# Search result: Catalogue data in Spring Semester 2023

Mathematics Master | ||||||||||||||||||

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. Credits from other application areas cannot be recognised for further application areas. | ||||||||||||||||||

Systems Design | ||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||
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151-0530-00L | Nonlinear Dynamics and Chaos II | W | 4 credits | 4G | G. Haller | |||||||||||||

Abstract | The internal structure of chaos; Hamiltonian dynamical systems; Normally hyperbolic invariant manifolds; Geometric singular perturbation theory; Finite-time dynamical systems | |||||||||||||||||

Objective | The course introduces the student to advanced, comtemporary concepts of nonlinear dynamical systems analysis. | |||||||||||||||||

Content | I. 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 notes | Handwritten instructor's notes and typed lecture notes will be downloadable from Moodle. | |||||||||||||||||

Literature | Books will be recommended in class | |||||||||||||||||

Prerequisites / Notice | Nonlinear Dynamics I (151-0532-00) or equivalent | |||||||||||||||||

363-0588-00L | Complex Networks | W | 4 credits | 2V + 1U | G. Casiraghi | |||||||||||||

Abstract | The 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 robustness of networked systems, (v) the study of network evolution, and (vi) data mining techniques for networked data sets. | |||||||||||||||||

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 * learn about statistical inference and data mining techniques for data on networked systems * learn methods and abstractions used in the growing literature on complex networks | |||||||||||||||||

Content | Networks 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 diffusion? And how can we 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, biology or (online) social networks. In a network approach, agents in such systems (like e.g. humans, computers, documents, power plants, biological or financial entities) are represented as nodes, whereas their interactions are represented as links. The first part of the course, "Introduction to networks: basic and advanced metrics", describes how networks can be represented mathematically and how the properties of their link structures can be quantified empirically. In a second part "Stochastic Models of Complex Networks" we address how analytical statements about crucial properties like connectedness or robustness can be made based on simple macroscopic stochastic models without knowing the details of a topology. In the third 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 diffusion processes as well as the existence of community structures. A fourth part "Network Optimisation and Inference" introduces models for the emergence of complex topological features which are due to stochastic optimization processes, as well as statistical methods to detect patterns in large data sets on networks. In a fifth part, we address "Network Dynamics", introducing models for the emergence of complex features that are due to (i) feedback phenomena in simple network growth processes or (iii) order correlations in systems with highly dynamic links. A final part "Research Trends" introduces recent research on the application of data mining and machine learning techniques to relational data. | |||||||||||||||||

Lecture notes | The lecture slides are provided as handouts - including notes and literature sources - to registered students only. All material is to be found on Moodle. | |||||||||||||||||

Literature | See handouts. Specific literature is provided for download - for registered students, only. | |||||||||||||||||

Prerequisites / Notice | There are no pre-requisites for this course. Self-study tasks (to be solved analytically and by means of computer simulations) are provided as home work. 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. | |||||||||||||||||

Competencies |
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363-0543-00L | Agent-Based Modelling of Social Systems | W | 3 credits | 2V + 1U | G. Vaccario | |||||||||||||

Abstract | Agent-based modeling is introduced as a bottom-up approach to understand the complex dynamics of social systems. The course is based on formal models of agents and their interactions. Computer simulations using Python allow the quantitative analysis of a wide range of social phenomena, e.g. cooperation and competition, opinion dynamics, spatial interactions and behaviour in social networks. | |||||||||||||||||

Objective | A successful participant of this course is able to - understand the rationale of agent-based models of social systems - understand the relation between rules implemented at the individual level and the emerging behavior at the global level - learn to choose appropriate model classes to characterize different social systems - grasp the influence of agent heterogeneity on the model output - efficiently implement agent-based models using Python and visualize the output | |||||||||||||||||

Content | This full-featured course on agent-based modeling (ABM) allows participants with no prior expertise to understand concepts, methods and tools of ABM, to apply them in their master or doctoral thesis. We focus on a formal description of agents and their interactions, to allow for a suitable implementation in computer simulations. Given certain rules for the agents, we are interested to model their collective dynamics on the systemic level. Agent-based modeling is introduced as a bottom-up approach to understand the complex dynamics of social systems. Agents represent the basic constituents of such systems. The are described by internal states or degrees of freedom (opinions, strategies, etc.), the ability to perceive and change their environment, and the ability to interact with other agents. Their individual (microscopic) actions and interactions with other agents, result in macroscopic (collective, system) dynamics with emergent properties, which we want to understand and to analyze. The course is structured in three main parts. The first two parts introduce two main agent concepts - Boolean agents and Brownian agents, which differ in how the internal dynamics of agents is represented. Boolean agents are characterized by binary internal states, e.g. yes/no opinion, while Brownian agents can have a continuous spectrum of internal states, e.g. preferences and attitudes. The last part introduces models in which agents interact in physical space, e.g. migrate or move collectively. Throughout the course, we will discuss a wide variety of application areas, such as: - opinion dynamics and social influence, - cooperation and competition, - online social networks, - systemic risk - emotional influence and communication - swarming behavior - spatial competition While the lectures focus on the theoretical foundations of agent-based modeling, weekly exercise classes provide practical skills. Using the Python programming language, the participants implement agent-based models in guided and in self-chosen projects, which they present and jointly discuss. | |||||||||||||||||

Lecture notes | The lecture slides will be available on the Moodle platform, for registered students only. | |||||||||||||||||

Literature | See handouts. Specific literature is provided for download, for registered students only. | |||||||||||||||||

Prerequisites / Notice | Participants of the course should have some background in mathematics and an interest in formal modeling and in computer simulations, and should be motivated to learn about social systems from a quantitative perspective. Prior knowledge of Python is not necessary. Self-study tasks are provided as home work for small teams (2-4 members). Weekly exercises (45 min) are used to discuss the solutions and guide the students. The examination will account for 70% of the grade and will be conducted electronically. The "closed book" rule applies: no books, no summaries, no lecture materials. The exam questions and answers will be only in English. The use of a paper-based dictionary is permitted. The group project to be handed in at the beginning of July will count 30% to the final grade. |

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