Search result: Catalogue data in Spring Semester 2022
Mathematics Master  | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
 Seminars and Semester Papers | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Seminars NOTICE: The number of seminar places is limited, and the special selection procedure should help to allocate the places not primarily according to the registration time. Everybody is waitlisted first when he/she tries to register for a seminar in myStudies. Moreover: Only one mathematics seminar can be chosen per semester. In case you need to attend 2 seminars in this semester, please take contact with the Study Administration (email: studiensekretariat@math.ethz.ch ). | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| 401-3370-17L | Arithmetic of Quadratic Forms  Number of participants limited to 12. Registration to this seminar is closed, the participants have been selected. There is no waiting list. | W | 4 credits | 2S | M. Akka Ginosar | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Abstract | Introductory seminar about rational quadratic forms. P-adic numbers, Hasse's local to global principle and the finiteness of the genus will be discussed. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Learning objective | Quadratic forms and the numbers they represent have been of interest to mathematicians for a long time. For example, which integers can be expressed as a sum of two squares of integers? Or as a sum of three squares? Lagrange's four-squares theorem for instance states that any positive integer can be expressed as a sum of four squares. Such questions motivated the development of many aspects of algebraic number theory. In this seminar we follow the beautiful monograph of Cassels "Rational quadratic forms" and will treat the fundamental results concerning quadratic forms over the integers and the rationals such as Hasse's local to global principle and finiteness of the genus. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Content | The seminar will mostly follow the book "Rational quadratic forms" by J.W.S. Cassels, particularly Chapters 1-9. Exercises in this book are an integral part of the seminar. Towards the end of the semester additional topics may be treated. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lecture notes | Cassels, John William Scott. Rational quadratic forms. Vol. 13. Academic Pr, 1978. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literature | Main reference: Cassels, John William Scott. Rational quadratic forms. Vol. 13. Academic Pr, 1978. Additional references: Kitaoka, Yoshiyuki. Arithmetic of quadratic forms. Vol. 106. Cambridge University Press, 1999. Schulze-Pillot, Rainer. "Representation by integral quadratic forms - a survey." Contemporary Mathematics 344 (2004): 303-322. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Prerequisites / Notice | The student is assumed to have attended courses on linear algebra and algebra (as taught at ETH for instance). Previous knowledge on p-adic numbers is not assumed. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3830-22L | Seminar on Minimal Surfaces (an Invitation to Geometric Analysis) 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Learning 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 some 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 | The three basic references that we will mostly refer to are the following ones: [Whi16] B. White, Introduction to minimal surface theory. Geometric analysis, 387-438, IAS/Park City Math. Ser., 22. American Mathematical Society, Providence, RI, 2016. [CM11] T. Colding, W. Minicozzi, A course in minimal surfaces. Graduate Studies in Mathematics, 121. American Mathematical Society, Providence, RI, 2011. xii+313 pp. [Oss86] 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 introductory lectures. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Prerequisites / Notice | In addition to the first four semesters of the Bachelor program in Mathematics (in particular all courses in Real and Complex Analysis, Measure Theory, Topology), some background in Differential and Riemannian Geometry is certainly a must. At the very least, students are expected to have taken Differential Geometry 1, and possibly be enrolled in the follow-up course Differential Geometry 2. In addition, some prior exposure to partial differential equations (primarily of elliptic type, and especially on basic topics like Schauder estimates and the maximum principle), although not strictly necessary, may certainly help. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-4420-22L | Topics in Harmonic Analysis | W | 4 credits | 2S | J. P. Gonçalves Ramos | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Abstract | The objective of the seminar will be to continue exploring important results and techniques in the area of Harmonic Analysis, mainly related to oscillatory integrals, singular integrals, multiplier theorems, restriction theory and other geometric problems in Fourier analysis. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Learning objective | A main objective in this seminar is to develop a solid knowledge in classical and modern harmonic analysis. At the same time, we aim to reach many current topics in research in harmonic analysis, building thus a solid toolbox for those whose are interested in tackling problems in the modern theory of singular integrals, the Fourier restriction conjecture and other central problems in harmonic analysis. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Prerequisites / Notice | It is expected that students have knowledge on classical Fourier/Harmonic analysis, such as basic theory of the Fourier transform, maximal functions, interpolation theorems, Fourier series, singular integrals and Littlewood-Paley decompositions. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-4820-22L | An Introduction to Mean-Field Limits for Vlasov Equations Limited number of participants. | W | 4 credits | 2S | M. Iacobelli, A. Rege | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Abstract | This course aims to give an introductory description of the classical approaches to the problem of the mean-field limit in mathematical analysis. In particular, the intent is to learn essential tools and techniques for studying Partial Differential Equations while applying them to Vlasov equations. