# Suchergebnis: Katalogdaten im Herbstsemester 2021

Doktorat Departement Mathematik

Die Liste der Lehrveranstaltungen (samt der zugehörigen Anzahl Kreditpunkte) für Doktoratsstudentinnen und Doktoratsstudenten wird jedes Semester im Newsletter der ZGSM veröffentlicht.
ACHTUNG: Kreditpunkte fürs Doktoratsstudium sind nicht mit ECTS-Kreditpunkten zu verwechseln!
Offizielle Website der Zurich Graduate School in Mathematics:
NummerTitelTypECTSUmfangDozierende
401-3628-14LBayesian StatisticsW4 KP2VF. Sigrist
KurzbeschreibungIntroduction to the Bayesian approach to statistics: decision theory, prior distributions, hierarchical Bayes models, empirical Bayes, Bayesian tests and model selection, empirical Bayes, Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods.
LernzielStudents understand the conceptual ideas behind Bayesian statistics and are familiar with common techniques used in Bayesian data analysis.
InhaltTopics that we will discuss are:

Difference between the frequentist and Bayesian approach (decision theory, principles), priors (conjugate priors, noninformative priors, Jeffreys prior), tests and model selection (Bayes factors, hyper-g priors for regression),hierarchical models and empirical Bayes methods, computational methods (Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods)
SkriptA script will be available in English.
LiteraturChristian Robert, The Bayesian Choice, 2nd edition, Springer 2007.

A. Gelman et al., Bayesian Data Analysis, 3rd edition, Chapman & Hall (2013).

Additional references will be given in the course.
Voraussetzungen / BesonderesFamiliarity with basic concepts of frequentist statistics and with basic concepts of probability theory (random variables, joint and conditional distributions, laws of large numbers and central limit theorem) will be assumed.
401-4889-00LMathematical Finance W11 KP4V + 2UD. Possamaï
- semimartingales and general stochastic integration
- absence of arbitrage and martingale measures
- fundamental theorem of asset pricing
- option pricing and hedging
- hedging duality
- optimal investment problems
LernzielAdvanced course on mathematical finance, presupposing good knowledge in probability theory and stochastic calculus (for continuous processes)
InhaltThis is an advanced course on mathematical finance for students with a good background in probability. We want to give an overview of main concepts, questions and approaches, and we do this mostly in continuous-time models.

Topics include
- semimartingales and general stochastic integration
- absence of arbitrage and martingale measures
- fundamental theorem of asset pricing
- option pricing and hedging
- hedging duality
- optimal investment problems
- and probably others
SkriptThe course is based on different parts from different books as well as on original research literature.

Lecture notes will not be available.
Literatur(will be updated later)
Voraussetzungen / BesonderesPrerequisites are the standard courses
- Probability Theory (for which lecture notes are available)
- Brownian Motion and Stochastic Calculus (for which lecture notes are available)
Those students who already attended "Introduction to Mathematical Finance" will have an advantage in terms of ideas and concepts.

This course is the second of a sequence of two courses on mathematical finance. The first course "Introduction to Mathematical Finance" (MF I), 401-3888-00, focuses on models in finite discrete time. It is advisable that the course MF I is taken prior to the present course, MF II.

For an overview of courses offered in the area of mathematical finance, see Link.
402-0861-00LStatistical PhysicsW10 KP4V + 2UM. Sigrist
KurzbeschreibungThis lecture covers the concepts of classical and quantum statistical physics. Several techniques such as second quantization formalism for fermions, bosons, photons and phonons as well as mean field theory and self-consistent field approximation. These are used to discuss phase transitions, critical phenomena and superfluidity.
LernzielThis lecture gives an introduction in the basic concepts and applications of statistical physics for the general use in physics and, in particular, as a preparation for the theoretical solid state physics education.
InhaltKinetic approach to statistical physics: H-theorem, detailed balance and equilibirium conditions.
Classical statistical physics: microcanonical ensembles, canonical ensembles and grandcanonical ensembles, applications to simple systems.
Quantum statistical physics: density matrix, ensembles, Fermi gas, Bose gas (Bose-Einstein condensation), photons and phonons.
Identical quantum particles: many body wave functions, second quantization formalism, equation of motion, correlation functions, selected applications, e.g. Bose-Einstein condensate and coherent state, phonons in elastic media and melting.
One-dimensional interacting systems.
Phase transitions: mean field approach to Ising model, Gaussian transformation, Ginzburg-Landau theory (Ginzburg criterion), self-consistent field approach, critical phenomena, Peierls' arguments on long-range order.
Superfluidity: Quantum liquid Helium: Bogolyubov theory and collective excitations, Gross-Pitaevskii equations, Berezinskii-Kosterlitz-Thouless transition.
SkriptLecture notes available in English.
LiteraturNo specific book is used for the course. Relevant literature will be given in the course.
402-0830-00LGeneral Relativity
Fachstudierende UZH müssen das Modul PHY511 direkt an der UZH buchen.
W10 KP4V + 2UC. Anastasiou
KurzbeschreibungIntroduction to the theory of general relativity. The course puts a strong focus on the mathematical foundations of the theory as well as the underlying physical principles and concepts. It covers selected applications, such as the Schwarzschild solution and gravitational waves.
LernzielBasic understanding of general relativity, its mathematical foundations (in particular the relevant aspects of differential geometry), and some of the phenomena it predicts (with a focus on black holes).
InhaltIntroduction to the theory of general relativity. The course puts a strong focus on the mathematical foundations, such as differentiable manifolds, the Riemannian and Lorentzian metric, connections, and curvature. It discusses the underlying physical principles, e.g., the equivalence principle, and concepts, such as curved spacetime and the energy-momentum tensor. The course covers some basic applications and special cases, including the Newtonian limit, post-Newtonian expansions, the Schwarzschild solution, light deflection, and gravitational waves.
LiteraturSuggested textbooks:

