FB 6 Mathematik/Informatik

Institut für Mathematik

# Hauptinhalt

## Wochenprogramm

### 29.11.2017 um 17:15 Uhr in Raum 69/125

#### Kaie Kubjas (Aalto University, Finland)

##### Geometry of Nonnegative and Positive Semidefinite Rank

One of many definitions gives the rank of an mxn matrix M as the smallest natural number r such that M can be factorized as AB, where A and B are mxr and rxn matrices respectively.
In many applications, we are interested in factorizations of a particular form. For example, factorizations with nonnegative entries define the nonnegative rank and are closely related to mixture models in statistics.
Another rank I will consider in my talk is the positive semidefinite (psd) rank. Both nonnegative and psd rank also appear in optimization and complexity theory.
Nonnegative and psd rank have geometric characterizations using nested polytopes.
I will explain how to use these characterizations to study the semialgebraic geometry of the set of matrices of given nonnegative or psd rank.

### 27.11.2017 um 14:15 Uhr in Raum 69/E23

#### Robert Kunsch (Universität Osnabrück)

##### Optimal sample size selection for Monte Carlo integration at a priori unknown dispersion

Abstract: We aim to compute the expected value of a random variable from i.i.d. samples. Under certain assumptions it is possible to select a sample size on ground of a variance estimation, or - more generally - an estimation on a central absolute \$p\$-moment, such that we can guarantee a small absolute error with high probability. The expected cost of the method depends on the dispersion of the random variable, namely the p-moment, which can be arbitrarily large. Lower bounds show that - up to constants - the cost of the algorithm is optimal in terms of accuracy, confidence level and dispersion of the particular input random variable. Problems of this type are uncommon for numerical analysis where the complexity of an integration problem is usually given by the minimal number of samples needed in order to guarantee a small error for the whole input class. This quantity would be infinite for the discussed setting.
Joint work with Erich Novak (Jena) and Daniel Rudolf (Göttingen)

### 22.11.2017 um 16:15 Uhr in 69/125:

#### Ilia Pirashvili (Universität Osnabrück)

##### From Toric Varieties to Topos Points - A brief Introduction to the Geometry of Monoids

In this talk, we will give a very short and non-formal introduction to toric varieties and then move on to monoid schemes, which enable us to generalise the former. It will end with the introduction of a new type of "geometry" that one can do with monoids, where prime ideals will be replaced with topos points. These two construcitons agree in the finitely generated case, but already in the simplest non-finitely generated case, there are significantly more topos points than prime ideals. This might be especially interesting for the multiplicative monoids of commutative rings.

### 24.11.2017 um 16:15 Uhr in 69/125:

#### Prof. Dr. Walter Ferrer (University of the Republic, Uruguay)

##### Observability: From Subgroups to Actions and Adjunctions

The exploration of the notion of observability exhibits transparently the rich interplay between
algebraic and geometric ideas in geometric invariant theory. We will talk about the concept of observable subgroup introduced in the 1960s with the purpose of studying extensions of representations from an affine algebraic subgroup to the whole group. It can be considered as an intermediate step in the notion of reductivity (semisimplicity) and it has been recently generalized to the concept of observable action of an affine algebraic group on an affine variety. We will also talk about the related concept of observable adjunction.

### 28.11.2017 um 16:15 Uhr in 69/125:

#### Timo de Wolff (TU Berlin)

##### Discrete Structures Related to Nonnegativity

Deciding nonnegativity of real polynomials is a fundamental problem in real algebraic geometry and polynomial optimization, which has countless applications. Since this problem is extremely hard, one usually restricts to sufficient conditions (certificates) for nonnegativity, which are easier to check. For example, since the 19th century the standard certificates for nonnegativity are sums of squares (SOS), which motivated Hilbert’s 17th problem. A maybe surprising fact is that both polynomial nonnegativity and nonnegativity certificates re closely related to different discrete structures such as polytopes and point configurations. In this talk, I will give an introduction to nonnegativity of real polynomial with a focus on the combinatorial point of view.

### 28.11.2017 um 14:15 Uhr in 69/E15:

#### Dr. Peter Arndt (Heinrich-Heine-Universität Düsseldorf)

##### From GAG to NAG: On Durov's and Haran's approaches to F1-geometry

I will give a quick introduction to Durov's "Generalized Algebraic Geometry" (GAG) and Haran's "Non-additive Geometry" (NAG), both of which are generalizations of algebraic geometry in which one glues more general objects than rings. I will present the compactifications of Spec Z that both authors construct. There is an inclusion from Durov's into Haran's F1-schemes, known to both authors. I will construct a left adjoint to this inclusion and compute it for Haran's compactification of Spec Z.