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| Content | Content of the course: Transport equations Characteristic method Weak solutions to conservative transport equations Kinetic theory of Plasmas Mean field limit From particles model to Vlasov-Poisson Dobrushin’s stability theorem | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Prerequisites / Notice | Required: Notions in functional analysis, differential equations and Lebesgue integration Optional: Distribution theory, Sobolev spaces, notions in elliptic PDEs | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3650-22L | Numerical Analysis Seminar: Deep Neural Network Methods for PDEs Number of Participants: limited to seven. Participation by consent of instructor. Closed for further registrations. | W | 4 credits | 2S | C. Schwab | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Abstract | The seminar will review recent _mathematical results_ on approximation power of deep neural networks (DNNs). The focus will be on mathematical proof techniques to obtain approximation rate estimates (in terms of neural network size and connectivity) on various classes of input data including, in particular, selected types of PDE solutions. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| Content | Deep Neural Networks (DNNs) have recently attracted substantial interest and attention due to outperforming the best established techniques in a number of tasks (Chess, Go, Shogi, autonomous driving, language translation, image classification, etc.). In big data analysis, DNNs achieved remarkable performance in computer vision, speech recognition and natural language processing. In many cases, these successes have been achieved by heuristic implementations combined with massive compute power and training data. For a (bird's eye) view, see https://doi.org/10.1017/9781108860604 and, more mathematical and closer to the seminar theme, https://doi.org/10.1109/TIT.2021.3062161 The seminar will review recent _mathematical results_ on approximation power of deep neural networks (DNNs). The focus will be on mathematical proof techniques to obtain approximation rate estimates (in terms of neural network size and connectivity) on various classes of input data including, in particular, selected types of PDE solutions. Mathematical results support that DNNs can equalize or outperform the best mathematical results known to date. Particular cases comprise: high-dimensional parametric maps, analytic and holomorphic maps, maps containing multi-scale features which arise as solution classes from PDEs, classes of maps which are invariant under group actions. Format of the Seminar: The seminar format will be oral student presentations, combined with written report. Student presentations will be based on a recent research paper selected in two meetings at the start of the semester. Grading of the Seminar: Passing grade will require a) 1hr oral presentation _via Zoom_ with Q/A from the seminar group, in early May 2022 and b) typed seminar report (``Ausarbeitung'') of several key aspects of the paper under review. Each seminar topic will allow expansion to a semester or a master thesis in the MSc MATH or MSc Applied MATH. Disclaimer: The seminar will _not_ address recent developments in DNN software, eg. TENSORFLOW, and algorithmic training heuristics, or programming techniques for DNN training in various specific applications. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| 401-4490-22L | Topology Optimization of Engineering Systems | W | 4 credits | 2S | F. Feppon | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Abstract | The goal of the course is to offer a rather exhaustive introduction on the field of topology optimization and its most recent emerging trends. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Learning objective | Topology optimization is a field focusing on the computation of optimized designs for various engineering applications, including weight reduction of mechanical structure, aerodynamic designs, or enhanced cooling in thermomechanical systems. In this course, we will mainly focus on nonparametric shape optimization based on the boundary variation method of Hadamard. Three sessions will be dedicated to project presentations made by the students, either on the presentation of some recent work related to the field, or on their own applicative project. Project will be written in FreeFEM based on a ready-to-use code. Webpage of the course: https://people.math.ethz.ch/~ffeppon/topopt_course/index.html | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Content | The lecture will outline as follows. 0. Topology optimization and automated generative design : perspectives and applications in the context of additive manufacturing (2 hours) 1. Common physical models in mechanical and aeronautic engineering. PDE and variational forms. Formulation of shape optimization problems. (2 hours) 2. Nonlinear constrained optimization. Nullspace gradient flows. (2 hours) 3. General results about shape optimization. Homogenization, relaxed designs. The SIMP method (2 hours). 4. Shape differential calculus. Shape derivatives and shape gradients. The adjoint method. (4 hours). 5. Numerical shape evolution algorithms : moving mesh methods, implicit surfaces and body-fitted meshes (2 hours). 6. Projects : numerical implementation in FreeFEM++. (4 hours) 7. Advanced methods : geometric constraints. (2 hours). 8. Project presentations. (6 hours) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lecture notes | Lecture notes will be provided for this course. The course material will be partially based on https://hal.archives-ouvertes.fr/hal-03207863/document | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Prerequisites / Notice | The students should have knowledge about weak formulations of partial differential equations and the finite element method. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3940-22L | Student Seminar in Mathematics and Data: Matrix Discrepancy Number of participants limited to 12. | W | 4 credits | 2S | A. Bandeira, A. Maillard | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Abstract | This student seminar will focus on an Open Problem in Matrix Discrepancy, often referred to as Matrix Spencer. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| Content | The student seminar will focus on Open Problem 4.3. here https://people.math.ethz.ch/~abandeira/TenLecturesFortyTwoProblems.pdf Each week a student will either present a related paper or thoughts on a particular case for the problem. When the Spring 2022 section of forum.math.ethz.ch opens up, more information will be posted there (keep an eye out for it). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Prerequisites / Notice | If you would like to participate in the student seminar, sign up and send Antoine Maillard <antoine.maillard@math.ethz.ch> (i) your transcript (a strong background in probability and linear algebra is needed) (ii) An argument with an upper bound where the constant C can depend on n, any dependency is fine. You can also send other thoughts on the problem. