# Search result: Catalogue data in Autumn Semester 2021

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

Electives For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Electives: Applied Mathematics and Further Application-Oriented Fields ¬ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Selection: Probability Theory, Statistics | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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401-3627-00L | High-Dimensional Statistics | W | 4 credits | 2V | P. L. Bühlmann | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | "High-Dimensional Statistics" deals with modern methods and theory for statistical inference when the number of unknown parameters is of much larger order than sample size. Statistical estimation and algorithms for complex models and aspects of multiple testing will be discussed. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Knowledge of methods and basic theory for high-dimensional statistical inference | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Lasso and Group Lasso for high-dimensional linear and generalized linear models; Additive models and many smooth univariate functions; Non-convex loss functions and l1-regularization; Stability selection, multiple testing and construction of p-values; Undirected graphical modeling | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | Peter Bühlmann and Sara van de Geer (2011). Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer Verlag. ISBN 978-3-642-20191-2. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Knowledge of basic concepts in probability theory, and intermediate knowledge of statistics (e.g. a course in linear models or computational statistics). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-4623-00L | Time Series AnalysisDoes not take place this semester. | W | 6 credits | 3G | F. Balabdaoui | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | The course offers an introduction into analyzing times series, that is observations which occur in time. The material will cover Stationary Models, ARMA processes, Spectral Analysis, Forecasting, Nonstationary Models, ARIMA Models and an introduction to GARCH models. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The goal of the course is to have a a good overview of the different types of time series and the approaches used in their statistical analysis. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | This course treats modeling and analysis of time series, that is random variables which change in time. As opposed to the i.i.d. framework, the main feature exibited by time series is the dependence between successive observations. The key topics which will be covered as: Stationarity Autocorrelation Trend estimation Elimination of seasonality Spectral analysis, spectral densities Forecasting ARMA, ARIMA, Introduction into GARCH models | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | The main reference for this course is the book "Introduction to Time Series and Forecasting", by P. J. Brockwell and R. A. Davis | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Basic knowledge in probability and statistics | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3612-00L | Stochastic SimulationDoes not take place this semester. | W | 5 credits | 3G | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | This course provides an introduction to statistical Monte Carlo methods. This includes applications of simulations in various fields (Bayesian statistics, statistical mechanics, operations research, financial mathematics), algorithms for the generation of random variables (accept-reject, importance sampling), estimating the precision, variance reduction, introduction to Markov chain Monte Carlo. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Stochastic simulation (also called Monte Carlo method) is the experimental analysis of a stochastic model by implementing it on a computer. Probabilities and expected values can be approximated by averaging simulated values, and the central limit theorem gives an estimate of the error of this approximation. The course shows examples of the many applications of stochastic simulation and explains different algorithms used for simulation. These algorithms are illustrated with the statistical software R. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Examples of simulations in different fields (computer science, statistics, statistical mechanics, operations research, financial mathematics). Generation of uniform random variables. Generation of random variables with arbitrary distributions (quantile transform, accept-reject, importance sampling), simulation of Gaussian processes and diffusions. The precision of simulations, methods for variance reduction. Introduction to Markov chains and Markov chain Monte Carlo (Metropolis-Hastings, Gibbs sampler, Hamiltonian Monte Carlo, reversible jump MCMC). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | A script will be available in English. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | P. Glasserman, Monte Carlo Methods in Financial Engineering. Springer 2004. B. D. Ripley. Stochastic Simulation. Wiley, 1987. Ch. Robert, G. Casella. Monte Carlo Statistical Methods. Springer 2004 (2nd edition). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Familiarity with basic concepts of probability theory (random variables, joint and conditional distributions, laws of large numbers and central limit theorem) will be assumed. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Selection: Financial and Insurance Mathematics In the Master's programmes in Mathematics resp. Applied Mathematics 401-3913-01L Mathematical Foundations for Finance is eligible as an elective course, but only if 401-3888-00L Introduction to Mathematical Finance isn't recognised for credits (neither in the Bachelor's nor in the Master's programme). For the category assignment take contact with the Study Administration Office (Link) after having received the credits. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3925-00L | Non-Life Insurance: Mathematics and Statistics | W | 8 credits | 4V + 1U | M. V. Wüthrich | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | The lecture aims at providing a basis in non-life insurance mathematics which forms a core subject of actuarial science. It discusses collective risk modeling, individual claim size modeling, approximations for compound distributions, ruin theory, premium calculation principles, tariffication with generalized linear models and neural networks, credibility theory, claims reserving and solvency. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The student is familiar with the basics in non-life insurance mathematics and statistics. This includes the basic mathematical models for insurance liability modeling, pricing concepts, stochastic claims reserving models and ruin and solvency considerations. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | The following topics are treated: Collective Risk Modeling Individual Claim Size Modeling Approximations for Compound Distributions Ruin Theory in Discrete Time Premium Calculation Principles Tariffication Generalized Linear Models and Neural Networks Bayesian Models and Credibility Theory Claims Reserving Solvency Considerations | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | M.V. Wüthrich, Non-Life Insurance: Mathematics & Statistics Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | M.V. Wüthrich, M. Merz. Statistical Foundations of Actuarial Learning and its Applications Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | The exams ONLY take place during the official ETH examination period. This course will be held in English and counts towards the diploma of "Aktuar SAV". For the latter, see details under Link. Prerequisites: knowledge of probability theory, statistics and applied stochastic processes. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Competencies |
