Search result: Catalogue data in Spring Semester 2021
|Doctoral Department of Mathematics |
More Information at: https://www.ethz.ch/en/doctorate.html
The list of courses (together with the allocated credit points) eligible for doctoral students is published each semester in the newsletter of the ZGSM.
WARNING: Do not mistake ECTS credits for credit points for doctoral studies!
| Graduate School|
Official website of the Zurich Graduate School in Mathematics:
|401-5002-21L||Cluster Algebras and Cluster Categories via Surfaces||W||0 credits||2V||K. Baur|
|Content||Twenty years of research in cluster theory have established deep links between cluster algebras, surface geometry and representation theory. Cluster structures can be defined using surface geometry, with curves corresponding to cluster variables and to rigid objects in associated categories. The classical Ptolemy relations give rise to exchange phenomena or mutation in the associated cluster structures.|
In these lectures I will focus on the geometric approach to cluster algebras and cluster categories. Topics discussed will include cluster structures on the Grassmannian, friezes, laminations, Postnikov diagrams, dimer models, root combinatorics.
|401-5004-21L||Prescribing Scalar Curvature in Conformal Geometry||W||0 credits||2V||A. Malchiodi|
|Content||We will consider the classical problem, proposed by Kazdan and Warner in the 70’s, of prescribing the scalar curvature of a Riemannian manifold via conformal deformations of the metric. This amounts to solving an elliptic nonlinear PDE with critical exponent, presenting difficulties due to lack of compactness. There are in general obstructions, but still several contributions to the existence theory have been given using different tehniques. These include Direct Methods of the Calculus of Variations, blow-up analysis, Liouville theorems, gluing constructions and topological or Morse-theoretical tools. We will give a general presentation of the subject, describing the principal contributions in the literature and arriving to more recent developments.|
|401-5006-21L||Random Graphs||W||0 credits||2V||M. Krivelevich|
|Content||Random graphs is a generic name for discrete probability spaces, whose ground sets are composed of graphs, or more generally of discrete structures. Since its inception by Erdős and Rényi some sixty years ago, random graphs have grown to be one of the key disciplines in modern Combinatorics, at the same time serving as an indispensable tool for other branches of Combinatorics and Computer Science.|
The course will serve as an introduction to random graphs, covering classical topics as well as addressing recent developments and tools.
Course syllabus (tentative):
Models of random graphs and of random graph processes; illustrative examples; random regular graphs, configuration model; small subgraphs; long paths and Hamiltonicity; hitting time results; coloring problems; extremal problems in random graphs; pseudo-random graphs.
Working knowledge of Graph Theory, familiarity with basic notions of Probability and Linear Algebra.
|401-3109-65L||Probabilistic Number Theory||W||8 credits||4G||E. Kowalski|
|Abstract||The course presents some results of probabilistic number theory in a unified manner, including distribution properties of the number of prime divisors of integers, probabilistic properties of the zeta function and statistical distribution of exponential sums.|
|Objective||The goal of the course is to present some results of probabilistic number theory in a unified manner.|
|Content||The main concepts will be presented in parallel with the proof of a few main theorems:|
(1) the Erdős-Wintner and Erdős-Kac theorems concerning the distribution of values of arithmetic functions;
(2) the distribution of values of the Riemann zeta function, including Selberg's central limit theorem for the Riemann zeta function on the critical line;
(3) the Chebychev bias for primes in arithmetic progressions;
(4) functional limit theorems for the paths of partial sums of families of exponential sums.
|Lecture notes||The lecture notes for the class are available at|
|Prerequisites / Notice||Prerequisites: Complex analysis, measure and integral, and at least the basic language of probability theory (the main concepts, such as convergence in law, will be recalled).|
Some knowledge of number theory is useful but the main results will also be summarized.
