Search result: Catalogue data in Autumn Semester 2019
Mathematics Bachelor | ||||||
First Year | ||||||
» First Year Compulsory Courses | ||||||
» Minor Courses | ||||||
» GESS Science in Perspective | ||||||
First Year Compulsory Courses | ||||||
First Year Examination Block 1 | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
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401-1151-00L | Linear Algebra I | O | 7 credits | 4V + 2U | T. H. Willwacher | |
Abstract | Introduction to the theory of vector spaces for students of mathematics or physics: Basics, vector spaces, linear transformations, solutions of systems of equations, matrices, determinants, endomorphisms, eigenvalues, eigenvectors. | |||||
Objective | - Mastering basic concepts of Linear Algebra - Introduction to mathematical methods | |||||
Content | - Basics - Vectorspaces and linear maps - Systems of linear equations and matrices - Determinants - Endomorphisms and eigenvalues | |||||
Literature | - R. Pink: Lineare Algebra I und II. Summary. Link: Link - G. Fischer: Lineare Algebra. Springer-Verlag 2014. Link: Link - K. Jänich: Lineare Algebra. Springer-Verlag 2004. Link: Link - H.-J. Kowalsky, G. O. Michler: Lineare Algebra. Walter de Gruyter 2003. Link: Link - S. H. Friedberg, A. J. Insel and L. E. Spence: Linear Algebra. Pearson 2003. Link - H. Schichl and R. Steinbauer: Einführung in das mathematische Arbeiten. Springer-Verlag 2012. Link: Link | |||||
402-1701-00L | Physics I | O | 7 credits | 4V + 2U | R. Grange | |
Abstract | This course gives a first introduction to Physics with an emphasis on classical mechanics. | |||||
Objective | Acquire knowledge of the basic principles regarding the physics of classical mechanics. Skills in solving physics problems. | |||||
252-0847-00L | Computer Science | O | 5 credits | 2V + 2U | M. Schwerhoff, F. Friedrich Wicker | |
Abstract | The course covers the fundamental concepts of computer programming with a focus on systematic algorithmic problem solving. Taught language is C++. No programming experience is required. | |||||
Objective | Primary educational objective is to learn programming with C++. After having successfully attended the course, students have a good command of the mechanisms to construct a program. They know the fundamental control and data structures and understand how an algorithmic problem is mapped to a computer program. They have an idea of what happens "behind the scenes" when a program is translated and executed. Secondary goals are an algorithmic computational thinking, understanding the possibilities and limits of programming and to impart the way of thinking like a computer scientist. | |||||
Content | The course covers fundamental data types, expressions and statements, (limits of) computer arithmetic, control statements, functions, arrays, structural types and pointers. The part on object orientation deals with classes, inheritance and polymorphism; simple dynamic data types are introduced as examples. In general, the concepts provided in the course are motivated and illustrated with algorithms and applications. | |||||
Lecture notes | English lecture notes will be provided during the semester. The lecture notes and the lecture slides will be made available for download on the course web page. Exercises are solved and submitted online. | |||||
Literature | Bjarne Stroustrup: Einführung in die Programmierung mit C++, Pearson Studium, 2010 Stephen Prata, C++ Primer Plus, Sixth Edition, Addison Wesley, 2012 Andrew Koenig and Barbara E. Moo: Accelerated C++, Addison-Wesley, 2000 | |||||
First Year Examination Block 2 | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
401-1261-07L | Analysis I | O | 10 credits | 6V + 3U | P. S. Jossen | |
Abstract | Introduction to the differential and integral calculus in one real variable: fundaments of mathematical thinking, numbers, sequences, basic point set topology, continuity, differentiable functions, ordinary differential equations, Riemann integration. | |||||
Objective | The ability to work with the basics of calculus in a mathematically rigorous way. | |||||
Literature | H. Amann, J. Escher: Analysis I Link J. Appell: Analysis in Beispielen und Gegenbeispielen Link R. Courant: Vorlesungen über Differential- und Integralrechnung Link O. Forster: Analysis 1 Link H. Heuser: Lehrbuch der Analysis Link K. Königsberger: Analysis 1 Link W. Walter: Analysis 1 Link V. Zorich: Mathematical Analysis I (englisch) Link A. Beutelspacher: "Das ist o.B.d.A. trivial" Link H. Schichl, R. Steinbauer: Einführung in das mathematische Arbeiten Link | |||||
Compulsory Courses | ||||||
Examination Block I In Examination Block I either the course unit 402-2883-00L Physics III or the course unit 402-2203-01L Classical Mechanics must be chosen and registered for an examination. (Students may also enrol for the other of the two course units; within the ETH Bachelor's programme in mathematics, this other course unit cannot be registered in myStudies for an examination nor can it be recognised for the Bachelor's degree.) | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
401-2303-00L | Complex Analysis | O | 6 credits | 3V + 2U | P. Biran | |
Abstract | Complex functions of one variable, Cauchy-Riemann equations, Cauchy theorem and integral formula, singularities, residue theorem, index of closed curves, analytic continuation, special functions, conformal mappings, Riemann mapping theorem. | |||||
Objective | Working knowledge of functions of one complex variables; in particular applications of the residue theorem. | |||||
Literature | B. Palka: "An introduction to complex function theory." Undergraduate Texts in Mathematics. Springer-Verlag, 1991. E.M. Stein, R. Shakarchi: Complex Analysis. Princeton University Press, 2010 Th. Gamelin: Complex Analysis. Springer 2001 E. Titchmarsh: The Theory of Functions. Oxford University Press D. Salamon: "Funktionentheorie". Birkhauser, 2011. (In German) L. Ahlfors: "Complex analysis. An introduction to the theory of analytic functions of one complex variable." International Series in Pure and Applied Mathematics. McGraw-Hill Book Co. K.Jaenich: Funktionentheorie. Springer Verlag R.Remmert: Funktionentheorie I. Springer Verlag E.Hille: Analytic Function Theory. AMS Chelsea Publications | |||||