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401-3922-00L | Life Insurance Mathematics | W | 4 credits | 2V | M. Koller | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | The classical life insurance model is presented together with the important insurance types (insurance on one and two lives, term and endowment insurance and disability). Besides that the most important terms such as mathematical reserves are introduced and calculated. The profit and loss account and the balance sheet of a life insurance company is explained and illustrated. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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401-3928-00L | Reinsurance Analytics | W | 4 credits | 2V | P. Antal, P. Arbenz | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | This course provides an introduction to reinsurance from an actuarial perspective. The objective is to understand the fundamentals of risk transfer through reinsurance and models for extreme events such as natural or man-made catastrophes. The lecture covers reinsurance contracts, Experience and Exposure pricing, natural catastrophe modelling, solvency regulation, and insurance linked securities | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | This course provides an introduction to reinsurance from an actuarial perspective. The objective is to understand the fundamentals of risk transfer through reinsurance and the mathematical approaches associated with low frequency high severity events such as natural or man-made catastrophes. Topics covered include: - Reinsurance Contracts and Markets: Different forms of reinsurance, their mathematical representation, history of reinsurance, and lines of business. - Experience Pricing: Modelling of low frequency high severity losses based on historical data, and analytical tools to describe and understand these models - Exposure Pricing: Loss modelling based on exposure or risk profile information, for both property and casualty risks - Natural Catastrophe Modelling: History, relevance, structure, and analytical tools used to model natural catastrophes in an insurance context - Solvency Regulation: Regulatory capital requirements in relation to risks, effects of reinsurance thereon, and differences between the Swiss Solvency Test and Solvency 2 - Insurance linked securities: Alternative risk transfer techniques such as catastrophe bonds | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | This course provides an introduction to reinsurance from an actuarial perspective. The objective is to understand the fundamentals of risk transfer through reinsurance and the mathematical approaches associated with low frequency high severity events such as natural or man-made catastrophes. Topics covered include: - Reinsurance Contracts and Markets: Different forms of reinsurance, their mathematical representation, history of reinsurance, and lines of business. - Experience Pricing: Modelling of low frequency high severity losses based on historical data, and analytical tools to describe and understand these models - Exposure Pricing: Loss modelling based on exposure or risk profile information, for both property and casualty risks - Natural Catastrophe Modelling: History, relevance, structure, and analytical tools used to model natural catastrophes in an insurance context - Solvency Regulation: Regulatory capital requirements in relation to risks, effects of reinsurance thereon, and differences between the Swiss Solvency Test and Solvency 2 - Insurance linked securities: Alternative risk transfer techniques such as catastrophe bonds | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Slides and lecture notes will be made available. An excerpt of last year's lecture notes is available here: Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Basic knowledge in statistics, probability theory, and actuarial techniques | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Competencies |
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401-3927-00L | Mathematical Modelling in Life Insurance | W | 4 credits | 2V | T. J. Peter | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | In life insurance, it is essential to have adequate mortality tables, be it for reserving or pricing purposes. The course provides the tools necessary to create mortality tables from scratch. Additionally, we study various guarantees embedded in life insurance products and learn to price them with the help of stochastic models. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The course's objective is to provide the students with the understanding and the tools to create mortality tables on their own. Additionally, students should learn to price embedded options in life insurance. Aside of the mere application of specific models, they should develop an intuition for the various drivers of the value of these options. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Following main topics are covered: 1. Guarantees and options embedded in life insurance products. - Stochastic valuation of participating contracts - Stochastic valuation of Unit Linked contracts 2. Mortality Tables: - Determining raw mortality rates - Smoothing techniques: Whittaker-Henderson, smoothing splines,... - Trends in mortality rates - Stochastic mortality model due to Lee and Carter - Neural Network extension of the Lee-Carter model - Integration of safety margins | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Lectures notes and slides will be provided | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | The exams ONLY take place during the official ETH examination period. The course counts towards the diploma of "Aktuar SAV". Good knowledge in probability theory and stochastic processes is assumed. Some knowledge in financial mathematics is useful. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Selection: Mathematical Physics, Theoretical Physics | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