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3600-22L | Student Seminar in Probability Theory Limited number of participants. Registration to the seminar will only be effective once confirmed by email from the organizers. | W | 4 credits | 2S | W. Werner, J. Bertoin, V. Tassion | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| 401-3620-22L | Student Seminar in Statistics: Causality Number of participants limited to 72. 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 | P. L. Bühlmann, M. Champion | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Abstract | Causality is dealing with fundamental questions about cause and effect. The student seminar covers statistical and mathematical aspects of causality ranging from fundamental formalization of concepts to practical algorithms and methods. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Learning objective | The participants of the seminar acquire knowledge about: concepts and formalization of statistical causality; methods, algorithms and corresponding assumptions for inferring causal relations from data; causal analysis in practice based on real data. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Prerequisites / Notice | Basic course in probability and statistics. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3900-16L | Advanced Topics in Discrete Optimization Number of participants limited to 12. | W | 4 credits | 2S | R. Zenklusen, R. Santiago Torres, V. Traub | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Abstract | In this seminar we will discuss selected topics in discrete optimization. The main focus is on mostly recent research papers in the field of Combinatorial Optimization. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Learning objective | The goal of the seminar is twofold. First, we aim at improving students' presentation and communication skills. In particular, students are to present a research paper to their peers and the instructors in a clear and understandable way. Second, students learn a selection of recent cutting-edge approaches in the field of Combinatorial Optimization by attending the other students' talks. A very active participation in the seminar helps students to build up the necessary skills for parsing and digesting advanced technical texts on a significantly higher complexity level than usual textbooks. A key goal is that students prepare their presentations in a concise and accessible way to make sure that other participants get a clear idea of the presented results and techniques. Students intending to do a project in optimization are strongly encouraged to participate. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Content | The selected topics will cover various classical and modern results in Combinatorial Optimization. Contrary to prior years, a very significant component of the seminar will be interactive discussions where active participation of the students is required. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literature | The learning material will be in the form of scientific papers. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Prerequisites / Notice | Requirements: We expect students to have a thorough understanding of topics covered in the course "Linear & Combinatorial Optimization". | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| 252-4102-00L | Seminar on Randomized Algorithms and Probabilistic Methods The deadline for deregistering expires at the end of the second week of the semester. Students who are still registered after that date, but do not attend the seminar, will officially fail the seminar. Number of participants limited to 24. | W | 2 credits | 2S | A. Steger | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Abstract | The aim of the seminar is to study papers which bring the students to the forefront of today's research topics. This semester we will study selected papers of the conference Symposium on Discrete Algorithms (SODA22). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Learning objective | Read papers from the forefront of today's research; learn how to give a scientific talk. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Prerequisites / Notice | The seminar is open for both students from mathematics and students from computer science. As prerequisite we require that you passed the course Randomized Algorithms and Probabilistic Methods (or equivalent, if you come from abroad). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 263-4203-00L | Geometry: Combinatorics and Algorithms The deadline for deregistering expires at the end of the second week of the semester. Students who are still registered after that date, but do not attend the seminar, will officially fail the seminar. | W | 2 credits | 2S | B. Gärtner, M. Hoffmann, E. Welzl, J. Cardinal, M. Wettstein | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Abstract | This seminar complements the course Geometry: Combinatorics & Algorithms. Students of the seminar will present original research papers, some classic and some of them very recent. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Learning objective | Each student is expected to read, understand, and elaborate on a selected research paper. To this end, (s)he should give a 45-min. presentation about the paper. The process includes * getting an overview of the related literature; * understanding and working out the background/motivation: why and where are the questions addressed relevant? * understanding the contents of the paper in all details; * selecting parts suitable for the presentation; * presenting the selected parts in such a way that an audience with some basic background in geometry and graph theory can easily understand and appreciate it. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Content | This seminar is held once a year and complements the course Geometry: Combinatorics & Algorithms. Students of the seminar will present original research papers, some classic and some of them very recent. The seminar is a good preparation for a master, diploma, or semester thesis in the area. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Prerequisites / Notice | Prerequisite: Successful participation in the course "Geometry: Combinatorics & Algorithms" (takes place every HS) is required. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3350-22L | A Survey of Geometric Group Theory Number of participants limited to 24. To sign up for this seminar, please e-mail Matthew Cordes <matthew.cordes@math.ethz.ch>. | W | 4 credits | 2S | M. Cordes | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Learning 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). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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