401-2333-00L | Methods of Mathematical Physics I | O | 6 credits | 3V + 2U | G. Felder | |
Abstract | Fourier series. Linear partial differential equations of mathematical physics. Fourier transform. Special functions and eigenfunction expansions. Distributions. Selected problems from quantum mechanics. | |||||
Objective | ||||||
402-2883-00L | Physics III | W | 7 credits | 4V + 2U | U. Keller | |
Abstract | Introductory course on quantum and atomic physics including optics and statistical physics. | |||||
Objective | A basic introduction to quantum and atomic physics, including basics of optics and equilibrium statistical physics. The course will focus on the relation of these topics to experimental methods and observations. | |||||
Content | Evidence for Quantum Mechanics: atoms, photons, photo-electric effect, Rutherford scattering, Compton scattering, de-Broglie waves. Quantum mechanics: wavefunctions, operators, Schrodinger's equation, infinite and finite square well potentials, harmonic oscillator, hydrogen atoms, spin. Atomic structure: Perturbation to basic structure, including Zeeman effect, spin-orbit coupling, many-electron atoms. X-ray spectra, optical selection rules, emission and absorption of radiation, including lasers. Optics: Fermat's principle, lenses, imaging systems, diffraction, interference, relation between geometrical and wave descriptions, interferometers, spectrometers. Statistical mechanics: probability distributions, micro and macrostates, Boltzmann distribution, ensembles, equipartition theorem, blackbody spectrum, including Planck distribution | |||||
Lecture notes | Lecture notes will be provided electronically during the course. | |||||
Literature | Quantum mechanics/Atomic physics/Molecules: "The Physics of Atoms and Quanta", H. Hakan and H. C. Wolf, ISBN 978-3-642-05871-4 Optics: "Optics", E. Hecht, ISBN 0-321-18878-0 Statistical mechanics: "Statistical Physics", F. Mandl 0-471-91532-7 | |||||
402-2203-01L | Classical Mechanics | W | 7 credits | 4V + 2U | M. Gaberdiel | |
Abstract | A conceptual introduction to theoretical physics: Newtonian mechanics, central force problem, oscillations, Lagrangian mechanics, symmetries and conservation laws, spinning top, relativistic space-time structure, particles in an electromagnetic field, Hamiltonian mechanics, canonical transformations, integrable systems, Hamilton-Jacobi equation. | |||||
Objective | Fundamental understanding of the description of Mechanics in the Lagrangian and Hamiltonian formulation. Detailed understanding of important applications, in particular, the Kepler problem, the physics of rigid bodies (spinning top) and of oscillatory systems. | |||||
252-0851-00L | Algorithms and Complexity | O | 4 credits | 2V + 1U | J. Lengler, A. Steger | |
Abstract | Introduction: RAM machine, data structures; Algorithms: sorting, median, matrix multiplication, shortest paths, minimal spanning trees; Paradigms: divide & conquer, dynamic programming, greedy algorithms; Data Structures: search trees, dictionaries, priority queues; Complexity Theory: P and NP, NP-completeness, Cook's theorem, reductions. | |||||
Objective | After this course students know some basic algorithms as well as underlying paradigms. They will be familiar with basic notions of complexity theory and can use them to classify problems. | |||||
Content | Die Vorlesung behandelt den Entwurf und die Analyse von Algorithmen und Datenstrukturen. Die zentralen Themengebiete sind: Sortieralgorithmen, Effiziente Datenstrukturen, Algorithmen für Graphen und Netzwerke, Paradigmen des Algorithmenentwurfs, Klassen P und NP, NP-Vollständigkeit, Approximationsalgorithmen. | |||||
Lecture notes | Ja. Wird zu Beginn des Semesters verteilt. | |||||
Examination Block II | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
401-2003-00L | Algebra I | O | 7 credits | 4V + 2U | R. Pink | |
Abstract | Introduction and development of some basic algebraic structures - groups, rings, fields. | |||||
Objective | Introduction to basic notions and results of group, ring and field theory. | |||||
Content | Group Theory: basic notions and examples of groups, subgroups, factor groups, homomorphisms, group actions, Sylow theorems, applications Ring Theory: basic notions and examples of rings, ring homomorphisms, ideals, factor rings, euclidean rings, principal ideal domains, factorial rings, applications Field Theory: basic notions and examples of fields, field extensions, algebraic extensions, applications | |||||
Literature | Karpfinger-Meyberg: Algebra, Spektrum Verlag S. Bosch: Algebra, Springer Verlag B.L. van der Waerden: Algebra I und II, Springer Verlag S. Lang, Algebra, Springer Verlag A. Knapp: Basic Algebra, Springer Verlag J. Rotman, "Advanced modern algebra, 3rd edition, part 1" Link J.F. Humphreys: A Course in Group Theory (Oxford University Press) G. Smith and O. Tabachnikova: Topics in Group Theory (Springer-Verlag) M. Artin: Algebra (Birkhaeuser Verlag) R. Lidl and H. Niederreiter: Introduction to Finite Fields and their Applications (Cambridge University Press) | |||||
Core Courses | ||||||
Core Courses: Pure Mathematics | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
401-3531-00L | Differential Geometry I At most one of the three course units (Bachelor Core Courses) 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. | W | 10 credits | 4V + 1U | U. Lang | |
Abstract | Introduction to differential geometry and differential topology. Contents: Curves, (hyper-)surfaces in R^n, geodesics, curvature, Theorema Egregium, Theorem of Gauss-Bonnet. Hyperbolic space. Differentiable manifolds, immersions and embeddings, Sard's Theorem, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem. | |||||
Objective | ||||||
Lecture notes | Partial lecture notes are available from Link | |||||
Literature | Differential geometry in R^n: - Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces - Wolfgang Kühnel: Differentialgeometrie. Kurven-Flächen-Mannigfaltigkeiten - Christian Bär: Elementare Differentialgeometrie Differential topology: - Dennis Barden & Charles Thomas: An Introduction to Differential Manifolds - Victor Guillemin & Alan Pollack: Differential Topology - Morris W. Hirsch: Differential Topology | |||||
