Search result: Catalogue data in Spring Semester 2021

Mathematics Bachelor Information
Electives
Core Courses and Electives (Mathematics Master)
» Electives (Mathematics Master)
Further Courses Suitable for the Second Year
NumberTitleTypeECTSHoursLecturers
401-2334-00LMethods of Mathematical Physics II Information W6 credits3V + 2UT. H. Willwacher
AbstractGroup theory: groups, representation of groups, unitary and orthogonal groups, Lorentz group. Lie theory: Lie algebras and Lie groups. Representation theory: representation theory of finite groups, representations of Lie algebras and Lie groups, physical applications (eigenvalue problems with symmetry).
Objective
401-2684-00LMathematics of Machine Learning Information W5 credits2V + 1UA. Bandeira, N. Zhivotovskii
AbstractIntroductory course to Mathematical aspects of Machine Learning, including Supervised Learning, Unsupervised Learning, Sparsity, and Online Learning.
ObjectiveIntroduction to Mathematical aspects of Machine Learning.
ContentMathematical aspects of Supervised Learning, Unsupervised Learning, Sparsity, and Online Learning. This course is a Mathematical course, with Theorems and Proofs.
Prerequisites / NoticeNote for non Mathematics students: this class requires a certain degree of mathematical maturity--including abstract thinking and the ability to understand and write proofs.
401-2140-21LSeminar in Algebraic Number Theory Restricted registration - show details
Number of participants limited to 12.
W4 credits2SR. Steiner
AbstractIn this seminar, you'll learn how various concepts of the integers, for example the prime factorisation, can be generalised to finite field extensions of the rational numbers. For this manner, the more robust theory of Dedekind rings is worked out and combined with Galois theory.
Objective- Understanding of Dedekind rings and factorisation of ideals as well as their class groups.

- Knowledge of how prime ideals may split under field extensions and how one may compute such a behaviour.

- Various insights into advanced algebraic, geometric, and analytic number theory, such as Kummer theory, Chebotarev's density theorem, Dirichlet's unit theorem, Dirichlet L-functions
Prerequisites / NoticeAlgebra I & II, where the latter may also be visited in parallel.
Seminars
This semester, many seminars have a waiting list with special selection procedure. If no other criteria apply, a definitive registration will be granted first of all to students who haven't got another seminar registration. Here is the best procedure for dealing with two waiting lists: first choose your preferred seminar and a few minutes later choose an alternative seminar.
IMPORTANT: Do not waitlist yourself for more than two seminars!
NumberTitleTypeECTSHoursLecturers
401-2140-21LSeminar in Algebraic Number Theory Restricted registration - show details
Number of participants limited to 12.
W4 credits2SR. Steiner
AbstractIn this seminar, you'll learn how various concepts of the integers, for example the prime factorisation, can be generalised to finite field extensions of the rational numbers. For this manner, the more robust theory of Dedekind rings is worked out and combined with Galois theory.
Objective- Understanding of Dedekind rings and factorisation of ideals as well as their class groups.

- Knowledge of how prime ideals may split under field extensions and how one may compute such a behaviour.

- Various insights into advanced algebraic, geometric, and analytic number theory, such as Kummer theory, Chebotarev's density theorem, Dirichlet's unit theorem, Dirichlet L-functions
Prerequisites / NoticeAlgebra I & II, where the latter may also be visited in parallel.
401-3110-21LStudent Seminar in Number Theory: Modular Forms Restricted registration - show details
Number of participants limited to 26.
W4 credits2SM. Schwagenscheidt
AbstractSeminar on the basic theory of classical elliptic modular forms
ObjectiveIn the seminar we will learn about the basic theory of classical elliptic modular forms. We start with the action of the modular group on the complex upper half-plane by Moebius transformations and describe its fundamental domain. As first examples of modular forms, we will investigate Eisenstein series, Ramanujan's Delta function, the Dedekind eta function, and the modular j-invariant. We will show that the space of modular forms of a fixed weight is finite dimensional, and determine its dimension. We will also study Hecke operators and the Petersson inner product on spaces of modular forms, and the L-functions associated with modular forms. Towards the end of the seminar we will discuss some more advanced topics, such as differential operators and quasimodular forms, the CM values of the j-function, and the periods of modular forms.
Lecture notesLink
LiteratureCohen, Strömberg: Modular Forms: A Classical Approach
Diamond, Shurman: A first course in modular forms
Koblitz: Introduction to elliptic curves and modular forms
Koecher, Krieg: Elliptische Funktionen und Modulformen
Lang: Introduction to modular forms
Miyake: Modular forms
Serre: A course in arithmetic
Zagier: The 1-2-3 of modular forms

Lecture notes on modular forms, available online: Link
Prerequisites / NoticeWe will need the fundamental results from complex analysis, and some elementary group theory.

