# Search result: Catalogue data in Autumn Semester 2018

Mathematics Bachelor | ||||||

Electives | ||||||

Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |
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401-3113-68L | Exponential Sums over Finite Fields | W | 8 credits | 4G | E. Kowalski | |

Abstract | Exponential sums over finite fields arise in many problems of number theory. We will discuss the elementary aspects of the theory (centered on the Riemann Hypothesis for curves, following Stepanov's method) and survey the formalism arising from Deligne's general form of the Riemann Hypothesis over finite fields. We will then discuss various applications, especially in analytic number theory. | |||||

Objective | The goal is to understand both the basic results on exponential sums in one variable, and the general formalism of Deligne and Katz that underlies estimates for much more general types of exponential sums, including the "trace functions" over finite fields. | |||||

Content | Examples of elementary exponential sums The Riemann Hypothesis for curves and its applications Definition of trace functions over finite fields The formalism of the Riemann Hypothesis of Deligne Selected applications | |||||

Lecture notes | Lectures notes from various sources will be provided | |||||

Literature | Kowalski, "Exponential sums over finite fields, I: elementary methods: Iwaniec-Kowalski, "Analytic number theory", chapter 11 Fouvry, Kowalski and Michel, "Trace functions over finite fields and their applications" | |||||

401-3100-68L | Introduction to Analytic Number Theory | W | 8 credits | 4G | I. N. Petrow | |

Abstract | This course is an introduction to classical multiplicative analytic number theory. The main object of study is the distribution of the prime numbers in the integers. We will study arithmetic functions and learn the basic tools for manipulating and calculating their averages. We will make use of generating series and tools from complex analysis. | |||||

Objective | The main goal for the course is to prove the prime number theorem in arithmetic progressions: If gcd(a,q)=1, then the number of primes p = a mod q with p<x is approximately (1/phi(q))*(x/log x), as x tends to infinity, where phi(q) is the Euler totient function. | |||||

Content | Developing the necessary techniques and theory to prove the prime number theorem in arithmetic progressions will lead us to the study of prime numbers by Chebyshev's method, to study techniques for summing arithmetic functions by Dirichlet series, multiplicative functions, L-series, characters of a finite abelian group, theory of integral functions, and a detailed study of the Riemann zeta function and Dirichlet's L-functions. | |||||

Lecture notes | Lecture notes will be provided for the course. | |||||

Literature | Multiplicative Number Theory by Harold Davenport Multiplicative Number Theory I. Classical Theory by Hugh L. Montgomery and Robert C. Vaughan Analytic Number Theory by Henryk Iwaniec and Emmanuel Kowalski | |||||

Prerequisites / Notice | Complex analysis Group theory Linear algebra Familiarity with the Fourier transform and Fourier series preferable but not required. | |||||

401-3059-00L | Combinatorics IIDoes not take place this semester. | W | 4 credits | 2G | N. Hungerbühler | |

Abstract | The course Combinatorics I and II is an introduction into the field of enumerative combinatorics. | |||||

Objective | Upon completion of the course, students are able to classify combinatorial problems and to apply adequate techniques to solve them. | |||||

Content | Contents of the lectures Combinatorics I and II: congruence transformation of the plane, symmetry groups of geometric figures, Euler's function, Cayley graphs, formal power series, permutation groups, cycles, Bunside's lemma, cycle index, Polya's theorems, applications to graph theory and isomers. | |||||

Selection: Geometry | ||||||

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

401-3057-00L | Finite Geometries II | W | 4 credits | 2G | N. Hungerbühler | |

Abstract | Finite geometries I, II: Finite geometries combine aspects of geometry, discrete mathematics and the algebra of finite fields. In particular, we will construct models of axioms of incidence and investigate closing theorems. Applications include test design in statistics, block design, and the construction of orthogonal Latin squares. | |||||

Objective | Finite geometries I, II: Students will be able to construct and analyse models of finite geometries. They are familiar with closing theorems of the axioms of incidence and are able to design statistical tests by using the theory of finite geometries. They are able to construct orthogonal Latin squares and know the basic elements of the theory of block design. | |||||