401-3461-00L | Functional Analysis I At most one of the three course units (Bachelor Core Courses) 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. | W | 10 credits | 4V + 1U | M. Struwe | |
Abstract | Baire category; Banach and Hilbert spaces, bounded linear operators; basic principles: Uniform boundedness, open mapping/closed graph theorem, Hahn-Banach; convexity; dual spaces; weak and weak* topologies; Banach-Alaoglu; reflexive spaces; compact operators and Fredholm theory; closed range theorem; spectral theory of self-adjoint operators in Hilbert spaces. | |||||
Objective | Acquire a good degree of fluency with the fundamental concepts and tools belonging to the realm of linear Functional Analysis, with special emphasis on the geometric structure of Banach and Hilbert spaces, and on the basic properties of linear maps. | |||||
Literature | We will be using the Lecture Notes on "Funktionalanalysis I" by Michael Struwe. Other useful, and recommended references include the following books: Haim Brezis: "Functional analysis, Sobolev spaces and partial differential equations". Springer, 2011. Manfred Einsiedler and Thomas Ward: "Functional Analysis, Spectral Theory, and Applications", Graduate Text in Mathematics 276. Springer, 2017. Peter D. Lax: "Functional analysis". Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. Elias M. Stein and Rami Shakarchi: "Functional analysis" (volume 4 of Princeton Lectures in Analysis). Princeton University Press, Princeton, NJ, 2011. Walter Rudin: "Functional analysis". International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991. Dirk Werner, "Funktionalanalysis". Springer-Lehrbuch, 8. Auflage. Springer, 2018 | |||||
Prerequisites / Notice | Solid background on the content of all Mathematics courses of the first two years of the undergraduate curriculum at ETH (most remarkably: fluency with measure theory, Lebesgue integration and L^p spaces). | |||||
401-3371-00L | Dynamical Systems I | W | 10 credits | 4V + 1U | W. Merry | |
Abstract | This course is a broad introduction to dynamical systems. Topic covered include topological dynamics, ergodic theory and low-dimensional dynamics. | |||||
Objective | Mastery of the basic methods and principal themes of some aspects of dynamical systems. | |||||
Content | Topics covered include: 1. Topological dynamics (transitivity, attractors, chaos, structural stability) 2. Ergodic theory (Poincare recurrence theorem, Birkhoff ergodic theorem, existence of invariant measures) 3. Low-dimensional dynamics (Poincare rotation number, dynamical systems on [0,1]) | |||||
Literature | The most relevant textbook for this course is Introduction to Dynamical Systems, Brin and Stuck, CUP, 2002. I will also produce full lecture notes, available from my website Link | |||||
Prerequisites / Notice | The material of the basic courses of the first two years of the program at ETH is assumed. In particular, you should be familiar with metric spaces and elementary measure theory. | |||||
401-3001-61L | Algebraic Topology I | W | 8 credits | 4G | A. Sisto | |
Abstract | This is an introductory course in algebraic topology, which is the study of algebraic invariants of topological spaces. Topics covered include: singular homology, cell complexes and cellular homology, the Eilenberg-Steenrod axioms. | |||||
Objective | ||||||
Literature | 1) A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. Book can be downloaded for free at: Link See also: Link 2) G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. 3) E. Spanier, "Algebraic topology", Springer-Verlag | |||||
Prerequisites / Notice | You should know the basics of point-set topology. Useful to have (though not absolutely necessary) basic knowledge of the fundamental group and covering spaces (at the level covered in the course "topology"). Some knowledge of differential geometry and differential topology is useful but not strictly necessary. Some (elementary) group theory and algebra will also be needed. | |||||
401-3114-69L | Introduction to Algebraic Number Theory | W | 8 credits | 3V + 1U | Ö. Imamoglu | |
Abstract | This is an introductory course in algebraic number theory covering algebraic integers, discriminant, ideal class group, Minkowski's theorem on the finiteness of the ideal class group, Dirichlet's unit theorem, ramification theory. | |||||
Objective | ||||||
Content | This is an introductory course in algebraic number theory covering algebraic integers, discriminant, ideal class group, Minkowski's theorem on the finiteness of the ideal class group, Dirichlet's unit theorem, ramification theory. | |||||
401-3132-00L | Commutative Algebra | W | 10 credits | 4V + 1U | E. Kowalski | |
Abstract | This course provides an introduction to commutative algebra as a foundation for and first steps towards algebraic geometry. | |||||
Objective | We shall cover approximately the material from --- most of the textbook by Atiyah-MacDonald, or --- the first half of the textbook by Bosch. Topics include: * Basics about rings, ideals and modules * Localization * Primary decomposition * Integral dependence and valuations * Noetherian rings * Completions * Basic dimension theory | |||||
Literature | Primary Reference: 1. "Introduction to Commutative Algebra" by M. F. Atiyah and I. G. Macdonald (Addison-Wesley Publ., 1969) Secondary Reference: 2. "Algebraic Geometry and Commutative Algebra" by S. Bosch (Springer 2013) Tertiary References: 3. "Commutative algebra. With a view towards algebraic geometry" by D. Eisenbud (GTM 150, Springer Verlag, 1995) 4. "Commutative ring theory" by H. Matsumura (Cambridge University Press 1989) 5. "Commutative Algebra" by N. Bourbaki (Hermann, Masson, Springer) | |||||
Prerequisites / Notice | Prerequisites: Algebra I (or a similar introduction to the basic concepts of ring theory). | |||||
» Core Courses: Pure Mathematics (Mathematics Master) | ||||||
Core Courses: Applied Mathematics and Further Appl.-Oriented Fields ¬ | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
401-3601-00L | Probability Theory At most one of the three course units (Bachelor Core Courses) 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. | W | 10 credits | 4V + 1U | A.‑S. Sznitman | |
Abstract | Basics of probability theory and the theory of stochastic processes in discrete time | |||||
