# Suchergebnis: Katalogdaten im Herbstsemester 2022

Doktorat Mathematik Mehr Informationen unter: Link | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Vertiefung Fachwissen Die Liste der Lehrveranstaltungen (samt der zugehörigen Anzahl Kreditpunkte) für Doktoratsstudentinnen und Doktoratsstudenten wird jedes Semester im Newsletter der ZGSM veröffentlicht. Link | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Graduate School Offizielle Website der Zurich Graduate School in Mathematics: Link | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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401-5003-72L | A PDE Approach to Mean-Field Disordered Systems Only for ETH D-MATH doctoral students and for doctoral students from the Institute of Mathematics at UZH. The latter need to send an email to Jessica Bolsinger (Link) with the course number. The email should have the subject „Graduate course registration (ETH)“. | W | 2 KP | 2V | J.‑C. Mourrat | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Nachdiplom lecture | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | The goal of statistical mechanics is to describe the large-scale behavior of collections of simple elements, often called spins, that interact through locally simple rules and are influenced by some amount of noise. We will discuss three classes of such models, of increasing difficulty, and will rely on a common PDE approach to study each of them. The first model we will study is the very simple Curie-Weiss model, in which every spin interacts with every other spin and has a preference for being aligned with the others. This model can be solved in a variety of ways, but will be used to develop our toolkit based on the study of certain Hamilton-Jacobi equations that naturally arise. We will then turn to a more challenging class of models coming from statistical inference. We will focus on a setup in which we observe a noisy version of a large rank-one matrix. We will compute the information-theoretic limit to the recovery of this matrix based on the PDE techniques introduced earlier. We will finally discuss spin-glass models, in which the local interactions between the spins are disordered. One of the core motivations for the development of the techniques presented here is to uncover the behavior of models in which spins can be of different types, such as for instance when the spins are organized over two layers, and only have direct interactions across layers. While the understanding of this class of models is still very limited, I will present some progress towards this goal. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | The prerequisites for these lectures are basic measure theory and probability theory. No prior knowledge of PDE theory will be assumed. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3059-00L | Kombinatorik IIFindet dieses Semester nicht statt. | W | 4 KP | 2G | N. Hungerbühler | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Der Kurs Kombinatorik I und II ist eine Einführung in die abzählende Kombinatorik. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | Die Studierenden sind in der Lage, kombinatorische Probleme einzuordnen und die adaequaten Techniken zu deren Loesung anzuwenden. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Inhalt der Vorlesungen Kombinatorik I und II: Kongruenztransformationen der Ebene, Symmetriegruppen von geometrischen Figuren, Eulersche Funktion, Cayley-Graphen, formale Potenzreihen, Permutationsgruppen, Zyklen, Lemma von Burnside, Zyklenzeiger, Saetze von Polya, Anwendung auf die Graphentheorie und isomere Molekuele. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3055-64L | Algebraic Methods in Combinatorics Findet dieses Semester nicht statt. | W | 6 KP | 2V + 1U | B. Sudakov | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. This course provides a gentle introduction to Algebraic methods, illustrated by examples and focusing on basic ideas and connections to other areas. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | The students will get an overview of various algebraic methods for solving combinatorial problems. We expect them to understand the proof techniques and to use them autonomously on related problems. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern theory has grown out of this early stage and often relies on deep, well-developed tools. One of the main general techniques that played a crucial role in the development of Combinatorics was the application of algebraic methods. The most fruitful such tool is the dimension argument. Roughly speaking, the method can be described as follows. In order to bound the cardinality of of a discrete structure A one maps its elements to vectors in a linear space, and shows that the set A is mapped to linearly independent vectors. It then follows that the cardinality of A is bounded by the dimension of the corresponding linear space. This simple idea is surprisingly powerful and has many famous applications. This course provides a gentle introduction to Algebraic methods, illustrated by examples and focusing on basic ideas and connections to other areas. The topics covered in the class will include (but are not limited to): Basic dimension arguments, Spaces of polynomials and tensor product methods, Eigenvalues of graphs and their application, the Combinatorial Nullstellensatz and the Chevalley-Warning theorem. Applications such as: Solution of Kakeya problem in finite fields, counterexample to Borsuk's conjecture, chromatic number of the unit distance graph of Euclidean space, explicit constructions of Ramsey graphs and many others. The course website can be found at Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | Lectures will be on the blackboard only, but there will be a set of typeset lecture notes which follow the class closely. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | Students are expected to have a mathematical background and should be able to write rigorous proofs. