FB 6 Mathematik/Informatik

Institut für Mathematik

# Hauptinhalt

## WS 2017/18

### 25.10.2017 um 17:15 Uhr in Raum 69/125

#### Elisa Gorla (Université de Neuchâtel)

##### Groebner Bases and their Applications to Cryptography

Groebner bases are a fundamental tool for computing with polynomial equations. In this talk I will introduce Groebner bases and discuss how they allow us to solve systems of polynomial equations. I will then discuss some applications of polynomial system solving to cryptography. Being able to efficiently solve a system of polynomial equations over a finite field allows us to crack certain cryptosystems. Therefore, one of the key questions in assessing the robustness of our cryptosystem is estimating how long it takes to compute the Groebner basis of a given system of polynomial equations.

### 01.11.2017 um 17:15 Uhr in Raum 69/125

#### Volker Betz (TU Darmstadt)

##### Geometry of Cycles in Spatial Random Permutations

Consider a square grid of N2 points, i.e. Λ = {0,1,...,N-1}2 . Let S be the set
of all permutations of π : Λ → Λ with the property that each x ∈ Λ is mapped
either to x or to a point y with |x − y| = 1, i.e. a neighboring point. An element
of S can be visualized as an ensemble of non-intersecting ’loops’, where we also
allow degenerate loops that only span two points.
The uniform distribution on S is surprisingly interesting and hard to under-
stand in the limit N → ∞. An exemplary question that we cannot answer is,
for example: let pN be the probability that the loop starting from the ’center’
of the box comes close (distance less than εN ) to the boundary. How does pN
behave as a function of N ? If we enforce such a loop to actually touch the
boundary, how will its shape typically look like?
In this talk, I will introduce several variants of the simple model above and
discuss what we know about the length and geometry of their loops. I will also
compare them to self-avoiding walks, for which a bit more is known rigorously,
and briefly indicate their significance for mathematical physics, which lies in
their relation to Bose-Einstein condensation.

### 08.11.2017 um 17:15 Uhr in Raum 69/125

#### Jan-Frederik Pietschmann (Universität Osnabrück)

##### On a Cross-Diffusion Model for Multiple Species with Nonlocal Interaction and Size Exclusion

We study a PDE model for two diffusing species interacting by local size exclusion and global attraction. This leads to a nonlinear degenerate cross-diffusion system, for which we provide a global existence result as well as additional regularity in the case of equal diffusivities. The analysis is motivated by the formulation of this system as a formal gradient flow for an appropriate energy functional consisting of entropic terms as well as quadratic nonlocal terms. Moreover, we investigate phase separation effects inherent in the cross-diffusion model by an analytical and numerical study of minimizers of the energy functional and their behaviour as the diffusivity tends to zero.

### 15.11.2017 um 17:15 Uhr in Raum 69/125

#### Sarah Scherotzke (WWU Münster)

##### The Chern Character and Categorification

The Chern character is a central construction with incarnations in algebraic topology, representation theory and algebraic geometry. It is an important tool to probe K-Theory, which is notoriously hard to compute. In my talk, I will explain, what the categorification of the Chern character is and how we can use it to show that certain classical constructions in algebraic geometry are of non-commutative origin. The category of motives plays the role of K-Theory in the categorified picture. The categorification leads also to the construction of higher invariants such as the secondary Chern characters and secondary K-Theory.

### 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.

### 06.12.2017 um 17:15 Uhr in Raum 69/125

#### Noam Berger (TU München)

##### G-Functions and G-Measures

Abstract: G-functions are the generalisation of Markov chains to the case where the chain has infinite memory. This notion was first introduced by Doblin and Fortet in 1938, and was further developed ever since. In the talk we will define and explain the notions of G-functions and G-measures, and discuss the issues      G-measure uniqueness and its relations to Ergodic Theory and Statistical Mechanics.