Objective | This course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned: Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains. | |||||
Content | This course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned: Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains. | |||||
Lecture notes | available, will be sold in the course | |||||
Literature | R. Durrett, Probability: Theory and examples, Duxbury Press 1996 H. Bauer, Probability Theory, de Gruyter 1996 J. Jacod and P. Protter, Probability essentials, Springer 2004 A. Klenke, Wahrscheinlichkeitstheorie, Springer 2006 D. Williams, Probability with martingales, Cambridge University Press 1991 | |||||
401-3621-00L | Fundamentals of Mathematical Statistics | W | 10 credits | 4V + 1U | S. van de Geer | |
Abstract | The course covers the basics of inferential statistics. | |||||
Objective | ||||||
401-3901-00L | Mathematical Optimization | W | 11 credits | 4V + 2U | R. Zenklusen | |
Abstract | Mathematical treatment of diverse optimization techniques. | |||||
Objective | The goal of this course is to get a thorough understanding of various classical mathematical optimization techniques with an emphasis on polyhedral approaches. In particular, we want students to develop a good understanding of some important problem classes in the field, of structural mathematical results linked to these problems, and of solution approaches based on this structural understanding. | |||||
Content | Key topics include: - Linear programming and polyhedra; - Flows and cuts; - Combinatorial optimization problems and techniques; - Equivalence between optimization and separation; - Brief introduction to Integer Programming. | |||||
Literature | - Bernhard Korte, Jens Vygen: Combinatorial Optimization. 6th edition, Springer, 2018. - Alexander Schrijver: Combinatorial Optimization: Polyhedra and Efficiency. Springer, 2003. This work has 3 volumes. - Ravindra K. Ahuja, Thomas L. Magnanti, James B. Orlin. Network Flows: Theory, Algorithms, and Applications. Prentice Hall, 1993. - Alexander Schrijver: Theory of Linear and Integer Programming. John Wiley, 1986. | |||||
Prerequisites / Notice | Solid background in linear algebra. | |||||
401-3622-00L | Statistical Modelling | W | 8 credits | 4G | C. Heinze-Deml | |
Abstract | In regression, the dependency of a random response variable on other variables is examined. We consider the theory of linear regression with one or more covariates, high-dimensional linear models, nonlinear models and generalized linear models, robust methods, model choice and nonparametric models. Several numerical examples will illustrate the theory. | |||||
Objective | Introduction into theory and practice of a broad and popular area of statistics, from a modern viewpoint. | |||||
Content | In der Regression wird die Abhängigkeit einer beobachteten quantitativen Grösse von einer oder mehreren anderen (unter Berücksichtigung zufälliger Fehler) untersucht. Themen der Vorlesung sind: Einfache und multiple Regression, Theorie allgemeiner linearer Modelle, Hoch-dimensionale Modelle, Ausblick auf nichtlineare Modelle. Querverbindungen zur Varianzanalyse, Modellsuche, Residuenanalyse; Einblicke in Robuste Regression. Durchrechnung und Diskussion von Anwendungsbeispielen. | |||||
Lecture notes | Lecture notes | |||||
Prerequisites / Notice | This is the course unit with former course title "Regression". Credits cannot be recognised for both courses 401-3622-00L Statistical Modelling and 401-0649-00L Applied Statistical Regression in the Mathematics Bachelor and Master programmes (to be precise: one course in the Bachelor and the other course in the Master is also forbidden). | |||||
252-0057-00L | Theoretical Computer Science | W | 7 credits | 4V + 2U | J. Hromkovic, H.‑J. Böckenhauer | |
Abstract | Concepts to cope with: a) what can be accomplished in a fully automated fashion (algorithmically solvable) b) How to measure the inherent difficulty of tasks (problems) c) What is randomness and how can it be useful? d) What is nondeterminism and what role does it play in CS? e) How to represent infinite objects by finite automata and grammars? | |||||
Objective | Learning the basic concepts of computer science along their historical development | |||||
Content | This lecture gives an introduction to theoretical computer science, presenting the basic concepts and methods of computer science in its historical context. We present computer science as an interdisciplinary science which, on the one hand, investigates the border between the possible and the impossible and the quantitative laws of information processing, and, on the other hand, designs, analyzes, verifies, and implements computer systems. The main topics of the lecture are: - alphabets, words, languages, measuring the information content of words, representation of algorithmic tasks - finite automata, regular and context-free grammars - Turing machines and computability - complexity theory and NP-completeness - design of algorithms for hard problems | |||||
Lecture notes | The lecture is covered in detail by the textbook "Theoretical Computer Science". | |||||
Literature | Basic literature: 1. J. Hromkovic: Theoretische Informatik. 5th edition, Springer Vieweg 2014. 2. J. Hromkovic: Theoretical Computer Science. Springer 2004. Further reading: 3. M. Sipser: Introduction to the Theory of Computation, PWS Publ. Comp.1997 4. J.E. Hopcroft, R. Motwani, J.D. Ullman: Introduction to Automata Theory, Languages, and Computation (3rd Edition), Addison-Wesley 2006. 5. I. Wegener: Theoretische Informatik. Teubner. More exercises and examples in: 6. A. Asteroth, Ch. Baier: Theoretische Informatik | |||||
Prerequisites / Notice | During the semester, two non-obligatory test exams will be offered. | |||||
252-0209-00L | Algorithms, Probability, and Computing | W | 8 credits | 4V + 2U + 1A | A. Steger, B. Gärtner, M. Ghaffari, D. Steurer | |
Abstract | Advanced design and analysis methods for algorithms and data structures: Random(ized) Search Trees, Point Location, Minimum Cut, Linear Programming, Randomized Algebraic Algorithms (matchings), Probabilistically Checkable Proofs (introduction). | |||||
