Julian Hollender

Graduate Lectures in Mathematics


This series of lectures aims at Master's and PhD students in mathematics and offers a first glimpse into topics which are not routinely taught in our MSc/PhD programme. The emphasis is to introduce new concepts and techniques, and not to present full mathematical details.


Stochastic Stability of Markov Models (Dr. Nikola Sandric)

The class of Markov models is one of the central objects in the theory of stochastic processes. These processes have strong connection with other areas of mathematics (e.g. harmonic analysis, PDEs, graph theory, geometry, etc.) and they are also used to model many phenomena appearing in nature and engineering (e.g. the classical model of enzyme activity, population processes, automatic speech recognition systems, information processing, etc.). In my three lectures I aim to provide an overview of the theory of Markov models with an emphasis on their (stochastic) stability properties (transience, recurrence and ergodicity).

Venue: Thursday, 13:00 - 14:30 in Willersbau A124
Dates: 09.07.2015, 16.07.2015, 23.07.2015
Diffuse Interface Models for Two-Phase Flow (Dr. Sebastian Aland)

Multi-phase flow systems can be found almost everywhere around us, from aerospace engineering down to the cells of our body. Mathematical modeling is an essential tool to predict how such systems behave. In this course we will derive the governing equations for two-phase flow by energy variation arguments. We will then focus on diffuse-interface (phase field) models to discuss various aspects of contact line dynamics, surface tension and discretization.

Venue: Thursday, 13:00 - 14:30 in Willersbau A124
Dates: 18.06.2015, 25.06.2015, 02.07.2015
Documents: Slides, Notes
Stochastic Geometry - Introduction, Overview, Applications (Dr. Lutz Muche)

Examples describing geometric probabilities are given to introduce the mathematical foundation of Stochastic Geometry. The Poisson process, some further frequently used point processes, and the Boolean model as important basics are discussed. Characteristics of random geometrical structures (nearest neighbour distribution function, contact and chord length distribution functions, K-function, pair correlation function) are given. An overview about random tessellations is made, especially Voronoi and Delaunay tessellations and their properties. Applications in natural sciences and engineering are discussed. As an important special case some results for distributional properties of the Poisson Voronoi and Poisson Delaunay tessellation are presented.

Venue: Thursday, 13:00 - 14:30 in Willersbau A124
Dates: 21.05.2015, 04.06.2015, 11.06.2015
Documents: Notes
Introduction to Jump Processes (Dr. Paolo Di Tella)

Jump processes, in particular semimartingales, play a fundamental role in stochastic analysis. Aim of this lectures is to introduce graduate and PhD students into the topic. To make the understanding easier, we concentrate on LÚvy processes (i.e. processes with homogeneous and independent increments), a special case of semimartingales. We consider the jump measure of a LÚvy process and define the stochastic integral relatively to it. Then we establish the canonical representation for semimartingales which are LÚvy processes, that is the It˘-LÚvy decomposition.

Venue: Thursday, 13:00 - 14:30 in Willersbau A124
Dates: 23.04.2015, 30.04.2015, 07.05.2015
Crash Course on K-Theory (Prof. Guillermo Corti˝as)

The aim of this series of 3 talks is to give an overview of basic aspects of algebraic and topological K-theory, and how they are related to each other and to other areas of mathematics. Topics that are going to be covered include middle algebraic and topological K-theory $K_0$ and $K_1$, higher and lower K-theory and the excision problem, as well as a comparison between K-theories.

Venue: Thursday, 13:00 - 14:30 in Willersbau A120
Dates: 19.02.2015, 26.02.2015, 05.03.2015
Levy Distributions in Atomic or Molecular Gases (Dr. Alexander Eisfeld)

First a brief introduction of heavy tailed stable Levy distributions is given. Then we show how such distributions can emerge for the energy and absorption spectrum of atoms/molecules in a cold gas. If time permits, finally the effect of these heavy tailed distributions on collective optical properties of the gas is discussed. No prior knowledge of quantum theory and atomic physics will be assumed.

Venue: Friday, 13:00 - 14:30 in Willersbau C203
Dates: 23.01.2015, 30.01.2015, 06.02.2015
Synchronization: Embryonic Development to Electronic Networks (Dr. Lucas Wetzel, David J. J÷rg, Alexandros Pollakis)

Both in biology and in engineering, synchronization of many autonomously oscillating parts is vital to support proper function of the whole; think of cardiac pacemaker cells, circadian rhythms, multi-core systems, or large antenna arrays. In this lecture series, we introduce mathematical models of coupled oscillators whose applicability ranges from biological to electronic systems.