C. Misner, K, Thorne and J. Wheeler: Gravitation
S. Carroll - Spacetime and Geometry: An Introduction to General Relativity
R. Wald - General Relativity
S. Weinberg - Gravitation and Cosmology
402-0843-00LQuantum Field Theory I
Fachstudierende UZH müssen das Modul PHY551 direkt an der UZH buchen.
W10 KP4V + 2UG. M. Graf
KurzbeschreibungThis course discusses the quantisation of fields in order to introduce a coherent formalism for the combination of quantum mechanics and special relativity.
Topics include:
- Relativistic quantum mechanics
- Quantisation of bosonic and fermionic fields
- Interactions in perturbation theory
- Scattering processes and decays
- Elementary processes in QED
LernzielThe goal of this course is to provide a solid introduction to the formalism, the techniques, and important physical applications of quantum field theory. Furthermore it prepares students for the advanced course in quantum field theory (Quantum Field Theory II), and for work on research projects in theoretical physics, particle physics, and condensed-matter physics.
SkriptWill be provided as the course progresses
Kompetenzen
 Fachspezifische Kompetenzen Konzepte und Theorien geprüft Verfahren und Technologien geprüft Methodenspezifische Kompetenzen Analytische Kompetenzen geprüft Entscheidungsfindung gefördert Medien und digitale Technologien gefördert Problemlösung geprüft Projektmanagement gefördert Soziale Kompetenzen Kommunikation gefördert Kooperation und Teamarbeit gefördert Kundenorientierung gefördert Menschenführung und Verantwortung gefördert Selbstdarstellung und soziale Einflussnahme gefördert Sensibilität für Vielfalt gefördert Verhandlung gefördert Persönliche Kompetenzen Anpassung und Flexibilität gefördert Kreatives Denken geprüft Kritisches Denken geprüft Integrität und Arbeitsethik gefördert Selbstbewusstsein und Selbstreflexion gefördert Selbststeuerung und Selbstmanagement gefördert
402-0897-00LIntroduction to String TheoryW6 KP2V + 1UJ. Brödel
KurzbeschreibungString theory is an attempt to quantise gravity and unite it with the other fundamental forces of nature. It is related to numerous interesting topics and questions in quantum field theory. In this course, an introduction to the basics of string theory is provided.
LernzielWithin this course, a basic understanding and overview of the concepts and notions employed in string theory shall be given. More advanced topics will be touched upon towards the end of the course briefly in order to foster further research.
Inhalt- mechanics of point particles and extended objects
- string modes and their quantisation; higher dimensions, supersymmetry
- D-branes, T-duality
- supergravity as a low-energy effective theory, strings on curved backgrounds
- two-dimensional field theories (classical/quantum, conformal/non-conformal)
LiteraturD. Lust, S. Theisen, Lectures on String Theory, Lecture Notes in Physics, Springer (1989).
M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory I, CUP (1987).
B. Zwiebach, A First Course in String Theory, CUP (2004).
J. Polchinski, String Theory I & II, CUP (1998).
Voraussetzungen / BesonderesRecommended: Quantum Field Theory I (in parallel)
252-0417-00LRandomized Algorithms and Probabilistic Methods W10 KP3V + 2U + 4AA. Steger
KurzbeschreibungLas Vegas & Monte Carlo algorithms; inequalities of Markov, Chebyshev, Chernoff; negative correlation; Markov chains: convergence, rapidly mixing; generating functions; Examples include: min cut, median, balls and bins, routing in hypercubes, 3SAT, card shuffling, random walks
LernzielAfter this course students will know fundamental techniques from probabilistic combinatorics for designing randomized algorithms and will be able to apply them to solve typical problems in these areas.
InhaltRandomized Algorithms are algorithms that "flip coins" to take certain decisions. This concept extends the classical model of deterministic algorithms and has become very popular and useful within the last twenty years. In many cases, randomized algorithms are faster, simpler or just more elegant than deterministic ones. In the course, we will discuss basic principles and techniques and derive from them a number of randomized methods for problems in different areas.
SkriptYes.
Literatur- Randomized Algorithms, Rajeev Motwani and Prabhakar Raghavan, Cambridge University Press (1995)
- Probability and Computing, Michael Mitzenmacher and Eli Upfal, Cambridge University Press (2005)
Takes place for the last time.
W9 KP3V + 2U + 3AM. Ghaffari, G. Zuzic
KurzbeschreibungThis is a graduate-level course on algorithm design (and analysis). It covers a range of topics and techniques in approximation algorithms, sketching and streaming algorithms, and online algorithms.
LernzielThis course familiarizes the students with some of the main tools and techniques in modern subareas of algorithm design.
InhaltThe lectures will cover a range of topics, tentatively including the following: graph sparsifications while preserving cuts or distances, various approximation algorithms techniques and concepts, metric embeddings and probabilistic tree embeddings, online algorithms, multiplicative weight updates, streaming algorithms, sketching algorithms, and derandomization.
Voraussetzungen / BesonderesThis course is designed for masters and doctoral students and it especially targets those interested in theoretical computer science, but it should also be accessible to last-year bachelor students.