402-0843-00L | Quantum Field Theory ISpecial Students UZH must book the module PHY551 directly at UZH. | W | 10 credits | 4V + 2U | G. M. Graf | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | This course discusses the quantisation of fields in order to introduce a coherent formalism for the combination of quantum mechanics and special relativity. Topics include: - Relativistic quantum mechanics - Quantisation of bosonic and fermionic fields - Interactions in perturbation theory - Scattering processes and decays - Elementary processes in QED - Radiative corrections | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The goal of this course is to provide a solid introduction to the formalism, the techniques, and important physical applications of quantum field theory. Furthermore it prepares students for the advanced course in quantum field theory (Quantum Field Theory II), and for work on research projects in theoretical physics, particle physics, and condensed-matter physics. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Will be provided as the course progresses | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Competencies |
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402-0861-00L | Statistical Physics | W | 10 credits | 4V + 2U | M. Sigrist | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | This lecture covers the concepts of classical and quantum statistical physics. Several techniques such as second quantization formalism for fermions, bosons, photons and phonons as well as mean field theory and self-consistent field approximation. These are used to discuss phase transitions, critical phenomena and superfluidity. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | This lecture gives an introduction in the basic concepts and applications of statistical physics for the general use in physics and, in particular, as a preparation for the theoretical solid state physics education. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Kinetic approach to statistical physics: H-theorem, detailed balance and equilibirium conditions. Classical statistical physics: microcanonical ensembles, canonical ensembles and grandcanonical ensembles, applications to simple systems. Quantum statistical physics: density matrix, ensembles, Fermi gas, Bose gas (Bose-Einstein condensation), photons and phonons. Identical quantum particles: many body wave functions, second quantization formalism, equation of motion, correlation functions, selected applications, e.g. Bose-Einstein condensate and coherent state, phonons in elastic media and melting. One-dimensional interacting systems. Phase transitions: mean field approach to Ising model, Gaussian transformation, Ginzburg-Landau theory (Ginzburg criterion), self-consistent field approach, critical phenomena, Peierls' arguments on long-range order. Superfluidity: Quantum liquid Helium: Bogolyubov theory and collective excitations, Gross-Pitaevskii equations, Berezinskii-Kosterlitz-Thouless transition. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Lecture notes available in English. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | No specific book is used for the course. Relevant literature will be given in the course. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

402-0830-00L | General Relativity Special Students UZH must book the module PHY511 directly at UZH. | W | 10 credits | 4V + 2U | C. Anastasiou | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Introduction to the theory of general relativity. The course puts a strong focus on the mathematical foundations of the theory as well as the underlying physical principles and concepts. It covers selected applications, such as the Schwarzschild solution and gravitational waves. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Basic understanding of general relativity, its mathematical foundations (in particular the relevant aspects of differential geometry), and some of the phenomena it predicts (with a focus on black holes). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Introduction to the theory of general relativity. The course puts a strong focus on the mathematical foundations, such as differentiable manifolds, the Riemannian and Lorentzian metric, connections, and curvature. It discusses the underlying physical principles, e.g., the equivalence principle, and concepts, such as curved spacetime and the energy-momentum tensor. The course covers some basic applications and special cases, including the Newtonian limit, post-Newtonian expansions, the Schwarzschild solution, light deflection, and gravitational waves. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | Suggested textbooks: C. Misner, K, Thorne and J. Wheeler: Gravitation S. Carroll - Spacetime and Geometry: An Introduction to General Relativity R. Wald - General Relativity S. Weinberg - Gravitation and Cosmology | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