Venue: Friday, 13:00 - 14:30 in Willersbau C203
Dates: 19.12.2014, 09.01.2015, 16.01.2015
Kinematics of Noisy Motion (Dr. Benjamin M. Friedrich)

This will be a hands-on course on the stochastic differential geometry of space curves, linking abstract mathematics and concrete examples. As a warm-up, we will discuss translational and rotational diffusion in the plane. Coupling translations and rotations, we will naturally encounter multiplicative noise, allowing us to discuss the subtleties of Ito versus Stratonovich calculus. We will then see how active translations and rotations can be conveniently described as a path on a Lie group called SE(3). No prior knowledge of Lie groups needed.

Venue: Friday, 13:00 - 14:30 in Willersbau C203 & Thursday, 15:30 - 17:00 in Willersbau A124
Dates: 28.11.2014, 04.12.2014, 12.12.2014
Documents: Notes
Indeterminate Moment Problems and Entire Functions (Prof. Dr. Christian Berg)

The indeterminate moment problem deals with probability distributions such as the log-normal distribution, which are not uniquely determined by their moments. A fundamental problem is to determine all other probability distributions having the same moments as the given distribution. Important tools for that are the corresponding orthonormal polynomials from which one can define a family of entire functions having common order, called the order of the moment problem. The orthonormal polynomials can be found from a recursion equation, and we will discuss how one can determine the order of the moment problem from the moments or from the recursion equation.

Venue: Tuesday, 13:00 - 14:30 in Willersbau A124 & Friday, 13:00 - 14:30 in Willersbau C203
Dates: 16.10.2014, 17.10.2014, 23.10.2014, 24.10.2014, 30.10.2014
Differentiable Ergodic Theory (Jun.-Prof. Dr. Kathrin Padberg-Gehle)

A dynamical system can be thought of as a rule for a time evolution \(T\) on a state space \(X\). The theory of dynamical systems is divided into several subfields such as differentiable dynamics, topological dynamics or ergodic theory, depending on the structures of \(T\) and \(X\). Ergodic theory considers the action of a given dynamical system on suitable measures and thus provides a powerful toolbox for the analysis and description of the dynamics. This short course introduces ergodic theoretical concepts for differentiable dynamical systems. In particular, we will deal with invariant measures, Lyapunov exponents and the corresponding ergodic theorems.

Venue: Tuesday, 16:40 - 18:10 in Willersbau C133 & Friday, 13:00 - 14:30 in Willersbau A124
Dates: 24.06.2014, 01.07.2014, 04.07.2014
Documents: Handout
The Periodic Table for Topological Insulators and Superconductors (Dr. Jens Bardarson)

Quantum mechanics is perhaps the most successful theory that physicists have ever conceived of. The quantum theory of solids, in particular, predicts and explains the difference between metals and insulators in terms of their so-called band structure. Surprisingly, it was only in the last few years that it was fully realized that not all insulators are the same. Rather, there are distinct classes of insulators that differ in the topology of their "electronic band structure". This has fundamental consequences: topological insulators are bulk insulators that have a stable and robust metallic surface. Similarly, topological superconductors realize exotic edge states that could potentially be used in future quantum computers. In these set of lectures we discuss some of the mathematical structures needed to classify all possible topological insulators and superconductors and thereby construct their "periodic table". The focus will be on the use of the concepts rather than attempting to formally and rigorously define them. No prior knowledge of quantum theory will be assumed and the few concepts needed are explained as they come along. The main mathematical constructs needed originate in group theory and, not surprisingly, topology.

Venue: Friday, 13:00 - 14:30 in Willersbau A124 & Tuesday, 16:40 - 18:10 in Willersbau C133
Dates: 06.06.2014, 17.06.2014, 20.06.2014
Documents: Notes
New Algorithms for the Discrete Logarithm Problem in Finite Fields of Small Characteristic (Dr. Jens Zumbrńgel)

The Discrete Logarithm Problem (DLP) has been a long-studied problem in number theory. Its importance has raised in particular since the introduction of Public-Key Cryptography and the Diffie-Hellman protocol in 1976, which relies on the hardness of the DLP. After very little algorithmic progress for more than 20 years, the DLP in finite fields of small characteristic underwent a dramatic development since 2013. At the heart of the new methods is the usage of higher splitting probabilities of certain polynomials over finite fields. These improved algorithms have a strong impact, e.g., on the security assumptions of some well known identity-based cryptosystems. This lecture series provides an overview of the history and the recent developments of DLP algorithms. We will discuss the mathematics behind the new algorithms as well as implementational aspects that lead to some recent significant record computations.

