Typ |
Veranstaltung |
Dozent |
Modulnummer |
V+Ü | Dynamische Systeme |
apl. Prof. Tarkhanov | A510, A710, A750, 771, 772, 781, 82j, MATVMD821-3, MATVMD921-3, MATVMD621-2 |
Umfang | 4h |
Inhalt | Dynamische Systeme sind mathematische Modelle für zeitabh{ängige Prozesse.
Die Zeitentwicklung kann kontinuierlich oder diskret sein.
Die Vorlesung soll dazu dienen, die wichtigsten Begriffe und Methoden aus diesem aktuellen
Teilgebiet der Mathematik kennenzulernen.
Die Theorie der dynamischen Systeme analysiert und charakterisiert das Verhalten f{ür
große Zeiten (Gleichgewicht, periodische Bahn, Attraktor, Stabilit{ät, Chaos, usw.).
Wir betrachten einerseits die strukturelle Stabilit{ät eines Systems gegen{über
St{örungen und andererseits Verzweigungen (Bifurkationen) bei {Änderungen von
Systemparametern.
Wir werden sehen, wie durch globale Verzweigungen komplizierte Dynamik (``Chaos'') entstehen
kann.
|
Literatur |
- Nikolai Tarkhanov, Mathematik f{ür Physiker, Universit{ät Potsdam, 2002
|
URL | http://www.tarkhanov-homepage.de/ |
Voraussetzungen | Analysis I u. II |
Zielgruppe | BSc, MSc, BEd, MEd |
Leistungsnachweis | Klausur |
Übungsleiter | apl. Prof. Tarkhanov |
Übungen | 2h |
|
V+S | Ergodentheorie |
Dr. Beckus, Michael Schwarz | MATVMD621-2, MATVMD631-2, MATVMD821-3, MATVMD831-3, MATVMD721 |
Umfang | 4h+2h |
Inhalt |
Die Ergodentheorie hat sich ursprünglich aus der Statistischen
Mechanik und der sogenannten Boltzmannschen Ergodenhypothese
entwickelt. Von zentraler Bedeutung in der Ergodentheorie ist das
durchschnittliche Langzeitverhalten eines dynamischen Systems. Ihr
Durchbruch als mathematische Disziplin liegt in den Arbeiten von
John von Neumanns und George Birkhoffs von 1931
begründet. Inzwischen findet die Ergodentheorie Anwendungen in
sehr unterschiedlichen Gebieten der Mathematik wie z.B. in der
Funktionalanalysis, Spektraltheorie, Zahlentheorie, Stochastik,
Algebra und Harmonischen Analysis.
In dieser Vorlesung werden die Grundlagen der Ergodentheorie wie
Ergodizität, Ergodensätze, dynamische Systeme, schwache und
starke Mischung entwickelt.
Die Vorlesung richtet sich an interessierte Studenten der Mathematik
mit soliden Vorkenntnissen in Analysis und Grundkenntnissen in der
Maß theorie. Die Vorlesung wird von einem Seminar begleitet. Sie
ist geeignet für ein fortgeschrittenes Bachelorstudium oder das
Masterstudium.
Ergodic theory has its foundation in the statistical mechanics and
the so called Boltzmann's Ergodic hypothesis. A central focus of
ergodic theory lies in the study of the long-term average behavior
of a dynamical system. It was established as a mathematical
discipline due to the groundbreaking works of John von Neumann and
George Birkhoff in 1931. Nowadays, ergodic theory plays a crucial
role in various mathematical fields such as functional analysis,
spectral theory, number theory, stochastic, algebra and harmonic
analysis.
During this lecture we will learn the basics of ergodic theory such
as ergodicity, ergodic theorems, dynamical systems, weak and
strongly mixing systems.
It targets students in mathematics with a solid background in
analysis and basic knowledge of measure theory. The lecture goes
along with a seminar. It is suitable for advanced bachelor students
and master students.
|
Literatur |
- K. Petersen, Ergodic Theory, Cambridge University Press, Cambridge, 1989.
- T. Eisner, B. Farkas, M. Haase, R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Graduate Texts in Mathematics, erscheint bei Springer-Verlag.
- M. Einsiedler, T. Ward, Ergodic Theory with a view towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag London, Ltd., London, 2011.
- H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981.