Content | Finite geometries I, II: finite fields, rings of polynomials, finite affine planes, axioms of incidence, Euler's thirty-six officers problem, design of statistical tests, orthogonal Latin squares, transformation of finite planes, closing theorems of Desargues and Pappus-Pascal, hierarchy of closing theorems, finite coordinate planes, division rings, finite projective planes, duality principle, finite Moebius planes, error correcting codes, block design | |||||

Literature | - Max Jeger, Endliche Geometrien, ETH Skript 1988 - Albrecht Beutelspacher: Einführung in die endliche Geometrie I,II. Bibliographisches Institut 1983 - Margaret Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press - Dembowski: Finite Geometries. | |||||

401-3111-68L | Elliptic Curves and Cryptography | W | 8 credits | 3V + 1U | L. Halbeisen | |

Abstract | Im ersten Teil der Vorlesung wird die algebraische Struktur von elliptischen Kurven behandelt. Insbesondere wird der Satz von Mordell bewiesen. Im zweiten Teil der Vorlesung werden dann Anwendungen elliptischer Kurven in der Kryptographie gezeigt, wie z.B. der Diffie-Hellman-Schluesselaustausch. | |||||

Objective | Rationale Punkte auf elliptischen Kurven, insbesondere Arithmetik auf elliptischen Kurven, Satz von Mordell, Kongruente Zahlen Anwendungen der elliptischen Kurven in der Kryptographie, wie zum Beispiel Diffie-Hellman-Schluesselaustausch, Pollard-Rho-Methode | |||||

Content | Im ersten Teil der Vorlesung wird die algebraische Struktur von elliptischen Kurven behandelt und die Menge der rationalen Punkte auf elliptischen Kurven untersucht. Insbesondere wird mit Hilfe von Saetzen aus der Algebra wie auch aus der projektiven Geometrie gezeigt, dass die Menge der rationalen Punkte auf einer elliptischen Kurven unter einer bestimmten Operation eine endlich erzeugte abelsche Gruppe bildet. Zudem werden elliptische Kurven untersucht, welche mit rationalen, rechtwinkligen Dreiecken mit ganzzahligem Flaecheninhalt zusammenhaengen. Im zweiten Teil der Vorlesung werden dann Anwendungen elliptischer Kurven in der Kryptographie gezeigt. Solche Anwendungen sind zum Beispiel ein auf elliptischen Kurven basierendes Kryptosystem oder ein Algorithmus zur Faktorisierung grosser Zahlen. | |||||

Literature | Joseph Silverman, John Tate: "Rational Points on Elliptic Curves", Undergraduate Texts in Mathematics, Springer-Verlag (1992) Ian Blake, Gadiel Seroussi, Nigel Smart: "Elliptic Curves in Cryptography", Lecture Notes Series 265, Cambridge University Press (2004) | |||||

Prerequisites / Notice | Voraussgesetzt werden Algebra I und Grundbegriffe der projektiven Geometrie. | |||||

Selection: Analysis | ||||||

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

401-4115-00L | Introduction to Geometric Measure Theory | W | 6 credits | 3V | U. Lang | |

Abstract | Introduction to Geometric Measure Theory from a metric viewpoint. Contents: Lipschitz maps, differentiability, area and coarea formula, rectifiable sets, introduction to the (de Rham-Federer-Fleming) theory of currents, currents in metric spaces after Ambrosio-Kirchheim, normal currents, relation to BV functions, slicing, compactness theorem for integral currents and applications. | |||||

Objective | ||||||

Content | Extendability and differentiability of Lipschitz maps, metric differentiability, rectifiable sets, approximate tangent spaces, area and coarea formula, brief survey of the (de Rham-Federer-Fleming) theory of currents, currents in metric spaces after Ambrosio-Kirchheim, currents with finite mass and normal currents, relation to BV functions, rectifiable and integral currents, slicing, compactness theorem for integral currents and applications. | |||||

Literature | - Pertti Mattila, Geometry of Sets and Measures in Euclidean Spaces, 1995 - Herbert Federer, Geometric Measure Theory, 1969 - Leon Simon, Introduction to Geometric Measure Theory, 2014, Link - Luigi Ambrosio and Bernd Kirchheim, Currents in metric spaces, Acta math. 185 (2000), 1-80 - Urs Lang, Local currents in metric spaces, J. Geom. Anal. 21 (2011), 683-742 | |||||