Venue: Tuesday, 16:40 - 18:10 in Willersbau C133 & Friday, 13:00 - 14:30 in Willersbau A124
Dates: 20.05.2014, 23.05.2014, 27.05.2014
Documents: Notes
Classical Infinitely Divisible Laws on \(\mathbb{R}^+\) (Prof. Dr. Thomas Simon)

Infinitely divisible laws on the set of positive real numbers are characterized by their Laplace transform. On the one hand, it can be difficult to guess whether an explicit density function with less explicit Laplace transform is infinitely divisible or not. On the other hand, investigating basic visual features of the density function of a positive infinitely divisible law with given Laplace transform might also turn out to be a difficult question. We will present several results and open problems on these two kinds of problems. Our presentation will be partly zoological, but not only and some of the methods we will display share a universal aspect.

Venue: Friday, 13:00 - 14:30 in Willersbau A124 & Tuesday, 16:40 - 18:10 in Willersbau C133
Dates: 11.04.2014, 15.04.2014, 22.04.2014, 29.04.2014
Introduction to the Theory of Anomalous Transport (Prof. Dr. Aleksei Chechkin)

Anomalous transport refers to non-equilibrium random processes that can not be described by using standard methods of statistical physics. This novel class of transport phenomena has recently been observed in a wide variety of complex systems such as amorphous semiconductors, fusion plasma, glassy and nano-materials, biological cells and epidemic spreading. The coherent description of anomalous transport in such a broad range of systems poses a fundamental challenge to a modern statistical physics of non-equilibrium state. This lecture series aims to provide an introductory overview of some basic concepts and tools of anomalous transport theory, such as continuous time random walk model, LÚvy flights, LÚvy walks, generalized stochastic Langevin equation, fractional motions and fractional kinetic equations. The presentation will be given at a 'physical' level of accuracy, with particular examples from different areas of science.

Venue: Thursday, 11:10 - 12:40 in Willersbau C203
Dates: 16.01.2014, 23.01.2014, 30.01.2014
Introduction to the Modern Theory of Functional Calculus for Closed Operators (Dr. Mihßly Kovßcs)

In several fields of mathematics (for example, numerical analysis, fractional calculus, probaility theory, maximal regularity problems for partial differential equations etc.) functional calculus techniques have been used rather successfully. At the heart of the theory is the problem of defining possibly unbounded functions of a closed operator on a Banach space in a consistent way. This enables one to manipulate ordinary functions, which is intuitive and simple, instead of operators. This short course gives a brief introduction to the modern theory which is based on an abstract, rather algebraic approach to functional calculus extensions. We will discuss the extension of the Hille-Phillips functional calculus for semigroup generators in detail. Some examples from the theory of fractional powers of operators and numerical analysis will also be discussed to show the power of the theory.

Venue: Friday, 13:00 - 14:30 in Willersbau A221
Dates: 24.01.2014, 31.01.2014, 07.02.2014
Notes: Additional Talk on 23.01.2014, 15:10-16:40, Willersbau A124
Documents: Handout
Faber-Krahn Inequalities for some Linear Problems and its Consequences (Prof. Dr. Marcello Lucia)

The classical isoperimetric inequality in the Euclidean space states that among all open sets of given volume, the ball minimizes the perimeter. This nice geometrical inequality allows to derive a lower bound on the first eigenvalue of the (negative) Laplacian with zero boundary condition, the so-called 'Faber-Krahn' inequality. Though such a lower bound fails for Neumann boundary condition, we will see that it holds for the first eigenvalue of the Laplacian acting on some other spaces of functions. Application to a class of nonlinear problem will be given, and if times permit a discussion of similar problems on the sphere will be given.

Venue: Friday, 13:00 - 14:30 in Willersbau A221
Dates: 29.11.2013, 06.12.2013, 13.12.2013
Ordinal Factor Analysis (Dr. Cynthia-Vera Glodeanu)

Factor Analysis is a commonly used complexity reduction technique for metric data. In the lectures, we present a factor analytical approach for qualitative data that can be represented by a formal context from Formal Concept Analysis. This approach is mainly developed here in Dresden, and it turns out that its expressiveness is similar to that of Factor Analysis based on metric data. We illustrate the technique with several examples taken from real world problems in the fields of psychology, medicine, social networks, etc.