- T. Tao, Ergodic Theory, see http://terrytao.wordpress.com/category/254a-ergodic-theory/ or http://terrytao.wordpress.com/books/poincares-legacies-course-notes-expository-articles-and-lecture-series-from-a-mathematical-blog/.
|
Voraussetzungen | Analysis, LAAG |
Zielgruppe | BSc, MSc, MEd |
Leistungsnachweis | mündliche Prüfung |
|
V+Ü | Nonparametric Statistics |
Dr. Mariucci | MATVMD837, MATVMD831, MATVMD731, MAT-DSAM2B, MATVMD931-2, 771, 772, 781, A710, A750, 83j, 9040 |
Umfang | 2h |
Inhalt |
This course aims at introducing the modern nonparametric techniques in statistical analysis. Statistical inference will be limited essentially to two models: density estimation and nonparametric regression models. The properties of classical nonparametric estimators such as kernel density estimators, local polynomial estimators and projection estimators will be recalled and an introduction to the minimax theory, model selection and adaptiveness will be provided. Finally, techniques to prove lower bounds in a minimax sense will be also presented.
The main idea of this course is to get students acquainted with the
fundamentals, basic properties and use of the most important recent
nonparametric techniques.
The lecture is supplemented by a 2 hours seminar with the same
name. |
Literatur |
- E. Giné and R. Nickl, Mathematical foundations of infinite-dimensional statistical models, Cambridge University Press 2015.
- A. B. Tsybakov, Introduction to Nonparametric Estimation, Springer 2009.
|
Voraussetzungen | Basic knowledge of probability, statistics and real analysis (random variables, expectation, limits and series, differentials and integrals, Taylor expansions and function spaces) |
Zielgruppe | MSc, M-Computational Science, MEd, M-Data Science, BSc |
Leistungsnachweis | Written or oral exam |
Übungsleiter | Dr. Mariucci |
Übungen | 2h |
|
V+Ü | Group actions in geometry and quantum field theory |
Prof. Paycha | 82j, MATVMD921-3 |
Umfang | 2h |
Inhalt | Symmetries, invariance, and conservation laws, are
constraints which play a central role in formulating physical
theories and models. In this course we shall focus on continuous
symmetries described in terms of Lie group actions, in which case
the correspondence between symmetries and conservation laws is given
by Emmy Noether's theorem which was published a century ago. In
quantum field theory, group actions arise in many disguises, only a
few of which we will highlight in this course while focusing on the
notion of anomaly. We also hope to touch on equivariant geometry and
localisation techniques used in a supersymmetric setup. This 14 hrs
course is only meant as a brief overview of different uses of group
actions without going into the depths of each of them.
Contents
- Emmy Noether's theorem [9, 3]
Lie groups and Lie algebras, Group actions and symmetries in physics
- From group actions to principle bundles
Smooth free and proper actions on finite dimensional
manifolds, Generalisation to Hilbert manifolds [11]
- The Faddeev-Popov procedure in quantum field theory [10]
The Faddeev-Popov procedure in the path integral formalism,
Anomalies [2], The Becchi-Rouet-Stora-Tseytlin procedure [7],
The Batalin-Vilkovisky procedure [6, 5]
- Equivariant localisation [1, 4, 12]
Stationary phase method, Duistermaat-Heckman localisation formula,
Localisation formulas and group actions, Equivariant differential
and BRST differential
|
Literatur |
Here are some references (most of which are available online) among
many other classical references:
- A. Alekseev, Notes on equivariant localisation, ESI 1999
https://www.esi.ac.at/static/esiprpr/esi744.pdf
- R. Bertlmann, Anomalies in Quantum Field Theory, Clarendon Press (2001)
- M. Ba nados, I. Reyes, A short review on Noether's theorems, gauge symmetries and boundary terms https://arxiv.org/pdf/1601.03616.pdf (2017)
- N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Springer Verlag (2004)
- P. Clavier, V.Dang Nguyen, Batalin nVilkovisky Formalism as a
Theory of Integration for Polyvectors, in Quantization,
Geometry and Noncommutative Structures in Mathematics and
Physics, Mathematical Physics Studies, Ed.: A. Cardona,
P. Morales H.Ocampo, S.Paycha Andr\`Es. Reyes (2016)
- D. Fiorenza, An introduction to the Batalin-Vilkovisky
formalism (2008) https://arxiv.org/pdf/math/0402057.pdf
- A. Fuster, M. Henneaux, A. Maas, BRST-antifield Quantization:
a Short Review (2005) arXiv:hep-th/0506098
- M. Henneaux, C.Teitelboim, Quantization of Gauge Systems,
Princeton University Press 1994
- E. Noether, Invariante Variationsprobleme (1918)