Sufficient comfort with both (A) Algorithm Design & Analysis and (B) Probability & Concentrations. E.g., having passed the course Algorithms, Probability, and Computing (APC) is highly recommended, though not required formally. If you are not sure whether you're ready for this class or not, please consult the instructor.
227-0447-00LImage Analysis and Computer Vision W6 KP3V + 1UL. Van Gool, E. Konukoglu, F. Yu
KurzbeschreibungLight and perception. Digital image formation. Image enhancement and feature extraction. Unitary transformations. Color and texture. Image segmentation. Motion extraction and tracking. 3D data extraction. Invariant features. Specific object recognition and object class recognition. Deep learning and Convolutional Neural Networks.
LernzielOverview of the most important concepts of image formation, perception and analysis, and Computer Vision. Gaining own experience through practical computer and programming exercises.
InhaltThis course aims at offering a self-contained account of computer vision and its underlying concepts, including the recent use of deep learning.
The first part starts with an overview of existing and emerging applications that need computer vision. It shows that the realm of image processing is no longer restricted to the factory floor, but is entering several fields of our daily life. First the interaction of light with matter is considered. The most important hardware components such as cameras and illumination sources are also discussed. The course then turns to image discretization, necessary to process images by computer.
The next part describes necessary pre-processing steps, that enhance image quality and/or detect specific features. Linear and non-linear filters are introduced for that purpose. The course will continue by analyzing procedures allowing to extract additional types of basic information from multiple images, with motion and 3D shape as two important examples. Finally, approaches for the recognition of specific objects as well as object classes will be discussed and analyzed. A major part at the end is devoted to deep learning and AI-based approaches to image analysis. Its main focus is on object recognition, but also other examples of image processing using deep neural nets are given.
SkriptCourse material Script, computer demonstrations, exercises and problem solutions
Voraussetzungen / BesonderesPrerequisites:
Basic concepts of mathematical analysis and linear algebra. The computer exercises are based on Python and Linux.
The course language is English.
227-0417-00LInformation Theory IW6 KP4GA. Lapidoth
KurzbeschreibungThis course covers the basic concepts of information theory and of communication theory. Topics covered include the entropy rate of a source, mutual information, typical sequences, the asymptotic equi-partition property, Huffman coding, channel capacity, the channel coding theorem, the source-channel separation theorem, and feedback capacity.
LernzielThe fundamentals of Information Theory including Shannon's source coding and channel coding theorems
InhaltThe entropy rate of a source, Typical sequences, the asymptotic equi-partition property, the source coding theorem, Huffman coding, Arithmetic coding, channel capacity, the channel coding theorem, the source-channel separation theorem, feedback capacity
LiteraturT.M. Cover and J. Thomas, Elements of Information Theory (second edition)
227-0423-00LNeural Network Theory W4 KP2V + 1UH. Bölcskei
KurzbeschreibungThe class focuses on fundamental mathematical aspects of neural networks with an emphasis on deep networks: Universal approximation theorems, capacity of separating surfaces, generalization, fundamental limits of deep neural network learning, VC dimension.
LernzielAfter attending this lecture, participating in the exercise sessions, and working on the homework problem sets, students will have acquired a working knowledge of the mathematical foundations of neural networks.
Inhalt1. Universal approximation with single- and multi-layer networks

2. Introduction to approximation theory: Fundamental limits on compressibility of signal classes, Kolmogorov epsilon-entropy of signal classes, non-linear approximation theory

3. Fundamental limits of deep neural network learning

4. Geometry of decision surfaces

5. Separating capacity of nonlinear decision surfaces

6. Vapnik-Chervonenkis (VC) dimension

7. VC dimension of neural networks

8. Generalization error in neural network learning
SkriptDetailed lecture notes are available on the course web page