Technical University Berlin

This course explores advanced principles of computer networks based on fundamentals of the topic. The topics are protocol mechanisms, principles of implementation, network algorithms, advanced network architectures, network simulation, network measurement as well as techniques of protocol specification and verification. Protocols mechanisms and techniques of protocols used in network protocols include signaling, separation of control and data channel, soft state and hard state, using of randomization, indirection, multiplexing of resources, localization of services, and network virtualization (overlays, VxLANs, peer-to-peer networks). The identification and study of principles that lead to the implementation of network protocols include system principles, reflections on efficiency, and caveats/ case studies. Network architecture examines “the big picture”. It identifies and studies principles that lead the design of network architectures. The course considers substantial questions rather than specific protocol and implementation tricks, which include internet design principles, lessons learned from the internet, architecture of telephone network, and circuit switching versus packet switching (revisited). Protocols cover network algorithms, self stabilization (examples of routing), Kelly's congestion control framework, and closed loop control on the example of TCP. Simulation, oblivious routing and routing in cryptocurrency networks includes principles of discrete event simulation, analysis of simulation results, packet versus flow models, bounding strategies (e.g., Chernoff bounds), and Gaussian distributions.

This course begins with a study of the most classical objects in algebraic geometry: conics and plane curves. Students spend time examining these examples to develop a feeling for how algebraic equations and geometric shapes interact and prove an early version of Bezout's theorem. The central part of the course develops the theory of sheaves and schemes, which provide the natural framework in which to formulate and generalize classical results. The course introduces morphisms of schemes and their fundamental properties, and it studies divisors and line bundles as fundamental tools for encoding geometric information. Students examine the local structure of schemes, including objects such as differential forms. The class also introduces Čech cohomology, both as a computational method and as a bridge to more advanced cohomological techniques. The course concludes with the Riemann-Roch theorem.

This course explores mathematical concepts that are useful and frequently used in machine learning. Students examine linear algebra (vector spaces, scalar products, orthogonal vectors, matrices as linear mappings, determinants, eigenvalue and eigenvectors), analysis (differentiation), and probability theory (multidimensional probability distributions, calculations with expected values and variances). The class also discusses some contemporary applications of mathematics in machine learning. 

This course examines groups, which are best understood as symmetries of mathematical objects. Students explore geometric group theory and the connection between the algebraic properties of a group and the geometric properties of the spaces it acts on.

In this course and through the DAMS Lab group (FG Big Data Engineering), students learn how to conduct research in areas of data engineering, data management, and machine learning systems. Students review scientific literature in these areas as well as how to design, implement, and evaluate prototypes. The lab group offers this project on large-scale data engineering. The course includes tasks in a wide range of components of data management and machine learning systems. Students will have the opportunity to make meaningful contributions to free open-source projects.