|401-4116-12L||Lectures on Drinfeld Modules||W||6 credits||3V||R. Pink|
|Abstract||Drinfeld modules: Basic theory, analytic uniformization, moduli spaces, good/bad/semistable reduction, Tate modules, Galois representations, endomorphism rings, etc.|
|Content||A central role in the arithmetic of fields of positive characteristic p is played by the Frobenius map x ---> x^p. The theory of Drinfeld modules exploits this map in a systematic fashion. Drinfeld modules of rank 1 can be viewed as analogues of the multiplicative group and are used in the class field theory of global function fields. Drinfeld modules of arbitrary rank possess a rich theory which has many aspects in common with that of elliptic curves, including analytic uniformization, moduli spaces, good/bad/semistable reduction, Tate modules, Galois representations.|
A full understanding of Drinfeld modules requires some knowledge in the arithmetic of function fields and, for comparison, the arithmetic of elliptic curves, which cannot all be presented in the framework of this course. Relevant results from these areas will be presented only cursorily when they are needed, but a fair amount of the theory can be developed without them.
|Literature||Drinfeld, V. G.: Elliptic modules (Russian), Mat. Sbornik 94 (1974), 594--627, translated in Math. USSR Sbornik 23 (1974), 561--592.|
Deligne, P., Husemöller, D: Survey of Drinfeld modules, Contemp. Math. 67, 1987, 25-91.
Goss, D.: Basic structures in function field arithmetic. Springer-Verlag, 1996.
Drinfeld modules, modular schemes and applications. Proceedings of the workshop held in Alden-Biesen, September 9¿14, 1996. Edited by E.-U. Gekeler, M. van der Put, M. Reversat and J. Van Geel. World Scientific Publishing Co., Inc., River Edge, NJ, 1997.
Thakur, Dinesh S.: Function field arithmetic. World Scientific Publishing Co., Inc., River Edge, NJ, 2004.
Further literature will be indicated during the course
|401-3002-12L||Algebraic Topology II||W||8 credits||4G||P. Biran|
|Abstract||This is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology including:|
cohomology of spaces, operations in homology and cohomology, duality.
|Literature||1) G. Bredon, "Topology and geometry",|
Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.
2) A. Hatcher, "Algebraic topology",
Cambridge University Press, Cambridge, 2002.
The book can be downloaded for free at:
3) E. Spanier, "Algebraic topology", Springer-Verlag
|Prerequisites / Notice||General topology, linear algebra, singular homology of topological spaces (e.g. as taught in "Algebraic topology I").|
Some knowledge of differential geometry and differential topology
is useful but not absolutely necessary.
|401-3226-00L||Symmetric Spaces||W||8 credits||4G||A. Iozzi|
|Abstract||* Generalities on symmetric spaces: locally and globally symmetric spaces, groups of isometries, examples|
* Symmetric spaces of non-compact type: flats and rank, roots and root spaces
* Iwasawa decomposition, Weyl group, Cartan decomposition
* Hints of the geometry at infinity of SL(n,R)/SO(n).
|Objective||Learn the basics of symmetric spaces|
|401-4206-17L||Groups Acting on Trees||W||6 credits||3G||B. Brück|
|Abstract||As a main theme, we will see how an action of a group on a tree enables us to break the group into smaller pieces, and thus gain better understanding of its structure.|
|Objective||Learn basics of Bass-Serre theory; get to know concepts from geometric group theory.|
|Content||As a mathematical object, a tree is a graph without any loops. It turns out that if a group acts on such an object, the algebraic structure of the group has a nice description in terms of the combinatorics of the graph. In particular, groups acting on trees can be decomposed in a certain way into simpler pieces.These decompositions can be described combinatorially, but are closely related to concepts from topology such as fundamental groups and covering spaces.|
This interplay between (elementary) concepts of algebra, combinatorics and geometry/topology is typical for geometric group theory. The course can also serve as an introduction to basic concepts of this field.