Objective | Studying and understanding of fundamental advanced concepts in algorithms, data structures and complexity theory. | |||||
Lecture notes | Will be handed out. | |||||
Literature | Introduction to Algorithms by T. H. Cormen, C. E. Leiserson, R. L. Rivest; Randomized Algorithms by R. Motwani und P. Raghavan; Computational Geometry - Algorithms and Applications by M. de Berg, M. van Kreveld, M. Overmars, O. Schwarzkopf. | |||||
» Core Courses: Applied Mathematics and Further Appl.-Oriented Fields (Mathematics Master) | ||||||
Core Courses: Further Application-Oriented Fields 402-0205-00L Quantum Mechanics I is eligible as an applied core course, but only if 402-0224-00L Theoretical Physics (offered for the last time in FS 2016) 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 | |
402-0205-00L | Quantum Mechanics I | W | 10 credits | 3V + 2U | G. Blatter | |
Abstract | Introduction to quantum theory: wave mechanics, Schroedinger equation, angular momentum, central force problems, potential scattering, spin. General structure: Hilbert space, states, obervables, equation of motion, density matrix, symmetries, Heisenberg- and interaction picture, approximate methods: perturbation theory, variational approach, quasi-classics. | |||||
Objective | Introduction to single-particle quantum mechanics. Familiarity with basic ideas and concepts (quantisation, operator formalism, symmetries, angular momentum, perturbation theory) and generic examples and applications (bound states, tunneling, hydrogen atom, harmonic oscillator). Ability to solve simple problems. | |||||
Content | Starting from Feynman's path-integral formulation, we develop the operator technique and introduce Dirac's notation. Quantum phenomena are developed by way of example for one-dimensional single particle problems (bound states, tunneling, scattering problems, resonances, periodic and disordered potentials). We introduce rotations and angular momenta and proceed with central symmetric problems, three dimensional scattering theory, spin, and the addition of angular momenta/spin. Various pictures (Schroedinger-, Heisenberg-, Dirac-) are explained and approximative methods such as variational techniques, perturbation theory, and quasi-classical formalism are introduced. | |||||
Lecture notes | Auf Moodle, in deutscher Sprache | |||||
Literature | G. Baim, Lectures on Quantum Mechanics E. Merzbacher, Quantum Mechanics L.I. Schiff, Quantum Mechanics R. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals J.J. Sakurai: Modern Quantum Mechanics A. Messiah: Quantum Mechanics I S. Weinberg: Lectures on Quantum Mechanics | |||||
Electives | ||||||
Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
401-3059-00L | Combinatorics II | 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. | |||||
401-3033-00L | Gödel's Theorems | W | 8 credits | 3V + 1U | L. Halbeisen | |
Abstract | Die Vorlesung besteht aus drei Teilen: Teil I gibt eine Einführung in die Syntax und Semantik der Prädikatenlogik erster Stufe. Teil II behandelt den Gödel'schen Vollständigkeitssatz Teil III behandelt die Gödel'schen Unvollständigkeitssätze | |||||
Objective | Das Ziel dieser Vorlesung ist ein fundiertes Verständnis der Grundlagen der Mathematik zu vermitteln. | |||||
Content | Syntax und Semantik der Prädikatenlogik Gödel'scher Vollständigkeitssatz Gödel'sche Unvollständigkeitssätze | |||||
Literature | Ergänzende Literatur wird in der Vorlesung angegeben. | |||||
Selection: Geometry | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
401-3057-00L | Finite Geometries II Does not take place this semester. | 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. | |||||
Selection: Analysis | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
401-4461-69L | Reading Course: Functional Analysis III, Unitary Representations Limited number of participants. Please contact Link | W | 3 credits | 6A | M. Einsiedler, further speakers | |
Abstract | ||||||
Objective | ||||||
Selection: Numerical Analysis No offering in this semester yet | ||||||
Selection: Probability Theory, Statistics | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
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 Analysis Does not take place this semester. | 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 "Statistical Modelling" 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. | |||||
401-3628-14L | Bayesian Statistics | W | 4 credits | 2V | F. Sigrist | |
Abstract | Introduction to the Bayesian approach to statistics: decision theory, prior distributions, hierarchical Bayes models, empirical Bayes, Bayesian tests and model selection, empirical Bayes, 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, noninformative priors, Jeffreys prior), tests and model selection (Bayes factors, hyper-g priors for 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 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 | |||||
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. 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. | |||||
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. | |||||
Prerequisites / Notice | Basic knowledge in statistics, probability theory, and actuarial techniques | |||||
Selection: Mathematical Physics, Theoretical Physics | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
402-0830-00L | General Relativity Special Students UZH must book the module PHY511 directly at UZH. | W | 10 credits | 4V + 2U | P. Jetzer | |
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 | |
401-3055-64L | 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 | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
252-0417-00L | Randomized Algorithms and Probabilistic Methods | W | 8 credits | 3V + 2U + 2A | 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) | |||||
252-1425-00L | Geometry: Combinatorics and Algorithms | W | 6 credits | 2V + 2U + 1A | B. Gärtner, 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. | |||||
263-4500-00L | Advanced Algorithms | W | 6 credits | 2V + 2U + 1A | M. Ghaffari, A. Krause | |
Abstract | This is an advanced course on the design and analysis of algorithms, covering a range of topics and techniques not studied in typical introductory courses on algorithms. | |||||