https://de.wikisource.org/wiki/Invariante_Variationsprobleme
- S. Paycha, The Faddeev-Popov Procedure and Application to
Bosonic Strings: An Infinite Dimensional Point of View,
Commun. Math. Phys. 147 (1992) 163-180
- S. Paycha, Basic prerequisites in differential geometry and
operator theory in view of applications to quantum field theory
(Unpublished notes)
- V. Pestun, Review of localization in geometry
(2016)arXiv:1608.02954
- V. Pestun et al, Localization techniques in quantum field
theories (2016) https://arxiv.org/abs/1608.02952
|
Voraussetzungen | Differential manifolds. Lie groups. Some acquaintance with quantum field theory is welcome yet not necessary |
Zielgruppe | MSc |
Leistungsnachweis | Exam |
Übungsleiter | Pierre Clavier |
Übungen | 2h |
|
V+Ü | Semi-Riemannsche Geometrie / Semi-Riemannian Geometry |
Dr. Roos | 261, 771, 772, 781, 81j, A710, A750, MATVMD611-2, MATVMD711, MATVMD811-5 |
Umfang | 4h |
Inhalt |
In der Vorlesung Semi-Riemannsche Geometrie studieren wir gekrümmte Räume beliebiger Dimension. Wir definieren die Messung von Längen und Winkeln mit Hilfe von semi-riemannschen Metriken. Wir führen einen Ableitungsbegriff für Vektorfelder ein und studieren lokal kürzeste Verbindungen zwischen zwei Punkten, sogenannte Geodätische. Anschließ end behandeln wir verschiedene Krümmungsbegriffe. Es ergeben sich viele weitergehende Fragen: Inwiefern wird die Topologie einer riemannschen Mannigfaltigkeit durch ihre Krümmung bestimmt? Welche Auswirkungen hat die Krümmung auf analytische Fragen, etwa die Lösung der Laplace-Gleichung oder der Wärmeleitungsgleichung? Was sind grundlegende Eigenschaften von gekrümmten Raumzeiten in der allgemeinen Relativitätstheorie? Wir werden je nach Wunsch der Studierenden einige dieser Fragen diskutieren.
In the lecture course Semi-Riemannian Geometry we study curved spaces of arbitrary dimensions. We use semi-Riemannian metrics to define lenghts and angles. We introduce a covariant derivative for vector fields and we study the locally shortest curves between two points, the so-called geodesics. Then we discuss several notions of curvature. This leads to several more advanced topics: In which way is the topology of the manifold determined by its curvature? What is the effect of curvature concerning analytical questions such as the solution of the Laplace equation or the heat equation? What are basic properties of curved space-times in general relativity? We will study some of these questions depending on the preferences of the audience.
|
Literatur |
- Bär: Differentialgeometrie, Skript, Potsdam 2013
- O'Neill: Semi-Riemannian Geometry, Academic Press, New York 2002
|
URL | https://www.math.uni-potsdam.de/professuren/geometrie/lehre/sommersemester-2019/vorlesung-semi-riemannsche-geometrie/ |
Voraussetzungen | Analysis 1 + 2 |
Zielgruppe | BSc, MSc, MEd, (lectures optional in English) |
Leistungsnachweis | Klausur |
Übungsleiter | Claudia Grabs |
Übungen | 2h |
|
V+Ü | Advanced Probability Theory |
Prof. Roelly | 771, 772, 83j, 82j, MATAMD621-2, MATVMD631-2, MATVMD831-3, MAT-DSAM8A |
Umfang | 4h |
Inhalt |
The purpose of this course is to treat in details selected fundamentals of modern probability theory.
The focus is in particular on limit theorems including the strong law of large numbers and Lindeberg's central limit theorem,
and on discrete-time processes like martingales, as well as basic results on Brownian motion. Various examples will be considered.
The participant is assumed to have a reasonable grasp of basic
probability, basic analysis and measure theory.
This lecture is appropriate for Master students or for advanced Bachelor students.
It is a natural application/extension of the course "Functional Analysis I".
It is part of both profiles "Mathematical modelling and data analysis" and "Structures of Mathematics
with physical background" in the course of studies Master of Science Mathematics.
The lecture also adresses to students of informatics and physics.
|
Literatur |
-
Durrett, R. Probability: theory and examples, Cambridge Series in Statistical and Probabilistic Mathematics 2010
|
URL | http://www.math.uni-potsdam.de/~roelly/sose19.html |
Voraussetzungen | Stochastik or Foundations of stochastics, if possible Functional Analysis 1 |
Zielgruppe | BSc, MSc, M Data Science, MWDT |
Leistungsnachweis | Written or oral exam |
Übungsleiter | Dr. Kosenkova |
Übungen | 2h |
|
V+Ü | Stastistical Machine Learning |
Dr. Suvorikova | MATVMD831-3, MATVMD931-3, 83j, MATVMD631-2, 781, MAT-DSAM2A |
Umfang | 4h |
Inhalt |
Machine learning is one of the fastest growing branches of modern data analysis.
