Search result: Catalogue data in Spring Semester 2014

Mathematics Master Information
Application Area
Only necessary and eligible for the Master degree in Applied Mathematics.
One of the application areas specified must be selected for the category Application Area for the Master degree in Applied Mathematics. At least 8 credits are required in the chosen application area.
Systems Design
NumberTitleTypeECTSHoursLecturers
151-0530-00LNonlinear Dynamics and Chaos II Information
Does not take place this semester.
W4 credits3GG. Haller
AbstractThe internal structure of chaos; Hamiltonian dynamical systems; Normally hyperbolic invariant manifolds; Geometric singular perturbation theory; Finite-time dynamical systems
ObjectiveThe course introduces the student to advanced, comtemporary concepts of nonlinear dynamical systems analysis.
ContentI. The internal structure of chaos: symbolic dynamics, Bernoulli shift map, sub-shifts of finite type; chaos is numerical iterations.

II.Hamiltonian dynamical systems: conservation and recurrence, stability of fixed points, integrable systems, invariant tori, Liouville-Arnold-Jost Theorem, KAM theory.

III. Normally hyperbolic invariant manifolds: Crash course on differentiable manifolds, existence, persistence, and smoothness, applications.
IV. Geometric singular perturbation theory: slow manifolds and their stability, physical examples. V. Finite-time dynamical system; detecting Invariant manifolds and coherent structures in finite-time flows
Lecture notesStudents have to prepare their own lecture notes
LiteratureBooks will be recommended in class
Prerequisites / NoticeNonlinear Dynamics I (151-0532-00) or equivalent
363-0588-00LComplex Networks Information W4 credits2V + 1UF. Schweitzer, D. Garcia Becerra, I. Scholtes
AbstractThe course provides an overview of the methods and abstractions used in (i) the quantitative study of complex networks, (ii) empirical network analysis, (iii) the study of dynamical processes in networked systems, (iv) the analysis of systemic risk in networked systems and (v) the study of network evolution.
Objective* the network approach to complex systems, where actors are represented as nodes and interactions are represented as links
* learn about structural properties of classes of networks
* learn about feedback mechanism in the formation of networks
* understand systemic risk as emergent property in networked systems
* learn methods and abstractions used in the growing literature on complex networks
ContentNetworks matter! This holds for social and economic systems, for technical infrastructures as well as for information systems. Increasingly, these networked systems are outside the control of a centralized authority but rather evolve in a distributed and self-organized way. How can we understand their evolution and what are the local processes that shape their global features? How does their topology influence dynamical processes like epidemic spreading, cascading failures or consensus? And how can you characterize the importance of specific nodes? This course provides a systematic answer to such questions, by developing methods and tools which can be applied to networks in diverse areas like infrastructure, communication, information systems or (online) social networks. In a network approach, agents in such systems (like e.g. humans, computers, documents, power plants or financial entities) are represented as nodes, whereas their interactions are represented as links.

The first part of the course, "Topology of Complex Networks", describes how networks can be represented mathematically and how the properties of their link structures can be quantified empirically. We further address how general statements about crucial properties like connectedness, robustness or efficiency can be made based on simple macroscopic stochastic models without knowing the details of a topology.

In the second part we address dynamical processes on complex networks. We show how a simple model for a random walk in networks can give insights into the authority of nodes, the efficiency of information diffusion processes as well as the existence of community structures. We further address the influence of the topology of complex networks on the spreading of epidemics and cascading failures as well as the emergence of synchronization and consensus.

In the third part "Network evolution" we introduce models for the emergence of complex topological features which are due to (i) stochastic optimization processes and heterogeneous node fitness, (ii) feedback phenomena in simple network growth processes or (iii) complex order correlations in systems with highly dynamic links.
Lecture notesThe lecture slides are provided as handouts - including notes and literature sources - to registered students only.
All material is to be found on Moodle at the following URL: Link
LiteratureSee handouts. Specific literature is provided for download - for registered students, only.
Prerequisites / NoticeThere are no pre-requisites for this course. Self-study tasks (to be solved analytically and by means of computer simulations) are provided as home. Weekly exercises (45 min) are used to discuss selected solutions. Active participation in the exercises is strongly suggested for a successful completion of the final exam.
363-0543-00LAgent-Based Modelling of Social Systems Information W3 credits2V + 1UF. Schweitzer, D. Garcia Becerra, N. Perony
AbstractAgent-based modelling is introduced as a bottom-up approach to understand the complex dynamics of social systems.
The course focuses on four different application areas, (I) opinion dynamics, (II) cooperation and competition, (III) spatial interaction, and (IV) online social networks. Emphasis is on formal modelling, quantitative analysis and computer simulation tools.
ObjectiveA successful participant of this course is able to
* understand the rationale of actor-centered models of social systems
* choose appropriate model classes to characterise social systems
* understand the relation between rules implemented at the individual level and the emerging behaviour at the global level
* grasp the influence of agent heterogeneity on the model output
* efficiently implement agent-based models using Python and visualise the output data
ContentAgent-based modelling provides a bottom-up approach to understand the complex dynamics of social systems. Agents have internal degrees of freedom (opinions, strategies), the ability to perceive, and to change, their environment, and to interact with other agents. Their (inter)actions result in collective dynamics with emergent properties that need to be analysed and understood quantitatively. As more, and more accurate, data about online and offline social systems become available, our formal understanding of these systems has to progress in the same manner. We focus on a parsimonious description of the agents' behaviour which relates individual interaction rules to the dynamics on the system's level and complements engineering and machine learning approaches to modelling.
The course focuses on four different application areas of agent-based models, (I) opinion dynamics, (II) cooperation and competition, (III) spatial interaction, and (IV) online social networks.
Whilst the lectures focus on the theoretical foundations of agent-based modelling, they are illustrated on a more practical level in weekly exercise classes. Using the Python programming language, the participants implement agent-based models in guided and autonomous projects, which they present and jointly discuss.
Lecture notesThe lecture slides will be available on the Moodle platform, for registered students only.
LiteratureSee handouts. Specific literature is provided for download, for registered students only.
Prerequisites / NoticeParticipants of the course should have some background in mathematics and a dedicated interest in formal modelling and computer simulations, and should be motivated to learn about social systems from a quantitative perspective.

Self-study tasks are provided as home work for small teams (3-5 members). Weekly exercises (45 min) are used to discuss the solutions, and guide the student. During the second half of the semester, teams have to complete a course project in which they will implement and discuss an agent-based model to characterise a system chosen jointly with the course organisers. This project will be evaluated, and its grade will count as 25% of the final grade.
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