Search result: Catalogue data in Autumn Semester 2019
Mathematics Master | ||||||
Seminars and Semester Papers | ||||||
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 | |
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401-4530-69L | Gauge Theory | W | 4 credits | 2S | W. Merry | |
Abstract | ||||||
Objective | The goal of the seminar is to understand Donaldson’s famous theorem that certain topological 4-manifolds do not admit a smooth structure. The idea is to study the topology of the moduli space of anti-self dual connections. | |||||
Content | What is gauge theory? Very roughly speaking, gauge theory (or Yang-Mills theory) is the study of the space of connections on a principal bundle. A Yang-Mills connection is an extremal of the Yang-Mills functional. These connections are important in both mathematics and physics. The most interesting case occurs when the base manifold is four-dimensional. Here one can also speak of instantons (or anti-self-dual connections). The instanton equations on four-manifolds were first studied in the late 70s. In 1981 as part of his PhD studies Simon Donaldson used these equations to make spectacular progress on (exotic) four-manifold topology. A rough outline of the topics intended to be covered is: 1. The classification of complex line bundles, and the classification of unitary connections (up to gauge equivalence) over them. 2. The Hodge Theorem. 3. Yang-Mills connections. 4. Anti-self dual connections and instantons (in dimension 4). 5. Uhlenbeck Compactness. 6. Donaldson’s Theorem. | |||||
Lecture notes | Lecture notes will be written by the participants! | |||||
Literature | An overview of the literature available will be posted on my forum. | |||||
Prerequisites / Notice | **THIS SEMINAR IS ONLY OPEN TO STUDENTS WHO HAVE PRE-REGISTERED WITH ME ON MY FORUM.** This is an advanced seminar. It is assumed you are familiar with: - Differential Geometry I - Differential Geometry II - Algebraic Topology I In addition it would be helpful if you knew: - Algebraic Topology II - Functional Analysis I | |||||
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. | |||||
401-3370-67L | 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. | |||||
401-3920-17L | 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 | ||||||
401-3650-68L | Numerical Analysis Seminar: Mathematics of Deep Neural Network Approximation Number of participants limited to 6. Consent of Instructor needed. | 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. | |||||
Objective | ||||||
Content | Presentation of the Seminar: 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 Link and, more mathematical and closer to the seminar theme, Link 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 with Q/A from the seminar group 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. | |||||
401-3660-69L | Numerical Analysis Seminar: Model Order Reduction and Reduced Bases for PDEs Number of participants limited to 5. Consent of Instructor needed. | W | 4 credits | 2S | C. Marcati | |
Abstract | Reduced Basis (RB) methods provide a technique to reduce the computational cost of problems described by partial differential equations which involve a wide range of parameters (parametric PDEs). Such problems are ubiquitous in science and engineering, both in the analysis of physical phenomena and in the design of new objects. | |||||
Objective | The aim of this seminar is to review recent mathematical results on theoretical aspects of Reduced Basis methods and to learn how model-order reduction techniques can be used to lower computational cost in the solution of parametric PDEs. | |||||
Content | Reduced Basis (RB) methods provide a technique to reduce the computational cost of problems described by partial differential equations which involve a wide range of parameters (parametric PDEs). Such problems are ubiquitous in science and engineering, both in the analysis of physical phenomena and in the design of new objects. Building on top of classical finite element approximations, RB methods split the work into a computationally heavy offline phase and an online phase—where only a reduced-order model needs to be solved—that can be executed almost in real-time. The first phase involves computing the high-fidelity solutions to the PDE on a carefully selected "training" set of parameters (so-called snapshots). The snapshots are then used as a reduced basis (hence the name of the method) for the solution of problems on new parameters. The methods used for the (quasi-)optimal selection of the basis are of independent interest and shared with other model-order reduction techniques in statistics, approximation, and data science. The estimates on a priori RB errors are linked with the approximability of the classes of solutions to the equations; furthermire, reduced approximations can be used as a theoretical tool in the analysis of other reduction techniques. | |||||
Literature | Introductory textbooks. [1] Jan S. Hesthaven, Gianluigi Rozza, and Benjamin Stamm, Certified reduced basis methods for parametrized partial differential equations, SpringerBriefs in Mathemat- ics, Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2016. [2] Alfio Quarteroni, Andrea Manzoni, and Federico Negri, Reduced basis methods for partial differential equations, Unitext, vol. 92, Springer, Cham, 2016. | |||||
Prerequisites / Notice | Format of the seminar The seminar format will be oral student presentations, combined with a written report. Student presentations will be based on a recent research paper selected in two meetings at the start of the semester. | |||||
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. | |||||
401-3920-69L | 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. | |||||
401-4910-69L | Topics in Mathematical Finance and Stochastic Analysis Number of participants limited to 24. | W | 4 credits | 2S | C. Czichowsky | |
Abstract | Backward stochastic differential equations (BSDEs) are an important tool of stochastic analysis. They appear naturally in applications of stochastic calculus in stochastic optimal control and mathematical finance. The seminar introduces students to the theory of BSDEs (rather than their applications) and covers different aspects of them. | |||||
Objective | The goal is to learn mathematical results in the theory of BSDEs. We will study chapters of the book “Backward Stochastic Differential Equations” by Jianfeng Zhang. | |||||
Literature | "Backward Stochastic Differential Equations" by Jiangfeng Zhang. | |||||
Prerequisites / Notice | Familiarity with measure-theoretic probability and stochastic calculus as in the standard D-MATH courses "Probability Theory" and "Brownian Motion and Stochastic Calculus" will be assumed. Textbook accounts can be found in the first two chapters of the book and the references therein. Participants are expected to attend the seminar and give a presentation. Topics will be assigned in 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). | |||||
Semester Papers There are several course units "Semester Paper" that are all equivalent. If, during your studies, you write several semester papers, choose among the different numbers in order to be able to obtain credits again. | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
401-3750-01L | Semester Paper Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required. For more information, see Link | W | 8 credits | 11A | Supervisors | |
Abstract | Semester Papers help to deepen the students' knowledge of a specific subject area. Students are offered a selection of topics. These papers serve to develop the students' ability for independent mathematical work as well as to enhance skills in presenting mathematical results in writing. | |||||
Objective | ||||||
Prerequisites / Notice | There are several course units "Semester Paper" that are all equivalent. If, during your studies, you write several semester papers, choose among the different numbers in order to be able to obtain credits again. | |||||
401-3750-02L | Semester Paper Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required. For more information, see Link | W | 8 credits | 11A | Supervisors | |
Abstract | Semester Papers help to deepen the students' knowledge of a specific subject area. Students are offered a selection of topics. These papers serve to develop the students' ability for independent mathematical work as well as to enhance skills in presenting mathematical results in writing. | |||||
Objective | ||||||
Prerequisites / Notice | There are several course units "Semester Paper" that are all equivalent. If, during your studies, you write several semester papers, choose among the different numbers in order to be able to obtain credits again. | |||||
401-3750-03L | Semester Paper Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required. For more information, see Link | W | 8 credits | 11A | Supervisors | |
Abstract | Semester Papers help to deepen the students' knowledge of a specific subject area. Students are offered a selection of topics. These papers serve to develop the students' ability for independent mathematical work as well as to enhance skills in presenting mathematical results in writing. | |||||
Objective | ||||||
Prerequisites / Notice | There are several course units "Semester Paper" that are all equivalent. If, during your studies, you write several semester papers, choose among the different numbers in order to be able to obtain credits again. |
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