It deals with a broad spectrum of methodologies which
aim to detect, extract, and process meaningful patterns in complex
(usually high-dimensional and non-linear) data sets.
As examples one may consider handwriting recognition,
classification of DNA sequences, or time series forecasting.
The goal of this course is to introduce widely-used methods of machine learning
from mathematical point of view using approaches of statistical learning.
The topics covered in the course are as follows: decision theory, linear classification, k-nearest neighbor algorithm, decision trees, neural networks, support vector machines, kernel methods, elements of Vapnik-Chervonenkis theory, Rademacher complexity and ensemble methods.
|
Literatur |
- S.Shalev-Shwartz and S.Ben-David: Understanding Machine Learning:
From Theory to Algorithms; published in 2014 by Cambridge University Press
- M.Mohri, A. Rostamizadeh and A. Talwalkar: Foundations of Machine Learning (2d edition); published in 2018 by MIT Press
|
Voraussetzungen | Stochastic I; Recommended: lectures in Statistics (e.g. Statistics I or Data Analysis) |
Zielgruppe | BSc, MSc, MEd, MDS |
Leistungsnachweis | Oral examination |
Übungsleiter | Dr. Suvorikova |
Übungen | 2h |
|
V+Ü | Zufällige Modelle |
Prof. Roelly | 771, 772, 781, 83j, A710 , A750, MATVMD631-2, MAT-VM-D731, MAT-VM-D831-4 |
Umfang | 4h |
Inhalt | In dieser Vorlesung werden grundlegende zufällige Modelle präsentiert.
Zunächst wird ein wichtiges Beispiel aus der statistischen Mechanik diskutiert: das Ising Modell. Anhand dieses Modells werden unter anderem die Begriffe Gibbsmaß{} und Phasenübergang eingeführt. Existenz- und Eindeutigkeitsergebnisse werden bewiesen.
Am Ende der Vorlesung wird Dr. Houdebert eine Einsicht in die Perkolationstheorie geben. Diese Theorie aus der stochastischen Geometrie beschreibt das Ausbilden von zusammenhängenden Gebieten (Clustern) bei zufallsbedingtem Besetzen von Strukturen.
|
Literatur |
-
Friedli, S. and Velenik Y. Statistical Mechanics of Lattice Systems:
a Concrete Mathematical Introduction, Cambridge University Press, 2017
-
Grimmett G. Percolation, Birkhäuser, 2000
-
Prum, T. and Fort, J.-C. Stochastic Processes on a Lattice and Gibbs Measures, Springer 1991
|
URL | http://www.math.uni-potsdam.de/~roelly/sose19.html |
Voraussetzungen | Stochastik |
Zielgruppe | BSc, MSc, MEd, MData Science |
Leistungsnachweis | Klausur |
Übungsleiter | Dr. Mazzonetto |
Übungen | 2h |
|
V+Ü | Bayesian inference and data assimilation |
Dr. Mariucci | MATVMD838, A710, A750, MATVMD84j, MATVMD83j, MATDAP01, VMD711-51 |
Umfang | 4h |
Inhalt |
This lecture introduces parametric Bayesian inference (prior models, forward model, coupling of measures) and its applications. Computational methods, such as Monte Carlo methods, importance sampling, Markov Processes and MCMC, are also discussed, as well as stochastic processes. Special attention will be paid to the connection of mathematical models and data assimilation (state estimation, Kalman filter, particle filters, variational methods), also in the high-dimensional setting. |
Literatur |
-
Sebastian Reich and Colin Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, 2015.
- Kody Law, Andrew Stuart and Konstantinos Zygalakis, Data Assimilation - A Mathematical Introduction, Springer-Verlag, 2015
|
Voraussetzungen | Basic knowledge of numerics, stochastics and dynamic processes. |
Zielgruppe | MSc, MCSc, MDSc |
Leistungsnachweis | Written exam |
Übungsleiter | Han Chen Lie |
Übungen | 2h |
|
V+Ü | Stochastic Processes (Theorie zeitabhängiger stochastischer und deterministischer Prozesse) |
Prof. Huisinga | MATVMD836, MATVMD834, MATVMD731, MATVMD631-32, MATDSAM3A, MATDSAM8A, 771, 772, 781, A510, A710, A750, 82j, 83j |
Umfang | 4h |
Inhalt | Stochastic processes play a key role in mathematics and many applied sciences. Example include eye moving during reading, locomotion of the social amoebae, biochemical reaction systems or spreading of diseases. The lecture gives an introduction to the important class of Markov processes in continuous and discrete time on discrete state spaces. Important concepts include recurrence/transience, invariant and stationary measures, reversibility and the strong law of large numbers, metastability, periodicity, master equation,
The lecture is part of the focus area 'applied mathematics: modelling and data analysis' (Profilrichtung 'Angewandte Mathematik: Modellierung und Datenanalyse'). |
Literatur |
- Bremaud. Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues
- Lasota and Mackey, Chaos, Fractals, and Noise, Springer, 1994.