Objective | This course is intended to familiarize students with (some of) the main tools and techniques developed over the last 15-20 years in algorithm design, which are by now among the key ingredients used in developing efficient algorithms. | |||||
Content | The lectures will cover a range of topics, 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. | |||||
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. | |||||
Selection: Further Realms | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
401-3503-69L | 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 | ||||||
401-3502-69L | 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 | ||||||
401-3504-69L | 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 | ||||||
401-0000-00L | Communication in Mathematics | W | 2 credits | 1V | W. Merry | |
Abstract | Don't hide your Next Great Theorem behind bad writing. This course teaches fundamental communication skills in mathematics: how to write clearly and how to structure mathematical content for different audiences, from theses, to preprints, to personal statements in applications. In addition, the course will help you establish a working knowledge of LaTeX. | |||||
Objective | Knowing how to present written mathematics in a structured and clear manner. | |||||
Content | Topics covered include: - Language conventions and common errors. - How to write a thesis (more generally, a mathematics paper). - How to use LaTeX. - How to write a personal statement for Masters and PhD applications. | |||||
Lecture notes | Full lecture notes will be made available on my website: Link | |||||
Prerequisites / Notice | There are no formal mathematical prerequisites. | |||||
401-0000-99L | Communication in Mathematics (Upgrade 2018 → 2019) This course unit is only for students who got 1 ECTS credit from last year's course unit 401-0000-00L CiM. (Registration now closed.) | W | 1 credit | 1V | W. Merry | |
Abstract | Don't hide your Next Great Theorem behind bad writing. This course teaches fundamental communication skills in mathematics: how to write clearly and how to structure mathematical content for different audiences, from theses, to preprints, to personal statements in applications. In addition, the course will help you establish a working knowledge of LaTeX. | |||||
Objective | Knowing how to present written mathematics in a structured and clear manner. | |||||
Content | Topics covered include: - Language conventions and common errors. - How to write a thesis (more generally, a mathematics paper). - How to use LaTeX. - How to write a personal statement for Masters and PhD applications. | |||||
Lecture notes | Full lecture notes will be made available on my website: Link | |||||
Prerequisites / Notice | There are no formal mathematical prerequisites. | |||||
Core Courses and Electives (Mathematics Master) | ||||||
» Core Courses (Mathematics Master) | ||||||
» Electives (Mathematics Master) | ||||||
Seminars Early enrolments for seminars in myStudies are encouraged, so that we will recognise need for additional seminars in a timely manner. Some seminars have waiting lists. Nevertheless, register for at most two mathematics seminars. | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
401-3370-67L | Homogeneous Dynamics and Counting Problems Number of participants limited to 12. Registration to the seminar will only be effective once confirmed by the organisers. Please contact Link. | W | 4 credits | 2S | P. Yang, further speakers | |
Abstract | Introductory seminar about the connection between counting problems and mixing properties for group actions. We discuss linear groups, Haar measures, measure preserving actions, ergodicity, the theorem of Howe-Moore and use these concepts to count integer points on certain affine varieties. | |||||
Objective | ||||||
Content | The goal behind the Gauss circle problem is to describe the asymptotics of the number of integer points in a given ball in Euclidean space as the radius of the ball goes to infinity. In this course we will study similar problems such as counting the number of integer matrices of a given determinant in large balls. In 1993 Duke, Rudnick and Sarnak solved counting problems of this kind by proving equidistribution of certain orbits in homogeneous spaces. Shortly thereafter, Eskin and McMullen gave an approach to proving the desired equidistribution result by exploiting mixing properties of certain group actions. In this seminar we develop the tools required for understanding the connection between mixing and counting for a selected number of explicit cases. Exercises are an integral part of the seminar. | |||||
Lecture notes | References will be provided. | |||||
Literature | Main references: M. Einsiedler, T. Ward Ergodic Theory with a view towards number theory, Springer. Further references will be provided. Additional references: W. Duke, Z. Rudnick, and P. Sarnak. Density of integer points on affine homogeneous varieties. Duke Math. J. Volume 71, Number 1 (1993), 143-179. A. Eskin and C. McMullen. Mixing, counting, and equidistribution in Lie groups. Duke Math. J. Volume 71, Number 1 (1993), 181-209. | |||||
Prerequisites / Notice | The students are expected to have mastered the content of the first two years taught at ETH. The seminar is mainly intended for Bachelor students. | |||||
401-3830-69L | Seminar on Minimal Surfaces The total number of students who may take this course for credit is limited to twenty; however further students are welcome to attend. | W | 4 credits | 2S | A. Carlotto | |
Abstract | This course is meant as an invitation to some key ideas and techniques in Geometric Analysis, with special emphasis on the theory of minimal surfaces. It is primarily conceived for advanced Bachelor or beginning Master students. | |||||
Objective | The goal of this course is to get a first introduction to minimal surfaces both in the Euclidean space and in Riemannian manifolds, and to see analytic tools in action to solve natural geometric problems. Students are guided through different types of references (standard monographs, surveys, research articles), encouraged to compare them and to critically prepare some expository work on a chosen topic. This course takes the form of a working group, where interactions among students, and between students and instructor are especially encouraged. | |||||