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-4785-00L | Mathematical and Computational Methods in PhotonicsFindet dieses Semester nicht statt. | W | 8 KP | 4G | H. Ammari | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | The aim of this course is to review new and fundamental mathematical tools, computational approaches, and inversion and optimal design methods used to address challenging problems in nanophotonics. The emphasis will be on analyzing plasmon resonant nanoparticles, super-focusing & super-resolution of electromagnetic waves, photonic crystals, electromagnetic cloaking, metamaterials, and metasurfaces | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | The field of photonics encompasses the fundamental science of light propagation and interactions in complex structures, and its technological applications. The recent advances in nanoscience present great challenges for the applied and computational mathematics community. In nanophotonics, the aim is to control, manipulate, reshape, guide, and focus electromagnetic waves at nanometer length scales, beyond the resolution limit. In particular, one wants to break the resolution limit by reducing the focal spot and confine light to length scales that are significantly smaller than half the wavelength. Interactions between the field of photonics and mathematics has led to the emergence of a multitude of new and unique solutions in which today's conventional technologies are approaching their limits in terms of speed, capacity and accuracy. Light can be used for detection and measurement in a fast, sensitive and accurate manner, and thus photonics possesses a unique potential to revolutionize healthcare. Light-based technologies can be used effectively for the very early detection of diseases, with non-invasive imaging techniques or point-of-care applications. They are also instrumental in the analysis of processes at the molecular level, giving a greater understanding of the origin of diseases, and hence allowing prevention along with new treatments. Photonic technologies also play a major role in addressing the needs of our ageing society: from pace-makers to synthetic bones, and from endoscopes to the micro-cameras used in in-vivo processes. Furthermore, photonics are also used in advanced lighting technology, and in improving energy efficiency and quality. By using photonic media to control waves across a wide band of wavelengths, we have an unprecedented ability to fabricate new materials with specific microstructures. The main objective in this course is to report on the use of sophisticated mathematics in diffractive optics, plasmonics, super-resolution, photonic crystals, and metamaterials for electromagnetic invisibility and cloaking. The book merges highly nontrivial multi-mathematics in order to make a breakthrough in the field of mathematical modelling, imaging, and optimal design of optical nanodevices and nanostructures capable of light enhancement, and of the focusing and guiding of light at a subwavelength scale. We demonstrate the power of layer potential techniques in solving challenging problems in photonics, when they are combined with asymptotic analysis and the elegant theory of Gohberg and Sigal on meromorphic operator-valued functions. In this course we shall consider both analytical and computational matters in photonics. The issues we consider lead to the investigation of fundamental problems in various branches of mathematics. These include asymptotic analysis, spectral analysis, mathematical imaging, optimal design, stochastic modelling, and analysis of wave propagation phenomena. On the other hand, deriving mathematical foundations, and new and efficient computational frameworks and tools in photonics, requires a deep understanding of the different scales in the wave propagation problem, an accurate mathematical modelling of the nanodevices, and fine analysis of complex wave propagation phenomena. An emphasis is put on mathematically analyzing plasmon resonant nanoparticles, diffractive optics, photonic crystals, super-resolution, and metamaterials. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3627-00L | High-Dimensional StatisticsFindet dieses Semester nicht statt. | W | 4 KP | 2V | P. L. Bühlmann | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | "High-Dimensional Statistics" deals with modern methods and theory for statistical inference when the number of unknown parameters is of much larger order than sample size. Statistical estimation and algorithms for complex models and aspects of multiple testing will be discussed. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | Knowledge of methods and basic theory for high-dimensional statistical inference | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Lasso and Group Lasso for high-dimensional linear and generalized linear models; Additive models and many smooth univariate functions; Non-convex loss functions and l1-regularization; Stability selection, multiple testing and construction of p-values; Undirected graphical modeling | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | Peter Bühlmann and Sara van de Geer (2011). Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer Verlag. ISBN 978-3-642-20191-2. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | Knowledge of basic concepts in probability theory, and intermediate knowledge of statistics (e.g. a course in linear models or computational statistics). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3628-14L | Bayesian StatisticsFindet dieses Semester nicht statt. | W | 4 KP | 2V | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Introduction 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | Students understand the conceptual ideas behind Bayesian statistics and are familiar with common techniques used in Bayesian data analysis. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Topics 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) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | A script will be available in English. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | Christian 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 / Besonderes | Familiarity 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

