# Search result: Catalogue data in Autumn Semester 2020

Mathematics Master | ||||||

Seminars and Semester Papers | ||||||

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 afterwards choose an alternative seminar. | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |
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401-3180-61L | Categories and Derived Functors Mathematics Bachelor or Master programmes with preference for Mathematics Bachelor 5th semester | W | 4 credits | 2S | R. Pink | |

Abstract | Categories, functors, natural transformations, limits, colimits, adjoint functors, additive & abelian categories, exact sequences, diagram lemmas, injectives, projectives, Mitchell's embedding theorem, complexes, homology, derived functors, acyclic resolutions, Tor, Ext, Yoneda-Ext, spectral sequence of filtered or double complexes & composite functors, group cohomology, derived functor of limits | |||||

Objective | ||||||

401-3110-70L | Student Seminar in Number Theory: Elliptic Curves Number of participants limited to 23. | W | 4 credits | 2S | M. Schwagenscheidt | |

Abstract | Seminar on the foundations of the theory of Elliptic Curves. | |||||

Objective | The participants learn the basics about elliptic curves, which will enable them to write a Bachelor's or Master's thesis in number theory. In addition to a talk, the writing of a short manuscript in latex will be required. | |||||

Content | We first study the basic properties of elliptic curves, such as the group law. Then we will proceed to study elliptic curves over the rationals and the question whether it has rational or integral points. One of the main goal of the seminar is the proof of the Mordell-Weil theorem, which states that the set of rational points of a rational elliptic curve is a finitely generated abelian group. Using the theory of elliptic functions we will show that an elliptic curve over the complex numbers can be viewed as a torus. As an outlook, we will sketch several deep results and conjectures about elliptic curves, such as Wiles' Modularity Theorem, which played an important role in the proof of Fermat's Last Theorem, and such as the Birch and Swinnerton-Dyer Conjecture. | |||||

Literature | Knapp: Elliptic Curves Koecher, Krieg: Elliptische Funktionen und Modulformen Milne: Elliptic Curves Silverman: The Arithmetic of Elliptic Curves Silverman, Tate: Rational Points on Elliptic Curves | |||||

Prerequisites / Notice | Basic knowledge of Algebra and Complex Analysis will be helpful. | |||||

401-3420-70L | Topics in Harmonic Analysis Number of participants limited to 20 | W | 4 credits | 2S | F. Da Lio, L. Kobel-Keller | |

Abstract | The aim of this seminar about harmonic analysis is to study the most important and most classical topics in that field, e.g. maximal functions, Marcinkiewicz interpolation, Fourier theory, distribution theory, singular integrals and Calderon-Zygmund theory. After an introduction delivered by the two organisers, each week participants will give a seminar talk (usually in groups of two). | |||||

Objective | The students will learn on one hand the most important concept in harmonic analysis and on the other hand improve their presentations skills (by delivering a seminar talk). | |||||

Literature | The main references are: E. Stein: "Singular integrals and differentiability properties of functions " E. Stein, G. Weiss: "Introduction to Fourier analysis on Euclidean spaces" L. Grafakos: "Modern Fourier Analysis" & "Classical Fourier Analysis" | |||||

401-3650-68L | Numerical Analysis Seminar: Mathematics of Deep Neural Network Approximation Number of participants limited to 6. Consent of Instructor needed. | W | 4 credits | 2S | C. Schwab | |

Abstract | The seminar will review recent _mathematical results_ on approximation power of deep neural networks (DNNs). The focus will be on mathematical proof techniques to obtain approximation rate estimates (in terms of neural network size and connectivity) on various classes of input data including, in particular, selected types of PDE solutions. | |||||

Objective | ||||||

Content | Presentation of the Seminar: Deep Neural Networks (DNNs) have recently attracted substantial interest and attention due to outperforming the best established techniques in a number of tasks (Chess, Go, Shogi, autonomous driving, language translation, image classification, etc.). In big data analysis, DNNs achieved remarkable performance in computer vision, speech recognition and natural language processing. In many cases, these successes have been achieved by heuristic implementations combined with massive compute power and training data. For a (bird's eye) view, see Link and, more mathematical and closer to the seminar theme, Link The seminar will review recent _mathematical results_ on approximation power of deep neural networks (DNNs). The focus will be on mathematical proof techniques to obtain approximation rate estimates (in terms of neural network size and connectivity) on various classes of input data including, in particular, selected types of PDE solutions. Mathematical results support that DNNs can equalize or outperform the best mathematical results known to date. Particular cases comprise: high-dimensional parametric maps, analytic and holomorphic maps, maps containing multi-scale features which arise as solution classes from PDEs, classes of maps which are invariant under group actions. Format of the Seminar: The seminar format will be oral student presentations, combined with written report. Student presentations will be based on a recent research paper selected in two meetings at the start of the semester. Grading of the Seminar: Passing grade will require a) 1hr oral presentation with Q/A from the seminar group and b) typed seminar report (``Ausarbeitung'') of several key aspects of the paper under review. Each seminar topic will allow expansion to a semester or a master thesis in the MSc MATH or MSc Applied MATH. Disclaimer: The seminar will _not_ address recent developments in DNN software, eg. TENSORFLOW, and algorithmic training heuristics, or programming techniques for DNN training in various specific applications. | |||||