Selection: Numerical Analysis | ||||||

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

401-4357-68L | On Deep Artificial Neural Networks and Partial Differential Equations | W | 4 credits | 2G | A. Jentzen | |

Abstract | In this lecture we rigorously analyse approximation capacities of deep artificial neural networks and prove that deep artificial neural networks do overcome the curse of dimensionality in the numerical approximation of solutions of partial differential equations (PDEs). | |||||

Objective | The aim of this course is to teach the students a decent knowledge on deep artificial neural networks and their approximation capacities. | |||||

Content | In recent years deep artificial neural networks (DNNs) have very successfully been used in numerical simulations for a series of computational problems ranging from computer vision, image classification, speech recognition, and natural language processing to computational advertisement. Such numerical simulations indicate that deep artificial neural networks seem to admit the fundamental power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named applications. In this lecture we rigorously analyse approximation capacities of deep artificial neural networks and prove that deep artificial neural networks do overcome the curse of dimensionality in the numerical approximation of solutions of partial differential equations (PDEs). In particular, this course includes (i) a rigorous mathematical introduction to artificial neural networks, (ii) an introduction to some partial differential equations, and (iii) results on approximation capacities of deep artificial neural networks. | |||||

Lecture notes | Lecture notes will be available as a PDF file. | |||||

Literature | Related literature: * Arnulf Jentzen, Diyora Salimova, and Timo Welti, A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients. arXiv:1809.07321 (2018), 48 pages. Available online at [Link]. * Philipp Grohs, Fabian Hornung, Arnulf Jentzen, and Philippe von Wurstemberger, A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations. arXiv:1809.02362 (2018), 124 pages. Available online at [Link]. * Andrew R. Barron, Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inform. Theory 39 (1993), no. 3, 930--945. | |||||

Prerequisites / Notice | Prerequisites: Analysis I and II, Elementary Probability Theory, and Measure Theory | |||||

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

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

401-3627-00L | High-Dimensional StatisticsDoes not take place this semester. | 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 Analysis | W | 6 credits | 3G | N. Meinshausen | |

Abstract | Statistical analysis and modeling of observations in temporal order, which exhibit dependence. Stationarity, trend estimation, seasonal decomposition, autocorrelations, spectral and wavelet analysis, ARIMA-, GARCH- and state space models. Implementations in the software R. | |||||

Objective | Understanding of the basic models and techniques used in time series analysis and their implementation in the statistical software R. | |||||

Content | This course deals with modeling and analysis of variables which change randomly in time. Their essential feature is the dependence between successive observations. Applications occur in geophysics, engineering, economics and finance. Topics covered: Stationarity, trend estimation, seasonal decomposition, autocorrelations, spectral and wavelet analysis, ARIMA-, GARCH- and state space models. The models and techniques are illustrated using the statistical software R. | |||||

Lecture notes | Not available | |||||

Literature | A list of references will be distributed during the course. | |||||

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

401-0625-01L | Applied Analysis of Variance and Experimental Design | W | 5 credits | 2V + 1U | L. Meier | |

Abstract | Principles of experimental design, one-way analysis of variance, contrasts and multiple comparisons, multi-factor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, split-plot designs, incomplete block designs, two-series factorials and fractional designs, power. | |||||

Objective | Participants will be able to plan and analyze efficient experiments in the fields of natural sciences. They will gain practical experience by using the software R. | |||||

Content | Principles of experimental design, one-way analysis of variance, contrasts and multiple comparisons, multi-factor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, split-plot designs, incomplete block designs, two-series factorials and fractional designs, power. | |||||

Literature | G. Oehlert: A First Course in Design and Analysis of Experiments, W.H. Freeman and Company, New York, 2000. | |||||

Prerequisites / Notice | The exercises, but also the classes will be based on procedures from the freely available, open-source statistical software R, for which an introduction will be held. | |||||

401-0649-00L | Applied Statistical Regression | W | 5 credits | 2V + 1U | M. Dettling | |