Content | The minimal surface equation, examples and basic questions. Parametrized surfaces, first variation of the area functional, different characterizations of minimality. The Gauss map, basic properties. The Douglas-Rado approach, basic existence results for the Plateau problem. Monotonicity formulae and applications, including the Farey-Milnor theorem on knotted curves. The second variation formula, stability and Morse index. The Bernstein problem and its solution in the two-dimensional case. Total curvature, curvature estimates and compactness theorems. Classification results for minimal surfaces of low Morse index. | |||||
Literature | Three basic references that we will mostly refer to are the following ones: 1) B. White, Lectures on minimal surface theory, Geometric analysis, 387–438, IAS/Park City Math. Ser., 22, Amer. Math. Soc., Providence, RI, 2016. 2) T. Colding, W. Minicozzi, A course in minimal surfaces. Graduate Studies in Mathematics, 121. American Mathematical Society, Providence, RI, 2011. xii+313 pp. 3) R. Osserman, A survey of minimal surfaces. Second edition. Dover Publications, Inc., New York, 1986. vi+207 pp. Further, more specific references will be listed during the first two introductory lectures. | |||||
Prerequisites / Notice | The content of the first two years of the Bachelor program in Mathematics, in particular all courses in Real and Complex Analysis, Measure Theory, Topology. Some familiarity with the language of Differential Geometry, although not a formal pre-requisite, might be highly helpful. Finally, a first course on elliptic equations (especially on basic topics like Schauder estimates and the maximum principle) might also be a plus. | |||||
401-4460-69L | Functional Analysis III, Unitary Representations Limited number of participants. Please contact Link | W | 4 credits | 2S | M. Einsiedler, further speakers | |
Abstract | The seminar is aimed at students having mastered (abelian) spectral theory and will discuss Unitary Representations and Unitary Duals. To get further into the theory the seminar is accompanied by a reading class with a second regular meeting every week. We will use the material Link | |||||
Objective | ||||||
Prerequisites / Notice | Prerequisites: Functional analysis II, spectral theory of abelian C*-algebras as discussed in the FA II course in spring 2019. The students are required to also take the reading course accompanying the seminar. | |||||
401-3920-17L | Numerical Analysis Seminar: Mathematics for Biomimetics Number of participants limited to 8. | W | 4 credits | 2S | H. Ammari, A. Vanel | |
Abstract | The aim of this seminar is to explore how we can learn from Nature to provide new approaches to solving some of the most challenging problems in sensing systems and materials science. An emphasis will be put on the mathematical foundation of bio-inspired perception algorithms in electrolocation and echolocation. | |||||
Objective | ||||||
401-3920-69L | Theory and Applications of Machine Learning Number of participants limited to 26. | W | 4 credits | 2S | P. Cheridito | |
Abstract | The seminar covers different aspects of machine learning. | |||||
Objective | The goal is to learn some of the mathematical methods used in machine learning. | |||||
Literature | Understanding Machine Learning: From Theory to Algorithms by Shalev-Shwartz and Ben-David | |||||
Prerequisites / Notice | Participants are required to attend and give a presentation. | |||||
401-3620-69L | Student Seminar in Statistics: The Art of Statistics Number of participants limited to 24 Mainly for students from the Mathematics Bachelor and Master Programmes who, in addition to the introductory course unit 401-2604-00L Probability and Statistics, have heard at least one core or elective course in statistics. Also offered in the Master Programmes Statistics resp. Data Science. | W | 4 credits | 2S | M. H. Maathuis | |
Abstract | We will study the book "The Art of Statistics: Learning from Data" by David Spiegelhalter. The focus of the book is not so much on technical aspects, but more on concepts, philosophical aspects, statistical thinking and communication. Chapters will be presented by pairs of students, followed by an open discussion with everyone in the class. | |||||
Objective | We will study roughly one chapter per week from the book "The Art of Statistics: Learning from Data" by David Spiegelhalter. The focus of the book is not so much on technical aspects, but more on concepts, philosophical aspects, statistical thinking and communication. This will also be the focus of the class, but we may occasionally look up additional information from references that are given in the book. Besides improving your statistical thinking, you will practice your self-studying, collaboration and presentation skills. | |||||
Literature | David Spiegelhalter (2019). The Art of Statistics: Learning from Data. UK: Pelican. ISBN: 978-0-241-39863-0 | |||||
Prerequisites / Notice | Besides an introductory course in Probability and Statistics, we require one subsequent Statistics course. We also expect some experience with the statistical software R. Topics will be assigned during the first meeting. | |||||
401-3200-69L | A Survey of Geometric Group Theory Does not take place this semester. Number of participants limited to 12. | W | 4 credits | 2S | ||
Abstract | In this class we will cover some of the tools, techniques, and groups central to the study of geometric group theory. After introducing the basic concepts (groups and metric spaces), we will branch out and sample different topics in geometric group theory based on the interest of the participants. | |||||
Objective | To learn and understand a wide range of tools and groups central to the field of geometric group theory. | |||||
Content | Possible topics include: properties of free groups and groups acting on trees, large scale geometric invariants (Dehn functions, hyperbolicity, ends of groups, asymptotic dimension, growth of groups), and examples of notable and interesting groups (Coxeter groups, right-angled Artin groups, lamplighter groups, Thompson's group, mapping class groups, and braid groups). | |||||