402-0861-00L | Statistical Physics | W | 10 KP | 4V + 2U | E. Demler | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | This 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | This 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Kinetic 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | Lecture notes available in English. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | No specific book is used for the course. Relevant literature will be given in the course. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

402-0830-00L | General Relativity Fachstudierende UZH müssen das Modul PHY511 direkt an der UZH buchen. | W | 10 KP | 4V + 2U | L. Senatore | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Introduction 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | Basic 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). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Introduction 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | Suggested 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

402-0843-00L | Quantum Field Theory IFachstudierende UZH müssen das Modul PHY551 direkt an der UZH buchen. | W | 10 KP | 4V + 2U | R. Renner | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | This 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 - Radiative corrections | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | The 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | Will be provided as the course progresses | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kompetenzen |
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402-0897-00L | Introduction to String TheoryFindet dieses Semester nicht statt. | W | 6 KP | 2V + 1U | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | String 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | Within 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) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | D. 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 / Besonderes | Recommended: Quantum Field Theory I (in parallel) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

252-0417-00L | Randomized Algorithms and Probabilistic Methods | W | 10 KP | 3V + 2U + 4A | A. Steger | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Las 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | After 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Randomized 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | Yes. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | - Randomized Algorithms, Rajeev Motwani and Prabhakar Raghavan, Cambridge University Press (1995) - Probability and Computing, Michael Mitzenmacher and Eli Upfal, Cambridge University Press (2005) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

227-0447-00L | Image Analysis and Computer Vision | W | 6 KP | 3V + 1U | E. Konukoglu, F. Yu | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Light 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | Overview of the most important concepts of image formation, perception and analysis, and Computer Vision. Gaining own experience through practical computer and programming exercises. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | This 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | Course material Script, computer demonstrations, exercises and problem solutions | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | Prerequisites: Basic concepts of mathematical analysis and linear algebra. The computer exercises are based on Python and Linux. The course language is English. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

227-0417-00L | Information Theory I | W | 6 KP | 4G | A. Lapidoth | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | This 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | The fundamentals of Information Theory including Shannon's source coding and channel coding theorems | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | The 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | T.M. Cover and J. Thomas, Elements of Information Theory (second edition) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

227-0423-00L | Neural Network Theory Findet dieses Semester nicht statt. | W | 4 KP | 2V + 1U | H. Bölcskei | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | The 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | After 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | 1. 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | Detailed lecture notes are available on the course web page Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | This course is aimed at students with a strong mathematical background in general, and in linear algebra, analysis, and probability theory in particular. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