### 20.12.2017 um 17:15 Uhr in Raum 69/125

#### Marko Lindner (TU Hamburg-Harburg)

##### Finite Sections of the Fibonacci Hamiltonian

The Fibonacci Hamiltonian is an infinite matrix with three diagonals. It is the standard model for quasicrystals in 1D. The finite section method aims to approximate the inverse of an infinite matrix A by the inverses of growing finite submatrices along the main diagonal of A. We show that this works, with arbitrary cut-off points, for the Fibonacci Hamiltonian A on the axis as well as on the half axis. This is joint work with Hagen Söding (Hamburg).

### 10.01.2018 um 17:15 Uhr in Raum 69/125

#### Dominic Schuhmacher (Universität Göttingen)

##### Optimal Transport with Applications in Statistics

In recent years optimal transport and Wasserstein metrics have become both an object of intensive study in mathematics and an important tool in applied statistics and machine learning for analyzing complex data. From image analysis and retrieval, over biometric applications, to the modelling of chemical reactions and cosmological models of the universe: whenever the available data can be interpreted as a mass distribution on some space, optimal transport ideas provide a natural and often powerful approach for the analysis.

In the light of such data sets, computing optimal transports between hundreds of thousands of particles still presents challenging mathematical and algorithmic problems. In this talk I give a general introduction into the theory of optimal transport and present three modern computational methods able to deal with large data sets. We also look at a paradigmatic example from statistics.

### 17.01.2018 um 17:15 Uhr in Raum 69/125

#### Sabine Jansen (University of Sussex)

##### Singularity Analysis, Generating Functions and Heavy-Tailed Random Variables

Generating functions play an important role in both analytic combinatorics and probability. Often it is possible to transfer knowledge on the generating function, notably its behavior near the dominant singularity, to information on the asymptotics in counting problems or the estimation of asymptotic probabilities. The talk explains how these techniques can be extended to a class of discrete heavy-tailed random variables with log-convex probability weights. The novel complex-analytic method combines singularity analysis, Lindel{\"o}f integrals, and bivariate saddle points. As an application, we prove three theorems on precise large and moderate deviations which provide a local variant of a result by S.~V.~Nagaev (1973). The theorems generalize five theorems by A.~V.~Nagaev (1968) on the stretched exponential laws $p(k) = c\exp( -k^\alpha)$ and apply to logarithmic hazard functions $c\exp( - (\log k)^\beta)$, $\beta>2$; they cover the big jump domain as well as the small steps domain. The analytic proof is complemented by clear probabilistic heuristics. Critical sequences are determined with a non-convex variational problem. The talk is based on joint work with N. Ercolani and D. Ueltschi (arXiv:1509.05199 [math.PR]).

### 31.01.2018 um 17:15 Uhr in Raum 69/125

#### Andreas Büchter (Universität Duisburg-Essen)

##### Herausforderungen und Lösungsansätze

Die öffentliche Diskussion über das sichere mathematische Wissen und Können von Studienanfängerinnen und Studienanfängern in Studiengängen mit relevanten Mathematikanteilen ist in der jüngeren Vergangenheit intensiv und teilweise verbittert geführt worden. Dabei scheint klar zu sein, dass das Abschlussprofil der Schule und die Eingangsprofile dieser Studiengänge keine besonders gute Passung aufweisen. Im Vortrag werden Befunde zum Übergang von der Schule in die Hochschule und mögliche Unterstützungsmaßnahmen für (angehende) Studierende in der Studieneingangsphase vorgestellt und diskutiert.

### 07.02.2018 um 17:15 Uhr in Raum 69/125

#### Prof. Moritz Kerz (Universität Regensburg)

##### Algebraic K-Theory and Blow-Ups

Algebraic K-theory deals with questions related to the existence and uniqueness
of a basis of a module in a systematic and abstract way. It provides important arithmetic
information on algebraic and geometric objects, but is notoriously hard to calculate. In
this talk I will explain how to obtain a better understanding of negative K-groups through
a new geometric techniques.

### 14.02.2018 um 17:15 Uhr in Raum 69/125

#### Erich Novak (Universität Jena)

##### A Universal Algorithm for Multivariate Integration

We present an algorithm for multivariate integration over cubes
that is unbiased and has optimal order of convergence (in the randomized
sense as well as in the worst case setting) for
many different Sobolev spaces.