- Meyn and Tweedie. Markov Chains and Stochastic Stability. Springer, Berlin, 1993.
Springer, New York, 1999.
|
Voraussetzungen | keine
|
Zielgruppe | BSc, BEd, MSc, MEd, COS |
Leistungsnachweis | Klausur |
Übungsleiter | Dr. Braunß, Enrico Reiß |
Übungen | 2h |
|
V+Ü | Introduction to theoretical systems biology |
Prof. Huisinga | MATMBIP05, MATVMD941-3, INFDSC4, MATDSAM3B, 84j |
Umfang | 2h |
Inhalt | The lecture gives an introduction the mathematical concepts, methods and approaches in modern systems biology. It focusses on the stochastic and deterministic formulation of biochemical reaction kinetics, illustrated in application to important biological signal transduction pathway and gene regulatory systems. Further topics include model order reduction of large-scale deterministic reaction systems and network motifs in gene regulatory networks.
|
Literatur |
- Klipp et al, Systems Biology: A textbook, Wiley-Blackwell, 2009
- Alon, An Introduction to Systems Biology. CRC Press, 2006
|
Voraussetzungen | keine |
Zielgruppe | MSc, Bioinformatik-M |
Leistungsnachweis | Klausur |
Übungsleiter | Daniel Seeler |
Übungen | 2h |
|
V+Ü | Introduction to Physiologically based Pharmacokinetic Modeling |
Prof. Huisinga | 84j, MATVMD941-3 |
Umfang | One week block course, for details see website below. |
Inhalt | The course introduces physiologically based pharmacokinetic concepts and modeling approaches with relevance to and application in drug discovery and development. We focus on mathematical models of the key ADME processes adsorption, distribution, metabolism and excretion, including ionization and (linear/saturable) protein binding, first-order and transit compartment models of absorption, a priori prediction of tissue-to-blood partition coefficients, hepatic metabolism and bilary excretion. Furthermore, the course establishes the link between detailed physiological based pharmacokinetic models and simple 1-/2-compartment models commonly used in late stage clinical phases via mathematical model reduction techniques (lumping approach). Finally, we introduce concepts of variability in physiological and anatomical parameters, extrapolation techniques to different species as well as from adults to children, and consider models of drug-drug interaction.
The course also includes a guest lecture illustrating the application of physiologically based pharmacokinetic modeling in the pharmaceutical industry.
|
Literatur | Will be announced at the beginning of the course
|
URL | http://www.pharmetrx.de |
Voraussetzungen | Online application via the graduate research training program PharMetrX: Pharmacometrics & Computational Disease Modeling
|
Zielgruppe | MSc, PhD |
Leistungsnachweis | Active participation |
|
V+Ü | Reinforcement learning |
Dr. de Wiljes | 851, 852, MATVMD1031-32, MATVMD1041-42 |
Umfang | Blockkurs vom 22.07-02.08.2019 |
Inhalt |
The course will cover the mathematical background of reinforcement learning and also focus on the different algorithmic approaches.
In order to deepen the theoretical and computational aspects we will look at various examples from a range of applicational areas and the
students will have the opportunity to implement the learned methods.
|
Literatur |
- Abraham/ Marsden: Foundations of Mechanics, American Mathematical Society, 2008
Arnol'd: Mathematische Methoden der Klassischen Mechanik
|
Voraussetzungen | |
Zielgruppe | BSc, MSc, MSc-Ph |
Leistungsnachweis | Projekt |
Übungsleiter | Dr. de Wiljes |
Übungen | 2h |
|
V+Ü | Wavelet-Kurs |
Prof. Holschneider | 771, 772, A710, A750 |
Umfang | 4h |
Inhalt | siehe unter: www.math.uni-potsdam.de/ hols
|
Voraussetzungen | keine |
Zielgruppe | BSc, BEd |
Leistungsnachweis | Klausur |
Übungsleiter | N.N. |
Übungen | 2h |
|