151-0563-01L | Dynamic Programming and Optimal Control | W | 4 KP | 2V + 1U | R. D'Andrea | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Introduction to Dynamic Programming and Optimal Control. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | Covers the fundamental concepts of Dynamic Programming & Optimal Control. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Dynamic Programming Algorithm; Deterministic Systems and Shortest Path Problems; Infinite Horizon Problems, Bellman Equation; Deterministic Continuous-Time Optimal Control. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | Dynamic Programming and Optimal Control by Dimitri P. Bertsekas, Vol. I, 3rd edition, 2005, 558 pages, hardcover. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | Requirements: Knowledge of advanced calculus, introductory probability theory, and matrix-vector algebra. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3225-DRL | Introduction to Lie Groups | W | 3 KP | 4G | M. Burger | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Topological groups and Haar measure. Definition of Lie groups, examples of local fields and examples of discrete subgroups; basic properties; Lie subgroups. Lie algebras and relation with Lie groups: exponential map, adjoint representation. Semisimplicity, nilpotency, solvability, compactness: Killing form, Lie's and Engel's theorems. Definition of algebraic groups and relation with Lie groups. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | The goal is to have a broad though foundational knowledge of the theory of Lie groups and their associated Lie algebras with an emphasis on the algebraic and topological aspects of it. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | A. Knapp: "Lie groups beyond an Introduction" (Birkhaeuser) A. Sagle & R. Walde: "Introduction to Lie groups and Lie algebras" (Academic Press, '73) F. Warner: "Foundations of differentiable manifolds and Lie groups" (Springer) H. Samelson: "Notes on Lie algebras" (Springer, '90) S. Helgason: "Differential geometry, Lie groups and symmetric spaces" (Academic Press, '78) A. Knapp: "Lie groups, Lie algebras and cohomology" (Princeton University Press) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | Topology and basic notions of measure theory. A basic understanding of the concepts of manifold, tangent space and vector field is useful, but could also be achieved throughout the semester. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3001-DRL | Algebraic Topology I | W | 3 KP | 4G | S. Kalisnik Hintz | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | This is an introductory course in algebraic topology, which is the study of algebraic invariants of topological spaces. Topics covered include: singular homology, cell complexes and cellular homology, the Eilenberg-Steenrod axioms. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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Literatur | 1) G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. 2) A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. Book can be downloaded for free at: Link See also: Link 3) E. Spanier, "Algebraic topology", Springer-Verlag | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | You should know the basics of point-set topology. Useful to have (though not absolutely necessary) basic knowledge of the fundamental group and covering spaces (at the level covered in the course "topology"). Some knowledge of differential geometry and differential topology is useful but not strictly necessary. Some (elementary) group theory and algebra will also be needed. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3533-DRL | Generalized Nonpositive Curvature | W | 3 KP | 3V | U. Lang | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | CAT(0) spaces, Busemann convex spaces, metric spaces with convex geodesic bicombings; injective metric spaces and injective hulls; Gromov hyperbolicity, Helly graphs and Helly groups; fixed points, barycenter constructions, and applications. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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Skript | Lectures notes will be provided. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | - M. R. Bridson, A. Haefliger: Metric Spaces of Non-Positive Curvature, Springer 1999 - A. Papadopoulos: Metric Spaces, Convexity and Nonpositive Curvature, EMS 2005 - U. Lang: Injective hulls of certain discrete metric spaces and groups, J. Topol. Anal. 5 (2013), 297-331 - D. Descombes, U. Lang: Convex geodesic bicombings and hyperbolicity, Geom. Dedicata 177 (2015), 367-384 - J. Chalopin et al.: Weakly Modular Graphs and Nonpositive Curvature, Memoirs AMS 268 (2020), no. 1309 - J. Chalopin et al.: Helly groups, arXiv:2002.06895v2 [math.GR] | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | Basic knowledge of Riemannian geometry and functional analysis will be assumed. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kompetenzen |
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401-4657-DRL | Numerical Solution of Stochastic Ordinary Differential Equations Alternative course titles: "Numerical Analysis of Stochastic Ordinary Differential Equations" / "Computational Methods for Quantitative Finance: Monte Carlo and Sampling Methods" Only for ZGSM (ETH D-MATH and UZH I-MATH) doctoral students. The latter need to register at myStudies and then send an email to Link with their name, course number and student ID. Please see Link | W | 3 KP | 3V + 1U | A. Stein | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | This course is on the numerical approximations of stochastic ordinary differential equations (SDEs) driven by Brownian motions and Lévy processes. SDEs have several applications, for example in financial engineering. The contents cover stochastic processes, stochastic calculus, well-posedness results for SDEs, strong and weak approximations of SDEs, and simulation via Monte Carlo methods. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | The aim of this course is to enable the students to carry out simulations and their mathematical convergence analysis for stochastic models originating from applications such as mathematical finance. For this the course teaches a decent knowledge of the different numerical methods, their underlying ideas, convergence properties and implementation issues. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Brownian motion and Lévy processes Stochastic integration and stochastic calculus Stochastic ordinary differential equations (SDEs) Numerical approximations of SDEs Stochastic simulation and Monte Carlo methods Applications to computational finance: Option valuation | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | There will be English, typed lecture notes for registered participants in the course. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | P. E. Kloeden and E. Platen: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin, 1992. P. Glassermann: Monte Carlo Methods in Financial Engineering. Springer-Verlag, New York, 2004. D. Applebaum: Lévy Processes and Stochastic Calculus. Cambridge University Press, 2009. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | Prerequisites: Mandatory: Probability and measure theory, basic numerical analysis and basics of MATLAB/Python programming. a) mandatory courses: Elementary Probability, Probability Theory I. b) recommended courses: Stochastic Processes. Start of lectures: Wednesday September 21, 2022. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-4944-DRL | Mathematics of Data Science | W | 2 KP | 4G | A. Bandeira | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Mostly self-contained, but fast-paced, introductory masters level course on various theoretical aspects of algorithms that aim to extract information from data. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | Introduction to various mathematical aspects of Data Science. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | These topics lie in overlaps of (Applied) Mathematics with: Computer Science, Electrical Engineering, Statistics, and/or Operations Research. Each lecture will feature a couple of Mathematical Open Problem(s) related to Data Science. The main mathematical tools used will be Probability and Linear Algebra, and a basic familiarity with these subjects is required. There will also be some (although knowledge of these tools is not assumed) Graph Theory, Representation Theory, Applied Harmonic Analysis, among others. The topics treated will include Dimension reduction, Manifold learning, Sparse recovery, Random Matrices, Approximation Algorithms, Community detection in graphs, and several others. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | The main mathematical tools used will be Probability, Linear Algebra (and real analysis), and a working knowledge of these subjects is required. In addition to these prerequisites, this class requires a certain degree of mathematical maturity--including abstract thinking and the ability to understand and write proofs. We encourage students who are interested in mathematical data science to take both this course and ``227-0434-10L Mathematics of Information'' taught by Prof. H. Bölcskei. The two courses are designed to be complementary. A. Bandeira and H. Bölcskei |

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