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## SS 2018

### 10.04.2018 um 16:15 Uhr in 69/125:

#### Arun Kumar (Universität Osnabrück)

##### Quaternionic Grassmannians

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 *n*th 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 *L _{1}*-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 **∞**.