Topics that will be covered in the lecture include:
- Trees and their automorphisms
- Different characterisations of free groups
- Amalgamated products and HNN extensions
- Graphs of groups
- Kurosh's theorem on subgroups of free (amalgamated) products
|Literature||J.-P. Serre, Trees. (Translated from the French by John Stillwell). Springer-Verlag, 1980. ISBN 3-540-10103-9|
O. Bogopolski. Introduction to group theory. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008. x+177 pp. ISBN: 978-3-03719-041-8
C. T. C. Wall. The geometry of abstract groups and their splittings. Revista Matemática Complutense vol. 16(2003), no. 1, pp. 5-101
|Prerequisites / Notice||Basic knowledge of group theory; being familiar with fundamental groups (e.g. the Seifert-van-Kampen Theorem) and covering theory is definitely helpful, although not strictly necessary.|
In particular, the standard material of the first two years of the Mathematics Bachelor is sufficient.
|401-3532-08L||Differential Geometry II||W||10 credits||4V + 1U||W. Merry|
|Abstract||This is a continuation course of Differential Geometry I.|
Topics covered include:
- Connections and curvature,
- Riemannian geometry,
- Gauge theory and Chern-Weil theory.
|Lecture notes||I will produce full lecture notes, available on my website: |
|Literature||There are many excellent textbooks on differential geometry. |
A friendly and readable book that contains everything covered in Differential Geometry I is:
John M. Lee "Introduction to Smooth Manifolds" 2nd ed. (2012) Springer-Verlag.
For Differential Geometry II, the textbooks:
- S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volume I (1963) Wiley,
- I. Chavel, "Riemannian Geometry: A Modern Introduction" 2nd ed. (2006), CUP,
are both excellent. The monograph
- A. L. Besse "Einstein Manifolds", (1987), Springer,
gives a comprehensive overview of the entire field, although it is extremely advanced. (By the end of the course you should be able to read this book.)
|Prerequisites / Notice||Familiarity with all the material from Differential Geometry I will be assumed (smooth manifolds, Lie groups, vector bundles, differential forms, integration on manifolds, principal bundles and so on). Lecture notes for Differential Geometry I can be found on my website.|
|401-3462-00L||Functional Analysis II||W||10 credits||4V + 1U||A. Carlotto|
|Abstract||Sobolev spaces, weak solutions of elliptic boundary value problems, basic results in elliptic regularity theory (including Schauder estimates), maximum principles.|
|Objective||Acquire fluency with Sobolev spaces and weak derivatives on the one hand, and basic elliptic regularity on the other. Apply these methods for studying elliptic boundary value problems.|
|Literature||Michael Struwe. Funktionalanalysis I und II. Lecture notes, ETH Zürich, 2013/14.|
Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.
Luigi Ambrosio, Alessandro Carlotto, Annalisa Massaccesi. Lectures on elliptic partial differential equations. Springer - Edizioni della Normale, Pisa, 2018.
David Gilbarg, Neil Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin, 2001.
Qing Han, Fanghua Lin. Elliptic partial differential equations. Second edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.
Michael Taylor. Partial differential equations I. Basic theory. Second edition. Applied Mathematical Sciences, 115. Springer, New York, 2011.
Lars Hörmander. The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Springer, Berlin, 2003.
|Prerequisites / Notice||Functional Analysis I plus a solid background in measure theory, Lebesgue integration and L^p spaces.|
|401-3052-10L||Graph Theory||W||10 credits||4V + 1U||B. Sudakov|
|Abstract||Basics, trees, Caley's formula, matrix tree theorem, connectivity, theorems of Mader and Menger, Eulerian graphs, Hamilton cycles, theorems of Dirac, Ore, Erdös-Chvatal, matchings, theorems of Hall, König, Tutte, planar graphs, Euler's formula, Kuratowski's theorem, graph colorings, Brooks' theorem, 5-colorings of planar graphs, list colorings, Vizing's theorem, Ramsey theory, Turán's theorem|
|Objective||The students will get an overview over the most fundamental questions concerning graph theory. We expect them to understand the proof techniques and to use them autonomously on related problems.|
|Lecture notes||Lecture will be only at the blackboard.|
|Literature||West, D.: "Introduction to Graph Theory"|
Diestel, R.: "Graph Theory"
Further literature links will be provided in the lecture.
