nuñez Betancourt.
08.05.2018 um 16:15 Uhr in 69/125:
Lisa Seccia (University of Genova, Italy)
An Invitation to Stanley-Reisner Theory
The Stanley-Reisner theory provides a bridge between combinatorics/topology on one side and commutative algebra on the other. The central link between these two worlds is given by the Stanley-Reisner correspondence that allows us to translate algebraic information into combinatorial properties.
This talk is supposed to be an introduction to this theory: in the first part I will recall some basic notions of combinatorics and commutative algebra and explain the Stanley-Reisner correspondence; then I will state the Hochster’s formula and other important results in Stanley-Reisner theory; finally I will present some open problems related to this topic.
15.05.2018 um 16:15 Uhr in 69/125:
Robert Kunsch (Universität Osnabrück)
The Difficulty of Monte Carlo Approximation of Multivariate Monotone Functions
We aim to approximate d-variate monotone functions based on n function evaluations with the error measured in the L1-norm. Algorithms may be deterministic or randomized (Monte Carlo). In terms of the order of convergence of the error of optimal methods for fixed d and growing n, there is no difference between the deterministic and the Monte Carlo setting. Unfortunately, involved methods require n to be superexponentially large in d in order to achieve a given accuracy, which is rather unpractical. It can be shown that any deterministic method that achieves a prescribed accuracy possesses a cost which is at least exponential in d. This dependency on the dimension can be significantly reduces by Monte Carlo. Still, the problem is intractable, especially for small error tolerances best known methods are deterministic.
I will give geometric intuitions for all results.
29.05.2018 um 16:15 Uhr in 69/125:
Carina Betken (Universität Osnabrück)
Fluctuations in a general preferential attachment model via Stein's method
We look at the indegree of a uniformly chosen vertex in a preferential attachment random graph, where the probability that a newly arriving vertex connects to an older vertex is proportional to a sublinear function f of the indegree of the older vertex at that time. We provide rates of convergence for the total variation distance between this degree distribution and an asymptotic power-law distribution as the number of vertices tends to ∞.