401-4660-70L | Robustness of Deep Neural Networks Number of participants limited to 40 | W | 4 credits | 2S | R. Alaifari | |

Abstract | While deep neural networks have been very successfully employed in classification problems, their stability properties remain still unclear. In particular, the presence of so-called adversarial examples has demonstrated that state-of-the-art networks are extremely vulnerable to small perturbations in the data. | |||||

Objective | In this seminar, we will consider the state-of-the-art in adversarial attacks and defenses. | |||||

Prerequisites / Notice | Participants should already be familiar with the principles of deep neural networks. The course will also include programming that will require knowledge in using either PyTorch or Tensorflow. | |||||

401-3640-70L | Volume Integral Equations: Theory and Numerics Number of participants limited to 10. | W | 4 credits | 2S | R. Hiptmair | |

Abstract | The seminar covers recent research articles on the theory and numerical treatment of volume integral equations for acoustic and electromagnetic scattering problems. | |||||

Objective | Beside conveying knowledge about the functional analytic method for analyzing integral equations and a range of numerical methods, the seminar is also meant to practise scientific presentation skills. | |||||

Content | Topics (based on research articles) 1. VIE for acoustic scattering 2.The Operator Equations of Lippmann-Schwinger Type for Acoustic and Elec- tromagnetic Scattering Problems in L2 3. VIE for electromagnetic scattering at dielectric bodies 4. Fast Solvers for the Lippmann-Schwinger equation 5. Numerical Solution of the Lippmann–Schwinger Equation by “Approx- imate Approximations” 6. Higher-order Fourier approximation in scattering by two-dimensional, inho- mogeneous media 7. Fast convolution with free-space Green’s functions 8. Fast numerical solution of the electromagnetic medium scattering problem 9. VIE Methods for Time-Harmonic Solutions of Maxwell’s Equations: Discretization, Spectrum and Preconditioning 10. The Discrete Dipole Approximation: an overview and recent developments | |||||

401-3920-17L | Numerical Analysis Seminar: Mathematics for Biomimetics Number of participants limited to 8. | W | 4 credits | 2S | H. Ammari, A. Vanel | |

Abstract | The aim of this seminar is to explore how we can learn from Nature to provide new approaches to solving some of the most challenging problems in sensing systems and materials science. An emphasis will be put on the mathematical foundation of bio-inspired perception algorithms in electrolocation and echolocation. | |||||

Objective | ||||||

401-3620-70L | Student Seminar in Statistics: Multiple Testing for Modern Data Science Number of participants limited to 24 Mainly for students from the Mathematics Bachelor and Master Programmes who, in addition to the introductory course unit 401-2604-00L Probability and Statistics, have heard at least one core or elective course in statistics. Also offered in the Master Programmes Statistics resp. Data Science. | W | 4 credits | 2S | M. Löffler, A. Taeb | |

Abstract | The course encompasses a review of approaches to multiple testing. | |||||

Objective | The students understand the relevance of multiple testing in modern applications. Further, they learn about two commonly used measures -- namely family-wise-error-rate (FWER) and false discovery rate (FDR) -- and approaches to control for them. | |||||

Content | In modern statistical applications it is often desired to perform thousands of statistical tests simultaneously. Performing a test at a desired level (e.g. 0.05) for each variable separately will result in many false positives. In science this is known as the ‘reproducibility crisis’. In this seminar we will review and discuss approaches to deal with this issue. First, we will consider the strong notion of FWER and how to control it via Bonferroni correction, permutation tests, step-up and hierarchical procedures or Tukey’s higher criticism. In the second part of the seminar we will investigate the less conservative FDR, discussing the classical Benjamini-Hochberg procedure, as well as more modern methods such as Knockoffs and Bayesian approaches. Throughout, we highlight the utility of discussed methods for real world applications. | |||||

Literature | Lecture 1: Bonferroni and Simes Link Link Lecture 2: Permutation tests Link Link Lecture 3: Hierarchical testing Link Link Link Lecture 4: Higher criticism Methodology: Link and for theoretical reference Link Application: Link and for more reference Link Lecture 5: Benjamini-Hochberg (BH) with martingales Link, Link Lecture 6: FDR control under dependence Link Link Lecture 7: Empirical null distribution Link Link Lecture 8: Bayes FDR methods Link Link Lecture 9: SLOPE Link Link Lecture 10: Knockoffs Link Link Lecture 11: Generalization of FWER and connections to FDR Link Link Lecture 12: Exploratory testing Link Link | |||||

Prerequisites / Notice | Every lecture will consist of an oral presentation highlighting key ideas of selected papers by a pair of students. Another two students will be responsible for asking questions during the presentation and providing a discussion of the pros+cons of the papers 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. |

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