|Prerequisites / Notice||Students are expected to have a mathematical background and should be able to write rigorous proofs.|
|401-4422-21L||An Introduction to the Calculus of Variations||W||4 credits||2V||A. Figalli|
|Abstract||Calculus of variations is a fundamental tool in mathematical analysis, used to investigate the existence, uniqueness, and properties of minimizers to variational problems.|
Classic examples include, for instance, the existence of the shortest curve between two points, the equilibrium shape of an elastic membrane, and so on.
|Content||In the course, we will study both 1-dimensional and multi-dimensional problems.|
|Prerequisites / Notice||Basic knowledge of Sobolev spaces is important, so some extra additional readings would be required for those unfamiliar with the topic.|
|401-4816-21L||Mathematical Aspects of Classical and Quantum Field Theory||W||8 credits||4V||M. Schiavina, University lecturers|
|Abstract||The course will cover foundational topics in classical and quantum field theory from a mathematical standpoint.|
Starting from the example of classical mechanics, the relevant mathematical foundations that are necessary for a rigorous approach to field theory will be provided.
|Objective||The objective of this course is to expose master and graduate students in mathematics and physics to the mathematical foundations of classical and quantum field theory.|
The course will provide a solid mathematical foundation to essential topics in classical and quantum field theories, both useful to mathematics master and graduate students with an interest but no previous background in QFT, as well as for physics master and graduate students who want to focus on more formal aspects of field theory.
|Content||Abstract (long version)|
The course will cover foundational topics in classical and quantum field theory from a mathematical standpoint.
Starting from the example of classical mechanics, the relevant mathematical foundations that are necessary for a rigorous approach to field theory will be provided.
The course will feature relevant instances of field theories and sigma models, and it will provide a first introduction the the concepts of quantisation, from mechanics to field theory.
Using scalar field theory and quantum electrodynamics as guideline, the course will present an overview of quantum field theory, focusing on its more mathematical aspects, including, if time permits, a modern approach to gauge theories and the renormalisation group.
The course will start with an overview of geometric concepts that will be used throughout, such as graded differential geometry, as well as fiber and vector bundles.
After brief review of classical mechanics, interpreted as a first example of a field theory, a thorough discussion of classical, local, Lagrangian field theory will follow, covering topics such as Noether’s Theorems, local and global symmetries. We will then present and discuss a number of examples from gauge theory.
In the second part of the course, quantisation will be discussed. The main examples of the scalar field and electrodynamics will be used as a guideline for more general considerations on the quantisation of more general and involved field theories.
In the last part of the course, we plan the discussion of modern approaches to quantisation of field theories with symmetries, renormalisation, and of the open challenges that arise.
|402-0844-00L||Quantum Field Theory II|
UZH students are not allowed to register this course unit at ETH. They must book the corresponding module directly at UZH.
|W||10 credits||3V + 2U||N. Beisert|
|Abstract||The subject of the course is modern applications of quantum field theory with emphasis on the quantization of non-abelian gauge theories.|
|Objective||The goal of this course is to lay down the path integral formulation of quantum field theories and in particular to provide a solid basis for the study of non-abelian gauge theories and of the Standard Model|
|Content||The following topics will be covered:|
- path integral quantization
- non-abelian gauge theories and their quantization
- systematics of renormalization, including BRST symmetries, Slavnov-Taylor Identities and the Callan-Symanzik equation
- the Goldstone theorem and the Higgs mechanism
- gauge theories with spontaneous symmetry breaking and their quantization
- renormalization of spontaneously broken gauge theories and quantum effective actions
|Literature||M.E. Peskin and D.V. Schroeder, "An introduction to Quantum Field Theory", Perseus (1995).|
S. Pokorski, "Gauge Field Theories" (2nd Edition), Cambridge Univ. Press (2000)
P. Ramond, "Field Theory: A Modern Primer" (2nd Edition), Westview Press (1990)
S. Weinberg, "The Quantum Theory of Fields" (Volume 2), CUP (1996).
|402-0810-00L||Computational Quantum Physics|
Special Students UZH must book the module PHY522 directly at UZH.