Literature | The topics will be chosen from "Office Hours with a Geometric Group Theorist" edited by Matt Clay and Dan Margalit. | |||||
Prerequisites / Notice | One should be familiar with the basics of groups, metric spaces, and topology (should be familiar with the fundamental group). | |||||
» Seminars (Mathematics Master) | ||||||
Minor Courses | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
401-1511-00L | Geometry | W | 3 credits | 2V + 1U | L. Halbeisen | |
Abstract | Im Mittelpunkt dieser Vorlesung steht die euklidische und die projektive Geometrie. | |||||
Objective | Axiomatischer Aufbau der euklidischen Geometrie mit Hilfe der Axiome von Hilbert. Klassische Sätze der projektiven Geometrie. | |||||
Content | Im ersten Teil der Vorlesung wird die euklidische Geometrie axiomatisch aufgebaut. Das dazu verwendete Axiomensystem stammt von David Hilbert. Nach einer kurzen Einführung in die projektive Geometrie werden dann in einem zweiten Teil die klassischen Sätze der projektiven Geometrie bewiesen. Dazu gehören z.B. die Sätze von Desargues, Pappos, Menelaos, Ceva, Pascal und Brianchon. | |||||
Literature | Robin Hartshorne: "Geometry: Euclid and Beyond", Springer Verlag H.S.M. Coxeter: "Projective Geometry", Springer Verlag | |||||
402-0351-00L | Astronomy Does not take place this semester. | W | 2 credits | 2V | S. P. Quanz | |
Abstract | An overview on the important topics in modern astronomy: planets, sun, stars, milky way, galaxies, and cosmology | |||||
Objective | This lecture gives a general introduction to main topics in modern astronomy. The lecture provide a basis for the more advanced lectures in astrophysics. | |||||
Content | Planeten, Sonne, Sterne, Milchstrasse, Galaxien und Kosmologie. | |||||
Lecture notes | Kopien der Präsentationen werde zur Verfügung gestellt. | |||||
Literature | Der Neue Kosmos. A. Unsöld, B. Baschek, Springer Oder sonstige Grundlehrbücher zur Astronomie. | |||||
Bachelor's Thesis | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
401-2000-00L | Scientific Works in Mathematics Target audience: Third year Bachelor students; Master students who cannot document to have received an adequate training in working scientifically. | O | 0 credits | Ö. Imamoglu | ||
Abstract | Introduction to scientific writing for students with focus on publication standards and ethical issues, especially in the case of citations (references to works of others.) | |||||
Objective | Learn the basic standards of scientific works in mathematics. | |||||
Content | - Types of mathematical works - Publication standards in pure and applied mathematics - Data handling - Ethical issues - Citation guidelines | |||||
Prerequisites / Notice | Directive Link | |||||
401-2000-01L | Lunch Sessions – Thesis Basics for Mathematics Students Details and registration for the optional MathBib training course: Link | Z | 0 credits | Speakers | ||
Abstract | Optional MathBib training course | |||||
Objective | ||||||
401-3990-10L | Bachelor's Thesis Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required. For more information, see Link | O | 8 credits | 11D | Supervisors | |
Abstract | The purpose of the BSc thesis is to deepen knowledge in a certain subject chosen by the student. In their BSc thesis, students should demonstrate their ability to carry out independent work in mathematics and to organize results in a written report. | |||||
Objective | ||||||
GESS Science in Perspective | ||||||
Science in Perspective | ||||||
» see Science in Perspective: Type A: Enhancement of Reflection Capability | ||||||
» Recommended Science in Perspective (Type B) for D-MATH. | ||||||
Language Courses | ||||||
» see Science in Perspective: Language Courses ETH/UZH | ||||||
Additional Courses | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
401-5000-00L | Zurich Colloquium in Mathematics | E- | 0 credits | S. Mishra, P. L. Bühlmann, R. Pandharipande, University lecturers | ||
Abstract | The lectures try to give an overview of "what is going on" in important areas of contemporary mathematics, to a wider non-specialised audience of mathematicians. | |||||
Objective | ||||||
401-5990-00L | Zurich Graduate Colloquium | E- | 0 credits | 1K | University lecturers | |
Abstract | The Graduate Colloquium is an informal seminar aimed at graduate students and postdocs whose purpose is to provide a forum for communicating one's interests and thoughts in mathematics. | |||||
Objective | ||||||
401-5960-00L | Colloquium on Mathematics, Computer Science, and Education Subject didactics for mathematics and computer science teachers. | E- | 0 credits | N. Hungerbühler, M. Akveld, J. Hromkovic, H. Klemenz | ||
Abstract | Didactics colloquium | |||||
Objective | ||||||
402-0101-00L | The Zurich Physics Colloquium | E- | 0 credits | 1K | S. Huber, A. Refregier, University lecturers | |
Abstract | Research colloquium | |||||
Objective | ||||||
402-0800-00L | The Zurich Theoretical Physics Colloquium | E- | 0 credits | 1K | O. Zilberberg, University lecturers | |
Abstract | Research colloquium | |||||
Objective | The Zurich Theoretical Physics Colloquium is jointly organized by the University of Zurich and ETH Zurich. Its mission is to bring both students and faculty with diverse interests in theoretical physics together. Leading experts explain the basic questions in their field of research and communicate the fascination for their work. | |||||
251-0100-00L | Computer Science Colloquium | E- | 0 credits | 2K | Lecturers | |
Abstract | Invited talks, covering the entire scope of computer science. External Listeners are welcome at no charge. A detailed schedule is published at the beginning of each semester. | |||||
Objective | Top international computer scientists take the floor at the distinguished computer science colloquium. Our guest speakers present impacting topics across various areas of the discipline. The colloquium series is held every semester and also includes inaugural and farewell lectures of the department's professors. The colloquium is a noteworthy event for all graduate students. Outside attendance is equally welcome. | |||||
Content | Eingeladene Vorträge aus dem gesamten Bereich der Informatik, zu denen auch Auswärtige kostenlos eingeladen sind. Zu Semesterbeginn erscheint jeweils ein ausführliches Programm. |