402-0897-00L | Introduction to String Theory | W | 6 credits | 2V + 1U | J. Brödel | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | String theory is an attempt to quantise gravity and unite it with the other fundamental forces of nature. It is related to numerous interesting topics and questions in quantum field theory. In this course, an introduction to the basics of string theory is provided. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Within this course, a basic understanding and overview of the concepts and notions employed in string theory shall be given. More advanced topics will be touched upon towards the end of the course briefly in order to foster further research. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | - mechanics of point particles and extended objects - string modes and their quantisation; higher dimensions, supersymmetry - D-branes, T-duality - supergravity as a low-energy effective theory, strings on curved backgrounds - two-dimensional field theories (classical/quantum, conformal/non-conformal) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | D. Lust, S. Theisen, Lectures on String Theory, Lecture Notes in Physics, Springer (1989). M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory I, CUP (1987). B. Zwiebach, A First Course in String Theory, CUP (2004). J. Polchinski, String Theory I & II, CUP (1998). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Recommended: Quantum Field Theory I (in parallel) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Selection: Mathematical Optimization, Discrete Mathematics | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3055-64L | Algebraic Methods in Combinatorics | W | 6 credits | 2V + 1U | B. Sudakov | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. This course provides a gentle introduction to Algebraic methods, illustrated by examples and focusing on basic ideas and connections to other areas. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The students will get an overview of various algebraic methods for solving combinatorial problems. We expect them to understand the proof techniques and to use them autonomously on related problems. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern theory has grown out of this early stage and often relies on deep, well-developed tools. One of the main general techniques that played a crucial role in the development of Combinatorics was the application of algebraic methods. The most fruitful such tool is the dimension argument. Roughly speaking, the method can be described as follows. In order to bound the cardinality of of a discrete structure A one maps its elements to vectors in a linear space, and shows that the set A is mapped to linearly independent vectors. It then follows that the cardinality of A is bounded by the dimension of the corresponding linear space. This simple idea is surprisingly powerful and has many famous applications. This course provides a gentle introduction to Algebraic methods, illustrated by examples and focusing on basic ideas and connections to other areas. The topics covered in the class will include (but are not limited to): Basic dimension arguments, Spaces of polynomials and tensor product methods, Eigenvalues of graphs and their application, the Combinatorial Nullstellensatz and the Chevalley-Warning theorem. Applications such as: Solution of Kakeya problem in finite fields, counterexample to Borsuk's conjecture, chromatic number of the unit distance graph of Euclidean space, explicit constructions of Ramsey graphs and many others. The course website can be found at Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Lectures will be on the blackboard only, but there will be a set of typeset lecture notes which follow the class closely. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Students are expected to have a mathematical background and should be able to write rigorous proofs. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Auswahl: Theoretical Computer Science, Discrete Mathematics | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

263-4500-00L | Advanced Algorithms Takes place for the last time. | W | 9 credits | 3V + 2U + 3A | M. Ghaffari, G. Zuzic | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | This is a graduate-level course on algorithm design (and analysis). It covers a range of topics and techniques in approximation algorithms, sketching and streaming algorithms, and online algorithms. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | This course familiarizes the students with some of the main tools and techniques in modern subareas of algorithm design. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | The lectures will cover a range of topics, tentatively including the following: graph sparsifications while preserving cuts or distances, various approximation algorithms techniques and concepts, metric embeddings and probabilistic tree embeddings, online algorithms, multiplicative weight updates, streaming algorithms, sketching algorithms, and derandomization. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | This course is designed for masters and doctoral students and it especially targets those interested in theoretical computer science, but it should also be accessible to last-year bachelor students. Sufficient comfort with both (A) Algorithm Design & Analysis and (B) Probability & Concentrations. E.g., having passed the course Algorithms, Probability, and Computing (APC) is highly recommended, though not required formally. If you are not sure whether you're ready for this class or not, please consult the instructor. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

252-1425-00L | Geometry: Combinatorics and Algorithms | W | 8 credits | 3V + 2U + 2A | B. Gärtner, E. Welzl, M. Hoffmann, M. Wettstein | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Geometric structures are useful in many areas, and there is a need to understand their structural properties, and to work with them algorithmically. The lecture addresses theoretical foundations concerning geometric structures. Central objects of interest are triangulations. We study combinatorial (Does a certain object exist?) and algorithmic questions (Can we find a certain object efficiently?) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The goal is to make students familiar with fundamental concepts, techniques and results in combinatorial and computational geometry, so as to enable them to model, analyze, and solve theoretical and practical problems in the area and in various application domains. In particular, we want to prepare students for conducting independent research, for instance, within the scope of a thesis project. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Planar and geometric graphs, embeddings and their representation (Whitney's Theorem, canonical orderings, DCEL), polygon triangulations and the art gallery theorem, convexity in R^d, planar convex hull algorithms (Jarvis Wrap, Graham Scan, Chan's Algorithm), point set triangulations, Delaunay triangulations (Lawson flips, lifting map, randomized incremental construction), Voronoi diagrams, the Crossing Lemma and incidence bounds, line arrangements (duality, Zone Theorem, ham-sandwich cuts), 3-SUM hardness, counting planar triangulations. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | yes | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | Mark de Berg, Marc van Kreveld, Mark Overmars, Otfried Cheong, Computational Geometry: Algorithms and Applications, Springer, 3rd ed., 2008. Satyan Devadoss, Joseph O'Rourke, Discrete and Computational Geometry, Princeton University Press, 2011. Stefan Felsner, Geometric Graphs and Arrangements: Some Chapters from Combinatorial Geometry, Teubner, 2004. Jiri Matousek, Lectures on Discrete Geometry, Springer, 2002. Takao Nishizeki, Md. Saidur Rahman, Planar Graph Drawing, World Scientific, 2004. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Prerequisites: The course assumes basic knowledge of discrete mathematics and algorithms, as supplied in the first semesters of Bachelor Studies at ETH. Outlook: In the following spring semester there is a seminar "Geometry: Combinatorics and Algorithms" that builds on this course. There are ample possibilities for Semester-, Bachelor- and Master Thesis projects in the area. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