The website of the seminar can be found at
https://people.math.ethz.ch/~mschwagen/modularforms
401-3140-21LMonstrous Moonshine Information Restricted registration - show details
Number of participants limited to 12.
W4 credits2ST.‑H. Bülles, R. Pandharipande
AbstractWe study Monstrous Moonshine, the surprising connection between modular forms and the Monster group.
ObjectiveTo understand the equation 196884 = 196883 + 1.
Prerequisites / NoticeAlgebra I and II. Some familiarity with modular forms and Lie algebras is helpful, but not crucial: all necessary concepts will be introduced in the early talks.
401-3520-21LSphere Packings, Lattices and Codes Information Restricted registration - show details
Number of participants limited to 12.
W4 credits2SD. Radchenko
AbstractSeminar on Sphere Packings, Lattices and Codes
ObjectiveTo learn about the sphere packing problem and its connection to various other topics such as error-correcting codes, combinatorial and spherical designs, and modular forms.
ContentSome of the tentative topics include: sphere packing problem; the kissing number problem; error-correcting codes; Shannon capacity; finite projective planes; binary Golay code; spherical designs; theta functions of lattices; linear programming bounds for spherical codes and sphere packings.
LiteratureJ.H. Conway, N.J. Sloane, Sphere Packings, Lattices and Groups, 3rd edition, Springer-Verlag New York, 2017.
W. Ebeling, Lattices and Codes: A Course Partially Based on Lectures by Friedrich Hirzebruch, third edition, Springer Spektrum, Wiesbaden, 2013.
D. Zagier, Elliptic modular forms and their applications, in "The 1-2-3 of Modular Forms" (K. Ranestad, ed.), Universitext, Springer, Berlin, 2008.
C. Zong, Sphere Packings, Universitext, Springer-Verlag New York, 1999.
Prerequisites / NoticeMany of the topics are self-contained and require only basic knowledge of linear algebra and analysis. Some of the later talks require basic knowledge of complex analysis. Some degree of familiarity with modular forms is also helpful, but not strictly necessary.
401-3350-21LClassical Theory of Elliptic Partial Differential Equations Restricted registration - show details
Number of participants limited to 12.
W4 credits2SJ. Serra
AbstractFollowing the book "Elliptic Partial Differential Equations" of Qing Han and Fanhua Lin, the seminar will cover ---from an introductory perspective--- some important classical tools and results in the standard theory of Elliptic PDE
ObjectiveTo present some of the most useful classical tools and results in nonlinear Elliptic PDE (weak and viscosity solutions and their maximum principles, moving plane method, Bernstein's technique, De Giorgi-Nash-Moser Harnack Inequality, etc.)
Content(flexible depending on the background of the students)
-Review of harmonic functions
-Weak and viscosity solutions
-Maximum principles and barriers
-Moving plane method
-Bernstein's technique
-Schauder estimates (review)
-De Giorgi-Nash-Moser and Hölder continuity of gradients
LiteratureElliptic Partial Differential Equations: Second Edition
Qing Han and Fanghua Lin
Publication Year: 2011
ISBN-10: 0-8218-5313-9
ISBN-13: 978-0-8218-5313-9
Courant Lecture Notes, vol. 1.R
Prerequisites / NoticeAlthough many parts of the book are rather self-contained, it would be advisable to have followed before the bachelor course Functional Analysis II
401-3830-21LWave Equations on Black Hole Spacetimes Restricted registration - show details
Number of participants limited to 12.
W4 credits2SC. Kehle
AbstractIntroduction to Lorentzian geometry, to the notion of a black hole, and to the study of linear wave equations on such spacetimes.
ObjectiveWe will learn about the basics of Lorentzian geometry, the geometric framework which incorporates space and time as one geometric entity---spacetime. Then, we will briefly introduce the Einstein equations of General Relativity and study the Schwarzschild and Reissner--Nordström black holes solutions. We will further discuss tools to study linear wave equations on black holes and other spacetimes.
ContentBlack holes are among the central theoretical predictions of general relativity which is governed by the celebrated Einstein's equations. The notion of a black hole has a clean mathematical definition, and the concept is already exhibited by the simplest non-trivial solution of the Einstein vacuum equation: the Schwarzschild solution. These “black hole spacetimes” give rise to many natural mathematical problems in the analysis of (hyperbolic) PDE which in turn describe physical phenomena related to black holes. More specifically we will cover the following topics: Basic Lorentzian geometry, the Schwarzschild and Reissner-Nordström black hole, the wave equation on general Lorentzian manifolds, the wave equation on black hole backgrounds. We will also adapt the content to the prior knowledge of the students.
LiteratureMain reference: Lecture Notes of Mihalis Dafermos: https://www.dpmms.cam.ac.uk/~md384/ETH-Nachdiplom-temp.pdf