Abstract | This course offers a practically oriented introduction into regression modeling methods. The basic concepts and some mathematical background are included, with the emphasis lying in learning "good practice" that can be applied in every student's own projects and daily work life. A special focus will be laid in the use of the statistical software package R for regression analysis. | |||||

Objective | The students acquire advanced practical skills in linear regression analysis and are also familiar with its extensions to generalized linear modeling. | |||||

Content | The course starts with the basics of linear modeling, and then proceeds to parameter estimation, tests, confidence intervals, residual analysis, model choice, and prediction. More rarely touched but practically relevant topics that will be covered include variable transformations, multicollinearity problems and model interpretation, as well as general modeling strategies. The last third of the course is dedicated to an introduction to generalized linear models: this includes the generalized additive model, logistic regression for binary response variables, binomial regression for grouped data and poisson regression for count data. | |||||

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

Literature | Faraway (2005): Linear Models with R Faraway (2006): Extending the Linear Model with R Draper & Smith (1998): Applied Regression Analysis Fox (2008): Applied Regression Analysis and GLMs Montgomery et al. (2006): Introduction to Linear Regression Analysis | |||||

Prerequisites / Notice | The exercises, but also the classes will be based on procedures from the freely available, open-source statistical software package R, for which an introduction will be held. In the Mathematics Bachelor and Master programmes, the two course units 401-0649-00L "Applied Statistical Regression" and 401-3622-00L "Regression" are mutually exclusive. Registration for the examination of one of these two course units is only allowed if you have not registered for the examination of the other course unit. | |||||

401-4637-67L | On Hypothesis Testing | W | 4 credits | 2V | F. Balabdaoui | |

Abstract | This course is a review of the main results in decision theory. | |||||

Objective | The goal of this course is to present a review for the most fundamental results in statistical testing. This entails reviewing the Neyman-Pearson Lemma for simple hypotheses and the Karlin-Rubin Theorem for monotone likelihood ratio parametric families. The students will also encounter the important concept of p-values and their use in some multiple testing situations. Further methods for constructing tests will be also presented including likelihood ratio and chi-square tests. Some non-parametric tests will be reviewed such as the Kolmogorov goodness-of-fit test and the two sample Wilcoxon rank test. The most important theoretical results will reproved and also illustrated via different examples. Four sessions of exercises will be scheduled (the students will be handed in an exercise sheet a week before discussing solutions in class). | |||||

Literature | - Statistical Inference (Casella & Berger) - Testing Statistical Hypotheses (Lehmann and Romano) | |||||

401-3628-14L | Bayesian StatisticsDoes not take place this semester. | W | 4 credits | 2V | ||

Abstract | Introduction to the Bayesian approach to statistics: Decision theory, prior distributions, hierarchical Bayes models, Bayesian tests and model selection, empirical Bayes, computational methods, Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods. | |||||

Objective | Students understand the conceptual ideas behind Bayesian statistics and are familiar with common techniques used in Bayesian data analysis. | |||||

Content | Topics that we will discuss are: Difference between the frequentist and Bayesian approach (decision theory, principles), priors (conjugate priors, Jeffreys priors), tests and model selection (Bayes factors, hyper-g priors in regression),hierarchical models and empirical Bayes methods, computational methods (Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods) | |||||

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

Literature | Christian Robert, The Bayesian Choice, 2nd edition, Springer 2007. A. Gelman et al., Bayesian Data Analysis, 3rd edition, Chapman & Hall (2013). Additional references will be given in the course. | |||||

Prerequisites / Notice | Familiarity with basic concepts of frequentist statistics and 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 Bachelor's programme in 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 | |

401-3922-00L | 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. | |||||

Objective | ||||||

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 sciences. It discusses collective risk modeling, individual claim size modeling, approximations for compound distributions, ruin theory, premium calculation principles, tariffication with generalized linear models, 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 and Generalized Linear Models Bayesian Models and Credibility Theory Claims Reserving Solvency Considerations | |||||

