FB 6 Mathematik/Informatik/Physik

Institut für Mathematik

# Hauptinhalt

## SS 2022

### 12.04.2022 um 14:15 Uhr in 35/E25

#### Andreas Kretschmer vor (Otto von Guericke Universität Magdeburg)

##### When are symmetric ideals monomial?

In this talk I will formulate a conjecture about solution sets of general symmetric polynomial systems and prove special cases. For example, we will see that the only common zero to all cyclic permutations of a general homogeneous polynomial containing some power of a variable is the origin. Moreover, in characteristic zero, if a homogeneous polynomial has symmetric support and sufficiently general coefficients, then the radical of the ideal generated by its orbit is monomial. The proofs are geometric and in part make use of the representation theory of the symmetric groups.

### 26.04.2022 um 14:15 Uhr in 93/E06

#### Elima Shehu (Universität Osnabrück)

##### The Line multiview variety

Consider the problem of 3D reconstruction in Computer Vision. Given m images of a 3D scene, the reconstruction process consists of three steps: determining the camera's location, identifying points and lines on the m images that come from the same points and lines in 3D space, and then solving the triangulation problem, which produces a 3D model of the scene. In this talk, we will concentrate on the second part of reconstruction problem, where the object will be lines. I will give an algebraic understanding to line correspondences. Describe the smallest algebraic set containing the image lines coming from the same line in 3D, namely the line multiview variety, and discuss some of its geometric properties. In the end, I will give an illustration of its ideal using Macaulay2, in the case of 4 cameras. This is joint work with Paul Breiding, Felix Rydell and Angélica Torres.

### 03.05.2022 um 14:15 Uhr in 93/E06

#### Sam Fairchild (Universität Osnabrück)

##### Counting pairs of saddle connections

A translation surface is a collection of polygons in the plane with parallel sides identified by translation to form a Riemann surface with a singular Euclidean structure. A saddle connection is a special type of closed geodesic, and the set of saddle connections can be associated to a discrete subset of the complex plane. I will discuss recent work showing that for almost every translation surface the number of pairs of saddle connections with bounded virtual area has quadratic asymptotic growth. No previous knowledge of translation surfaces or counting problems will be assumed. This is based on joint work with Jayadev Athreya and Howard Masur.

### 10.05.2022 um 14:15 Uhr in 93/E06

#### Justus Bruckamp  (Universität Osnabrück)

##### Edge-Connected Subgraph Polytopes - Facet-Defining Inequalities

We study the 2-edge-connected subgraph polytope which is defined as follows: For a 2-edge-connected graph we consider all 2-edge-connected subgraphs and take the convex hull of their incidence vectors. We give several facet-defining inequalities for this polytope which can be used for solving the maximum weighted 2-edge-connected subgraph problem.

### 24.05.2022 um 14:15 Uhr in 93/E06

#### Marie-Charlotte Brandenburg (MPI Leipzig)

##### The positive tropicalization of low rank matrices

Given a (d x n)-matrix of fixed rank r, we can interpret the columns of the matrix as n points in d-dimensional space, lying on a common r-dimensional subspace. Similarly, given the tropicalization of this matrix, we obtain a configuration of points lying on a tropical linear space of dimension r. We consider such tropical point configurations, and introduce a combinatorial criterion ('triangle criterion') to characterize configurations which can be obtained from the tropicalization of matrices with positive entries. This is based on joint work in progress with Georg Loho and Rainer Sinn. No prior knowledge of tropical geometry will be assumed for this talk.

### 31.05.2022 um 14:15 Uhr in 35/E23-E24

#### Chiara Meroni (MPI Leipzig)

##### Combinatorics and semialgebraic geometry: intersection bodies

Intersection bodies are a popular construction in convex geometry. We will analyze some of their interesting features and then focus on the intersection bodies of polyotopes. They are always semialgebraic sets and are naturally related to hyperplane arrangements, which somehow describe their boundary structure. We will explore this connection via some examples and explain the ideas behind the main results. This is based on a joint work with Katalin Berlow, Marie-Charlotte Brandenburg and Isabelle Shankar.

### 07.06.2022 um 14:15 Uhr in 35/E23-E24

#### Tarek Emmrich (Universität Osnabrück)

##### Sparse signals on graphs and simplicial complexes

Signals with a sparse representation in a given basis as well as Laplacian eigenvectors of graphs play a big role in signal processing and machine learning. We put these topics together and look at signals on graphs that have a sparse representation in the basis of eigenvectors of the Laplacian matrix. We give explicit algorithms to recover those sums by sampling the signal only on few vertices, i.e. the number of required samples is independent on the total size of the graph and takes only local properties of the graph into account.

### 21.06.2022 um 14:15 Uhr in 35/E23-E24

#### Christian Ahring (Universität Osnabrück)

##### From stable vector bundles to stable modules

Stability of vector bundles over smooth projective curves was introduced by Mumford in the 1960’s and has been generalized to higher dimensional varieties by Gieseker, Takemoto and others. The notion of stability is essential, e.g., for the construction of moduli spaces of sheaves. We try to build a similar theory for filtered syzygy modules over a two-dimensional noetherian local ring and illustrate some difficulties that arise even in the simplest case, i.e. if the ring is regular.

### 28.06.2022 um 14:15 Uhr in 35/E23-E24

#### Daniel Otten  (Universität Osnabrück)

##### A brief introduction to the mathematical aspects of neural networks

Due to the rich application possibilities for neural networks, there is a wide range of literature primarily aimed at users. However, the mathematical aspects of learning with neural networks are often neglected. Many results of the research are presented on empirical grounds, although they can be explained mathematically as well.  The presentation is organized as follows: In the first section, the basic concepts are introduced to be able to describe learning with neural networks mathematically. This is followed by a consideration of overfitting and under fitting from the perspective of statistical learning theory. Thereby, the results are applied to neural networks.

### 14.07.2022 um 15:15 Uhr in 35/E23-E24

#### Serkan Hosten (San Francisco State University)

##### Logarithmic Voronoi Cells for Gaussian Models

We extend the theory of logarithmic Voronoi cells to Gaussian statistical models. In general, a logarithmic Voronoi cell at a point on a Gaussian model is a convex set contained in its log-normal spectrahedron. We show that for models of ML degree one and linear covariance models the two sets coincide. In particular, they are equal for both directed and undirected graphical models. We introduce decomposition theory of logarithmic Voronoi cells for the latter family. We also study covariance models, for which logarithmic Voronoi cells are, in general, strictly contained in log-normal spectrahedra. We give an explicit description of logarithmic Voronoi cells for the bivariate correlation model and show that they are semi-algebraic sets. Finally, we prove that boundaries of logarithmic Voronoi cells for unrestricted correlation models cannot be described by polynomials over the algebraic closure of the field of rational numers. (Joint work with Yulia Alexandr)

Abstract