Further references (going beyond the scope of the seminar):
- Dafermos, Mihalis, and Igor Rodnianski. "Lectures on black holes and linear waves." Clay Math. Proc 17 (2013): 97-205. (see also arXiv:0811.0354)
- Aretakis, Stefanos. "General Relativity". https://www.math.toronto.edu/aretakis/General%20Relativity-Aretakis.pdf
- Christodoulou, Demetrios. Mathematical problems of general relativity I. Vol. 1. European Mathematical Society, 2008.
Prerequisites / NoticeIdeally, participants have some familiarity with the basics of differential manifolds (definition of smooth manifolds, tangent space, vector fields, as well as the formal apparatus of Riemannian geometry: connections, curvature, geodesics) and basic functional analysis (Sobolev spaces, etc.).
401-3940-21LStudent Seminar in Mathematics and Data: Optimal Transport Restricted registration - show details
Number of participants limited to 12.
W4 credits2SA. Bandeira, G. Chinot
AbstractThe Seminar starts with a basic introduction to Optimal Transport (including but not limited to: Monge and Kantorovich formulations, duality, Wassertstein distance). After the introductory material, each week will be devoted to either a research article in the topic or a more advanced concept. Particular emphasis will be given to applications to statistics and data science.
Objective
Lecture notesMore information, including list of papers, will be available at Link
LiteratureMore information, including list of papers, will be available at Link
Prerequisites / NoticeThis seminar requires a certain degree of mathematical maturity--including abstract thinking and the ability to understand and write proofs. Probability theory and Linear Algebra is a required pre-requisite. Some basic familiarity with Optimization and Functional Analysis is beneficial.
401-3600-21LStudent Seminar in Probability Theory Restricted registration - show details
Limited number of participants.
Registration to the seminar will only be effective once confirmed by email from the organizers.
W4 credits2SW. Werner, J. Bertoin, V. Tassion
Abstract
Objective
401-3620-21LStudent Seminar in Statistics: Statistical Network Modeling Information Restricted registration - show details
Number of participants limited to 48.
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.
W4 credits2SP. L. Bühlmann, M. Azadkia
AbstractNetwork models can be used to analyze non-iid data because their structure incorporates interconnectedness between the individuals. We introduce networks, describe them mathematically, and consider applications.
ObjectiveNetwork models can be used to analyze non-iid data because their structure incorporates interconnectedness between the individuals. The participants of the seminar acquire knowledge to formulate and analyze network models and to apply them in examples.
LiteratureE. D. Kolaczyk and G. Csárdi. Statistical analysis of network data with R. Springer, Cham, Switzerland, second edition, 2020.

Tianxi Li, Elizaveta Levina, and Ji Zhu. Network cross-validation by edge sampling, 2020. Preprint arXiv:1612.04717.

Tianxi Li, Elizaveta Levina, and Ji Zhu. Community models for partially observed networks from surveys, 2020. Preprint arXiv:2008.03652.