252-0417-00L | Randomized Algorithms and Probabilistic Methods | W | 10 credits | 3V + 2U + 4A | A. Steger | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Las Vegas & Monte Carlo algorithms; inequalities of Markov, Chebyshev, Chernoff; negative correlation; Markov chains: convergence, rapidly mixing; generating functions; Examples include: min cut, median, balls and bins, routing in hypercubes, 3SAT, card shuffling, random walks | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | After this course students will know fundamental techniques from probabilistic combinatorics for designing randomized algorithms and will be able to apply them to solve typical problems in these areas. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Randomized Algorithms are algorithms that "flip coins" to take certain decisions. This concept extends the classical model of deterministic algorithms and has become very popular and useful within the last twenty years. In many cases, randomized algorithms are faster, simpler or just more elegant than deterministic ones. In the course, we will discuss basic principles and techniques and derive from them a number of randomized methods for problems in different areas. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Yes. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | - Randomized Algorithms, Rajeev Motwani and Prabhakar Raghavan, Cambridge University Press (1995) - Probability and Computing, Michael Mitzenmacher and Eli Upfal, Cambridge University Press (2005) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Selection: Further Realms | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-4944-20L | Mathematics of Data Science | W | 8 credits | 4G | A. Bandeira | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Mostly self-contained, but fast-paced, introductory masters level course on various theoretical aspects of algorithms that aim to extract information from data. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Introduction to various mathematical aspects of Data Science. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | These topics lie in overlaps of (Applied) Mathematics with: Computer Science, Electrical Engineering, Statistics, and/or Operations Research. Each lecture will feature a couple of Mathematical Open Problem(s) related to Data Science. The main mathematical tools used will be Probability and Linear Algebra, and a basic familiarity with these subjects is required. There will also be some (although knowledge of these tools is not assumed) Graph Theory, Representation Theory, Applied Harmonic Analysis, among others. The topics treated will include Dimension reduction, Manifold learning, Sparse recovery, Random Matrices, Approximation Algorithms, Community detection in graphs, and several others. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | The main mathematical tools used will be Probability, Linear Algebra (and real analysis), and a working knowledge of these subjects is required. In addition to these prerequisites, this class requires a certain degree of mathematical maturity--including abstract thinking and the ability to understand and write proofs. We encourage students who are interested in mathematical data science to take both this course and ``227-0434-10L Mathematics of Information'' taught by Prof. H. Bölcskei. The two courses are designed to be complementary. A. Bandeira and H. Bölcskei | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

227-0423-00L | Neural Network Theory | W | 4 credits | 2V + 1U | H. Bölcskei | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | The class focuses on fundamental mathematical aspects of neural networks with an emphasis on deep networks: Universal approximation theorems, capacity of separating surfaces, generalization, fundamental limits of deep neural network learning, VC dimension. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | After attending this lecture, participating in the exercise sessions, and working on the homework problem sets, students will have acquired a working knowledge of the mathematical foundations of neural networks. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | 1. Universal approximation with single- and multi-layer networks 2. Introduction to approximation theory: Fundamental limits on compressibility of signal classes, Kolmogorov epsilon-entropy of signal classes, non-linear approximation theory 3. Fundamental limits of deep neural network learning 4. Geometry of decision surfaces 5. Separating capacity of nonlinear decision surfaces 6. Vapnik-Chervonenkis (VC) dimension 7. VC dimension of neural networks 8. Generalization error in neural network learning | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Detailed lecture notes are available on the course web page Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | This course is aimed at students with a strong mathematical background in general, and in linear algebra, analysis, and probability theory in particular. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3502-71L | Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies. | W | 2 credits | 4A | Supervisors | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3503-71L | Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies. | W | 3 credits | 6A | Supervisors | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3504-71L | Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies. | W | 4 credits | 9A | Supervisors | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective |

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