Venue: Friday, 13:00 - 14:30 in Willersbau A221
Dates: 25.10.2013, 01.11.2013, 08.11.2013
Notes: Additional Talk on 14.11.2013, 11:10-12:40, Willersbau C203
Documents: Slides
Dualities for Algebraic and Topological Structures (Dr. Sebastian Kerkhoff)

According to the Encyclopaedia of mathematics and the Princeton companion to mathematics, Duality theory is "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics". The lecture series aims to provide an introductory overview of duality theory and focuses on those dualities that are constructed between algebraic and topological structures. These dualities have proven themselves to be particularly useful, with many applications in algebra, functional analysis, geometry and topology.

Dates: 25.06.2013, 02.07.2013, 09.07.2013
Documents: Notes
Gradient Systems and their Applications (Samuel Littig)

Many ODEs and PDEs appearing from models in science can reasonably be formulated as gradient systems. We will introduce the general framework and show how the notion can be applied to the Parabolic \(p\)-Laplace Problem, the Porous Medium Equation and the Fast Diffusion Equation to obtain existence, uniqueness and properties of the solutions (energy estimates, order preservation, asymptotic behaviour) in very elegant ways.

Dates: 04.06.2013, 11.06.2013, 18.06.2013
Copulas (Sebastian Fuchs)

Copulas are multivariate distribution functions on the unit hypercube whose one-dimensional marginals are uniformly distributed. We will introduce copulas from an analytic point of view and discuss their properties and applications.

Dates: 07.05.2013, 14.05.2013, 28.05.2013
Documents: Notes
Introduction to KAM Theory (Gabriel Fuhrmann)

Let \(M\) be an analytic manifold. Two endomorphisms \(f\colon M\to M\), \(\ g\colon M\to M\) are said to be (topologically) conjugate if there exists a homeomorphism \(h\colon M\to M\) such that \(h\circ f=g\circ h\). Conjugated maps share a lot of properties. Hence, it is a natural question to ask whether nearby endomorphisms are conjugated. KAM theory provides tools to give an affirmative answer to this question and moreover, yields conjugacies \(h\) of high regularity (e.g., analytic).We study the proof of Arnold's Theorem, which is one of the easiest KAM-proofs and still exhibits all the typical characteristics of a KAM proof.

Dates: 16.04.2013, 23.04.2013, 30.04.2013
Documents: Notes
Interacting Particle Systems (Dr. Anja Vo▀-B÷hme)

Interacting particle systems (IPSs) are stochastic models for the temporal evolution of spatially explicit systems of locally interacting entities (particles, cells, agents, etc). They prove to be important in many fields of application such as biological, physical and social systems. The lectures will cover the construction of IPSs as Markov processes with a particular state space, important techniques for the analysis of their long-time behavior, numerical and simulation methods for IPSs as well as applications to biological systems of interacting cell population.

Dates: 11.01.2013, 18.01.2013, 25.01.2013
Documents: Notes
Right-Continuous Functions with Left Limits (Dr. Bj÷rn B÷ttcher)

Right continuous functions with left limits, also known as cÓdlÓg functions, play a central role in the theory of stochastic processes. We will look at them from an analytic point of view and discuss their properties, related metrics and applications.

Dates: 30.11.2012, 07.12.2012, 14.12.2012
Documents: Handout
Viscosity Solutions of 2nd Order PDEs (Julian Hollender)

Scalar fully nonlinear partial differential equations of second order emerge in many applications, e.g. Hamilton-Jacobi-Bellman equations ins stochastic control theory and mathematical finance, Isaac's equation in stochastic game theory, or Monge-AmpŔre equations in differential geometry. We will introduce the concept of viscosity solutions and present clever ideas to prove uniqueness and existence of solutions for those problems.

Dates: 09.11.2012, 16.11.2012, 23.11.2012
Documents: Notes
Malliavin Calculus for Jump Processes (Dr. Felix Lindner)

Malliavin calculus or the stochastic calculus of variations is a differential calculus on the probability space underlying a given stochastic process. It allows one to make sense of such things as the derivative of a random variable and leads to deep results concerning, e.g., the existence and smoothness of density functions. This lecture series aims to provide an introductory overview of this fascinating theory, with an emphasis on jump processes.

Dates: 19.10.2012, 26.10.2012, 02.11.2012
Documents: Handout

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Author: Julian Hollender