Tianxi Li, Elizaveta Levina, and Ji Zhu. Prediction Models for Network-Linked Data, 2018. Preprint arXiv:1602.01192.
Prerequisites / NoticeEvery class will consist of an oral presentation highlighting key ideas of selected book chapters by a pair of students. Another two students will be responsible for asking questions during the presentation and providing a discussion of the the presented concepts and ideas, including pros+cons, at the end. Finally, an additional two students are responsible for giving an evaluation on the quality of the presentations/discussions and provide constructive feedback for improvement.
401-3620-20LStudent Seminar in Statistics: Inference in Non-Classical Regression Models Restricted registration - show details
Does not take place this semester.
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.
W4 credits2SF. Balabdaoui
AbstractReview of some non-standard regression models and the statistical properties of estimation methods in such models.
ObjectiveThe main goal is the students get to discover some less known regression models which either generalize the well-known linear model (for example monotone regression) or violate some of the most fundamental assumptions (as in shuffled or unlinked regression models).
ContentLinear regression is one of the most used models for prediction and hence one of the most understood in statistical literature. However, linearity might too simplistic to capture the actual relationship between some response and given covariates. Also, there are many real data problems where linearity is plausible but the actual pairing between the observed covariates and responses is completely lost or at partially. In this seminar, we review some of the non-classical regression models and the statistical properties of the estimation methods considered by well-known statisticians and machine learners. This will encompass:
1. Monotone regression
2. Single index model
3. Unlinked regression
4. Partially unlinked regression
Lecture notesNo script is necessary for this seminar
LiteratureIn the following is the material that will read and studied by each pair of students (all the items listed below are available through the ETH electronic library or arXiv):

1. Chapter 2 from the book "Nonparametric estimation under shape constraints" by P. Groeneboom and G. Jongbloed, 2014, Cambridge University Press

2. "Nonparametric shape-restricted regression" by A. Guntuoyina and B. Sen, 2018, Statistical Science, Volume 33, 568-594

3. "Asymptotic distributions for two estimators of the single index model" by Y. Xia, 2006, Econometric Theory, Volume 22, 1112-1137

4. "Least squares estimation in the monotone single index model" by F. Balabdaoui, C. Durot and H. K. Jankowski, Journal of Bernoulli, 2019, Volume 4B, 3276-3310

5. "Least angle regression" by B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani, 2004, Annals of Statsitics, Volume 32, 407-499.

6. "Sharp thresholds for high dimensional and noisy sparsity recovery using l1-constrained quadratic programming (Lasso)" by M. Wainwright, 2009, IEEE transactions in Information Theory, Volume 55, 1-19

7."Denoising linear models with permuted data" by A. Pananjady, M. Wainwright and T. A. Courtade and , 2017, IEEE International Symposium on Information Theory, 446-450.

8. "Linear regression with shuffled data: statistical and computation limits of permutation recovery" by A. Pananjady, M. Wainwright and T. A. Courtade , 2018, IEEE transactions in Information Theory, Volume 64, 3286-3300

9. "Linear regression without correspondence" by D. Hsu, K. Shi and X. Sun, 2017, NIPS

10. "A pseudo-likelihood approach to linear regression with partially shuffled data" by M. Slawski, G. Diao, E. Ben-David, 2019, arXiv.

11. "Uncoupled isotonic regression via minimum Wasserstein deconvolution" by P. Rigollet and J. Weed, 2019, Information and Inference, Volume 00, 1-27
401-3900-16LAdvanced Topics in Discrete Optimization Restricted registration - show details
Number of participants limited to 12.
W4 credits2SR. Zenklusen, R. Santiago Torres, V. Traub
AbstractIn 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.
ObjectiveThe 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.
ContentThe 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.
LiteratureThe learning material will be in the form of scientific papers.
Prerequisites / NoticeRequirements: We expect students to have a thorough understanding of topics covered in the course "Mathematical Optimization".
252-4102-00LSeminar on Randomized Algorithms and Probabilistic Methods Restricted registration - show details
Does not take place this semester.
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.
W2 credits2SA. Steger
AbstractThe 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 (SODA18).
ObjectiveRead papers from the forefront of today's research; learn how to give a scientific talk.
Prerequisites / NoticeThe 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).
263-4203-00LGeometry: Combinatorics and Algorithms Information
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.
W2 credits2SB. Gärtner, M. Hoffmann, E. Welzl, M. Wettstein
AbstractThis seminar complements the course Geometry: Combinatorics & Algorithms. Students of the seminar will present original research papers, some classic and some of them very recent.
ObjectiveEach 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.
ContentThis 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 / NoticePrerequisite: Successful participation in the course "Geometry: Combinatorics & Algorithms" (takes place every HS) is required.
» Seminars (Mathematics Master)
Minor Courses
NumberTitleTypeECTSHoursLecturers
401-1032-21LProofs and Basic Structures Information W4 credits2V + 1UW. Merry
Abstract
Objective
ContentAxiomatische Mengenlehre und mathematische Logik bilden die Fundamente, auf denen unser Fach aufgebaut ist.