|W||8 credits||2V + 2U||M. H. Fischer|
|Abstract||This course provides an introduction to simulation methods for quantum systems. Starting from the one-body problem, a special emphasis is on quantum many-body problems, where we cover both approximate methods (Hartree-Fock, density functional theory) and exact methods (exact diagonalization, matrix product states, and quantum Monte Carlo methods).|
|Objective||Through lectures and practical programming exercises, after this course:|
Students are able to describe the difficulties of quantum mechanical simulations.
Students are able to explain the strengths and weaknesses of the methods covered.
Students are able to select an appropriate method for a given problem.
Students are able to implement basic versions of all algorithms discussed.
|Lecture notes||A script for this lecture will be provided.|
|Literature||A list of additional references will be provided in the script.|
|Prerequisites / Notice||A basic knowledge of quantum mechanics, numerical tools (numerical differentiation and integration, linear solvers, eigensolvers, root solvers, optimization), and a programming language (for the teaching assignments, you are free to choose your preferred one).|
|401-3917-00L||Stochastic Loss Reserving Methods||W||4 credits||2V||R. Dahms|
|Abstract||Loss Reserving is one of the central topics in non-life insurance. Mathematicians and actuaries need to estimate adequate reserves for liabilities caused by claims. These claims reserves have influence all financial statements, future premiums and solvency margins. We present the stochastics behind various methods that are used in practice to calculate those loss reserves.|
|Objective||Our goal is to present the stochastics behind various methods that are used in prctice to estimate claim reserves. These methods enable us to set adequate reserves for liabilities caused by claims and to determine prediction errors of these predictions.|
|Content||We will present the following stochastic claims reserving methods/models:|
- Stochastic Chain-Ladder Method
- Bayesian Methods, Bornhuetter-Ferguson Method, Credibility Methods
- Distributional Models
- Linear Stochastic Reserving Models, with and without inflation
- Bootstrap Methods
- Claims Development Result (solvency view)
- Coupling of portfolios
|Literature||M. V. Wüthrich, M. Merz, Stochastic Claims Reserving Methods in Insurance, Wiley 2008.|
|Prerequisites / Notice||The exams ONLY take place during the official ETH examination periods.|
This course will be held in English and counts towards the diploma "Aktuar SAV".
For the latter, see details under www.actuaries.ch.
Basic knowledge in probability theory is assumed, in particular conditional expectations.
|401-3642-00L||Brownian Motion and Stochastic Calculus||W||10 credits||4V + 1U||W. Werner|
|Abstract||This course covers some basic objects of stochastic analysis. In particular, the following topics are discussed: construction and properties of Brownian motion, stochastic integration, Ito's formula and applications, stochastic differential equations and connection with partial differential equations.|
|Objective||This course covers some basic objects of stochastic analysis. In particular, the following topics are discussed: construction and properties of Brownian motion, stochastic integration, Ito's formula and applications, stochastic differential equations and connection with partial differential equations.|
|Lecture notes||Lecture notes will be distributed in class.|
|Literature||- J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer (2016).|
- I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, Springer (1991).
- D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer (2005).
- L.C.G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales, vol. 1 and 2, Cambridge University Press (2000).
- D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer (2006).
|Prerequisites / Notice||Familiarity with measure-theoretic probability as in the standard D-MATH course "Probability Theory" will be assumed. Textbook accounts can be found for example in |
- J. Jacod, P. Protter, Probability Essentials, Springer (2004).