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

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. We learn to create mortality tables from scratch. Additionally, we study various guarantees embedded in life insurace 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. Overview on guarantees & options in life insurance with a real-world example demonstrating their risks 2. Mortality tables - Determining raw mortality rates - Smoothing of raw mortality rates - Trends in mortality rates - Lee-Carter model - Integration of safety margins 3. Primer on Financial Mathematics - Ito integral - Black-Scholes and Hull-White model 4. Valuation of Unit linked contracts with embedded options 5. Valuation of Participating contracts | |||||

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. | |||||

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

Abstract | This course provides an actuarial introduction to reinsurance. The objective is to understand the fundamentals of risk transfer through reinsurance, and the mathematical 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 alternative risk transfer | |||||

Objective | This course provides an introduction to reinsurance from an actuarial point of view. 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 - Alternative Risk Transfer: Alternatives to traditional reinsurance such as insurance linked securities and catastrophe bonds | |||||

Content | This course provides an introduction to reinsurance from an actuarial point of view. 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 - Alternative Risk Transfer: Alternatives to traditional reinsurance such as insurance linked securities and catastrophe bonds | |||||

Lecture notes | Slides, lecture notes, and references to literature will be made available. | |||||

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

401-3905-68L | Convex Optimization in Machine Learning and Computational Finance | W | 4 credits | 2V | P. Cheridito, M. Baes | |

Abstract | ||||||

Objective | ||||||

Content | Part 1: Convex Analysis Lecture 1: General introduction, convex sets and functions Lecture 2: Semidefinite cone, Separation theorems (Application to the Fundamental Theorem of Asset Pricing) Lecture 3: Analytic properties of convex functions, duality (Application to Support Vector Machines) Lecture 4: Lagrangian duality, conjugate functions, support functions Lecture 5: Subgradients and subgradient calculus (Application to Automatic Differentiation and Lexicographic Differentiation) Lecture 6: Karush-Kuhn-Tucker Conditions (Application to Markowitz portfolio optimization) Part 2: Applications Lecture 7: Approximation, Lasso optimization, Covariance matrix estimation (Application: a politically optimal splitting of Switzerland) Lecture 8: Clustering and MaxCut problems, Optimal coalitions and Shapley Value Part 3: Algorithms Lecture 9: Intractability of Optimization, Gradient Method for convex optimization, Stochastic Gradient Method (Application to Neural Networks) Lecture 10: Fundamental flaws of Gradient Methods, Mirror Descent Method (Application to Multiplicative Weight Method and Adaboost) Lecture 11: Accelerated Gradient Method, Smoothing Technique (Application to large-scale Lasso optimization) Lecture 12: Newton Method and its fundamental drawbacks, Self-Concordant Functions Lecture 13: Interior-Point Methods | |||||

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

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

402-0830-00L | General Relativity | W | 10 credits | 4V + 2U | R. Renner | |

Abstract | Manifold, Riemannian metric, connection, curvature; Special Relativity; Lorentzian metric; Equivalence principle; Tidal force and spacetime curvature; Energy-momentum tensor, field equations, Newtonian limit; Post-Newtonian approximation; Schwarzschild solution; Mercury's perihelion precession, light deflection. | |||||

Objective | Basic understanding of general relativity, its mathematical foundations, and some of the interesting phenomena it predicts. | |||||

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 N. Straumann - General Relativity with applications to Astrophysics | |||||

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

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

401-3054-14L | Probabilistic Methods in Combinatorics | W | 6 credits | 2V + 1U | B. Sudakov | |

Abstract | This course provides a gentle introduction to the Probabilistic Method, with an emphasis on methodology. We will try to illustrate the main ideas by showing the application of probabilistic reasoning to various combinatorial problems. | |||||

Objective | ||||||

Content | The topics covered in the class will include (but are not limited to): linearity of expectation, the second moment method, the local lemma, correlation inequalities, martingales, large deviation inequalities, Janson and Talagrand inequalities and pseudo-randomness. | |||||

Literature | - The Probabilistic Method, by N. Alon and J. H. Spencer, 3rd Edition, Wiley, 2008. - Random Graphs, by B. Bollobás, 2nd Edition, Cambridge University Press, 2001. - Random Graphs, by S. Janson, T. Luczak and A. Rucinski, Wiley, 2000. - Graph Coloring and the Probabilistic Method, by M. Molloy and B. Reed, Springer, 2002. |

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