Der Kurs beginnt mit einer Einführung in die Zermelo-Fraenkel-Mengenlehre. Nebenbei werden wir beweisen, dass Zahlen (!) existieren - zuerst die natürlichen Zahlen, dann die reellen Zahlen und schliesslich andere "grosse" Kardinalzahlen. Wir diskutieren die Implikationen des Auswahlaxioms und der berühmten Kontinuumshypothese.

Sobald die grundlegenden Strukturen fest etabliert sind, gehen wir zur "Kunst des Beweises" über. Das Ziel ist es, Ihnen zu helfen, Beweise zu verstehen und zu konstruieren, und zu lernen, klare und prägnante Mathematik zu schreiben.

Ein wahres Ensemble von Themen aus der Kombinatorik, Algebra und Zahlentheorie (wenn es die Zeit erlaubt) wird vorgestellt - diese Themen sind so gewählt, dass sie gute Beispiele zur Veranschaulichung einer Reihe grundlegender Beweismethoden liefern und fundamentale Ideen vorstellen, die Teil des Standard-Toolkits eines jeden Mathematikers sind. Als besonderes Highlight werden wir eine Auswahl der grössten klassischen Beweise aller Zeiten sehen.
Lecture notesVollständige Vorlesungsnotizen werden zur Verfügung gestellt.
Prerequisites / NoticeEs gibt keine mathematischen Voraussetzungen.
351-1138-00LPRISMA Capstone - Rethinking Sustainable Cities and Communities
Bachelor students get preferential access to this course. All interested students must apply through a separate application process at: https://mtecethz.qualtrics.com/jfe/form/SV_cx4ZghhYhQAY3nT

Participation is subject to successful selection through this sign-up process.
W4 credits4VA. Cabello Llamas, M. Augsburger
AbstractThe goal of this intense one-week course is to bring students from different backgrounds together to make connections between disciplines and to build bridges to society. Supported by student coaches and experts, our student teams will use hands-on Design Thinking methods to address relevant challenges based on the UN sustainable development goals.
ObjectiveIn this intense 7-day block course students will be able to acquire and practice essential cross-disciplinary competencies as well as gaining an understanding of a human-centered innovation process. More specifically students will learn to:
- Work and think in a problem-based way.
- Put their own field into a broader context.
- Engage in collaborative ideation with a multidisciplinary team.
- Identify challenges related to relevant societal issues.
- Develop, prototype and plan innovative solutions for a range of different contexts.
- Innovate in a human-centered way by observing and interacting with key stakeholders.

The acquired methods and skills are based on the ETH competence framework and can be applied to tackle a broad range of problems in academia and society. Moving beyond traditional teaching approaches, this course allows students to engage creatively in a process of rethinking and redesigning aspects and elements of current and future urban areas, actively contributing towards fulfilling the UN SDG 11.
ContentThe course is divided in to three stages:

Warm-up and framing: The goal of this first stage is to get familiar with current problems faced by cities and communities as well as with the Design Thinking process and mindset. The students will learn about the working process, the teaching spaces and resources, as well as their fellow students and the lecturers.

Identifying challenges: The objective is to get to know additional methods and tools to identify a specific challenge relevant for urban areas through fieldwork and direct engagement with relevant stakeholders, resulting in the definition of an actionable problem statement that will form the starting point for the development of innovative solutions.

Solving challenges within current and future context: During this phase, students will apply the learned methods and tools to solve the identified challenge in a multi-disciplinary group by creating, developing and testing high-potential ideas. The ideas are presented to relevant academic, industry and societal stakeholders on the last day of the week.

To facilitate the fast-paced innovation journey, the multidisciplinary teams are supported throughout the week by experienced student coaches.

This course is a capstone for the student-lead initiative PRISMA. (https://www.prisma.ethz.ch/).
Prerequisites / NoticeBachelor students get preferential access to this course. All interested students must apply through a separate application process at: https://mtecethz.qualtrics.com/jfe/form/SV_cx4ZghhYhQAY3nT

Participation is subject to successful selection through this sign-up process.
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