- R. Durrett, Probability: Theory and Examples, Cambridge University Press (2010).
|401-4611-21L||Rough Path Theory||W||4 credits||2V||A. Allan, J. Teichmann|
|Abstract||The aim of this course is to provide an introduction to the theory of rough paths, with a particular focus on their integration theory and associated rough differential equations, and how the theory relates to and enhances the field of stochastic calculus.|
|Objective||Our first motivation will be to understand the limitations of classical notions of integration to handle paths of very low regularity, and to see how the rough integral succeeds where other notions fail. We will construct rough integrals and establish solutions of differential equations driven by rough paths, as well as the continuity of these objects with respect to the paths involved, and their consistency with stochastic integration and SDEs. Various applications and extensions of the theory will then be discussed.|
|Lecture notes||Lecture notes will be provided by the lecturer.|
|Literature||P. K. Friz and M. Hairer, A course on rough paths with an introduction to regularity structures, Springer (2014).|
P. K. Friz and N. B. Victoir. Multidimensional stochastic processes as rough paths, Cambridge University Press (2010).
|Prerequisites / Notice||The aim will be to make the course as self-contained as possible, but some knowledge of stochastic analysis is highly recommended. The course “Brownian Motion and Stochastic Calculus” would be ideal, but not strictly required.|
|263-4400-00L||Advanced Graph Algorithms and Optimization||W||8 credits||3V + 1U + 3A||R. Kyng, M. Probst|
|Abstract||This course will cover a number of advanced topics in optimization and graph algorithms.|
|Objective||The course will take students on a deep dive into modern approaches to|
graph algorithms using convex optimization techniques.
By studying convex optimization through the lens of graph algorithms,
students should develop a deeper understanding of fundamental
phenomena in optimization.
The course will cover some traditional discrete approaches to various graph
problems, especially flow problems, and then contrast these approaches
with modern, asymptotically faster methods based on combining convex
optimization with spectral and combinatorial graph theory.
|Content||Students should leave the course understanding key|
concepts in optimization such as first and second-order optimization,
convex duality, multiplicative weights and dual-based methods,
acceleration, preconditioning, and non-Euclidean optimization.
Students will also be familiarized with central techniques in the
development of graph algorithms in the past 15 years, including graph
decomposition techniques, sparsification, oblivious routing, and
spectral and combinatorial preconditioning.
|Prerequisites / Notice||This course is targeted toward masters and doctoral students with an|
interest in theoretical computer science.
Students should be comfortable with design and analysis of algorithms, probability, and linear algebra.
Having passed the course Algorithms, Probability, and Computing (APC) is highly recommended, but not formally required. If you are not
sure whether you're ready for this class or not, please consult the
|401-3902-21L||Network & Integer Optimization: From Theory to Application||W||6 credits||3G||R. Zenklusen|
|Abstract||This course covers various topics in Network and (Mixed-)Integer Optimization. It starts with a rigorous study of algorithmic techniques for some network optimization problems (with a focus on matching problems) and moves to key aspects of how to attack various optimization settings through well-designed (Mixed-)Integer Programming formulations.|
|Objective||Our goal is for students to both get a good foundational understanding of some key network algorithms and also to learn how to effectively employ (Mixed-)Integer Programming formulations, techniques, and solvers, to tackle a wide range of discrete optimization problems.|
|Content||Key topics include:|
- Matching problems;
- Integer Programming techniques and models;
- Extended formulations and strong problem formulations;
- Solver techniques for (Mixed-)Integer Programs;
- Decomposition approaches.
|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.
- Vanderbeck François, Wolsey Laurence: Reformulations and Decomposition of Integer Programs. Chapter 13 in: 50 Years of Integer Programming 1958-2008. Springer, 2010.
- Alexander Schrijver: Theory of Linear and Integer Programming. John Wiley, 1986.
|Prerequisites / Notice||Solid background in linear algebra. Preliminary knowledge of Linear Programming is ideal but not a strict requirement. Prior attendance of the course Mathematical Optimization is a plus.|
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