Content:
This is the first part of a three semester introduction into the basic mathematical field of Analysis. Differential and integral calculus in a real variable will be covered. Topics:

1. fundamentals, elementary logic, ordered pairs, relations, functions, domain and range of a function, inverse functions (injectivity, surjectivity)
2. numbers, induction, calculations in R, C
3. arrangement of R, maximum and minimum, supremum and infimum of real sets, supremum / infimum completeness of R, absolute value of a real number, Q is dense in R
4. sequences and series, limits, cauchy sequences, convergence criteria, series and basic principles of convergence
5. topological aspects of R, open, closed, and compact real sets
6. sequences of functions, series of functions, power series
7. properties of functions, boundedness, monotony, convexity
8. continuity, limits and continuity of functions, uniform continuity, intermediate value theorems, continuity and compactness
9. differentiability, concept of the derivative, differentiation rules, mean value theorem, local and global extrema, curvature, monotony, convexity
10. elementary functions, rational functions, root functions, exponential functions, angular functions, hyperbolic functions, real logarithm, inverse trigonometric functions, curve sketching
11. beginnings of integral calculus

Detailed Information can be found on the Homepage of 19202801 Analysis I.

Literature:

• Bröcker, Theodor: Analysis 1, Spektrum der Wissenschaft-Verlag.
• Forster, Otto: Analysis 1, Vieweg-Verlag.
• Spivak, Michael: Calculus, 4th Edition.

Viele Analysis Bücher sind auch über die Fachbibliothek der FU Berlin elektronisch verfügbar.

Bei Schwierigkeiten mit den Grundbegriffen Menge, Abbildung etc. ist die folgende Ausarbeitung empfehlenswert:

• Scheerer, Hans: Brückenkurs, Skript FU Berlin 2006.

Content:

• Basic terms/concepts: sets, maps, equivalence relations, groups, rings, fields
• Linear equation systems: solvability criteria, Gauss algorithm
• Vector spaces: linear independence, generating systems and bases, dimension, subspaces, quotient spaces, cross products in R^3,
• Linear maps: image and rank, relationship to matrices, behaviour under change of basis
• Dual vector spaces: multilinear forms, alternating and symmetric bilinear forms, relationship to matrices, change of basis
• Determinants: Cramer's rule, eigenvalues and eigenvectors

Prerequisites:

Participation in the preparatory course (Brückenkurs) is highly recommended.

• Siegfried Bosch, Lineare Algebra, 4. Auflage, Springer-Verlag, 2008;
• Gerd Fischer, Lernbuch Lineare Algebra und Analytische Geometrie, Springer-Verlag, 2017;
• Bartel Leendert van der Waerden, Algebra Volume I, 9th Edition, Springer 1993;

Zu den Grundlagen

• Kevin Houston, Wie man mathematisch denkt: Eine Einführung in die mathematische Arbeitstechnik für Studienanfänger, Spektrum Akademischer Verlag, 2012

• Whiteboard: Signing up for the course with the FU Whiteboard system is mandatory for participating. Whiteboard will also serve as a course homepage.
• Active participation: A problem set will be published every week. Each problem set has to be solved within one week. Submissions are allowed by groups of max. two students. For active participation, 50% of the total possible amount of points have to be reached. Moreover, we expect active participation in the tutorials.
• Regular participation: Participating in the lectures is recommended. Participation in the tutorials is required.
• Covid: The modus operandi of the course (online, hybrid, presence) will be determined at the beginning of the semester depending on the Covid situation. Please follow the Whiteboard homepage and announcements.

Inhalt:

Die Auswahl der behandelten numerischen Verfahren enthält Polynominterpolation, Newton-Cotes-Formeln zur numerische Integration und Euler-Verfahren für lineare Differentialgleichungen.

Studierende der Mathematik (Monobachelor und Lehramt) und Bioinformatik, sowie Numerikinteressierte aus Physik, Informatik und anderen Natur- und Geisteswissenschaften.

best approximation of functions, least squares problems, interpolation, numerical quadrature, computation of eigenvalues and eigenvectors, numerical methods for initiatl value problems (part 1)

Stoer, Josef und Roland Bulirsch: Numerische Mathematik - eine Einführung, Band 1. Springer, Berlin, 2005.

Content:

Selection from the following topics:

• Enumeration (twelvefold way, inclusion-exclusion, double counting, recursions, generating functions, inversion, Ramsey's Theorem, asymptotic counting)
• Discrete Structures (graphs, set systems, designs, posets, matroids)
• Graph Theory (trees, matchings, connectivity, planarity, colorings)

• J. Matousek, J. Nesetril (2002/2007): An Invitation to Discrete Mathematics, Oxford University Press, Oxford/Diskrete Mathematik, Springer Verlag, Berlin, Heidelberg.
• L. Lovasz, J. Pelikan, K. Vesztergombi (2003): Discrete Mathemtics - Elementary and Beyond/Diskrete Mathematik, Springer Verlag, New York.
• N. Biggs (2004): Discrete Mathematics. Oxford University Press, Oxford.
• M. Aigner (2004/2007): Diskrete Mathematik, Vieweg Verlag, Wiesbaden/Discrete Mathemattics, American Mathematical Society, USA.
• D. West (2011): Introduction to Graph Theory. Pearson Education, New York.

Target group:

BMS students, Master and Bachelor students

Content:

Selection from the following topics:

• Counting (basics, double counting, Pigeonhole Principle, recursions, generating functions, Inclusion-Exclusion, inversion, Polya theory)
• Discrete Structures (graphs, set systems, designs, posets, matroids)
• Graph Theory (trees, matchings, connectivity, planarity, colorings)
• Algorithms (asymptotic running time, BFS, DFS, Dijkstra, Greedy, Kruskal, Hungarian, Ford-Fulkerson)

Contents:

• Construction of stochastic processes;
• conditional expectation;
• martingales in discrete time: convergence, stopping theorems, inequalities;
• convergence types in stochastics;
• uniform integrability;
• Markov chains in discrete and continuous time: recurrence and transience, invariant measures;
• convergence in distribution of stochatic processes;
• Brownian motion and invariance principle.

Detailed Information can be found on the Homepage of 19212901 Basismodul: Stochastics II.

Literature:

• Klenke: Wahrscheinlichkeitstheorie
• Durrett: Probability. Theory and Examples.

Weitere Literatur wird im Lauf der Vorlesung bekannt gegeben.
Further literature will be given during the lecture.

Prerequisite: Stochastics I  and  Analysis I — III.

Inhalt

• This course is the sequel of the course of Stochastics I. The main objective is to go beyond the first principles in probability theory by introducing the general language of measure theory, and the application of this framework in a wide variety of probabilistic scenarios.
  More precisely, the course will cover the following aspects of probability theory:
• Measure theory and the Lebesgue integral
• Convergence of random variables and 0-1 laws
• Generating functions: branching processes and characteristic functions
• Markov chains
• Introduction to martingales

Content:
This is the first part of a three semester introduction into the basic mathematical field of Analysis. Differential and integral calculus in a real variable will be covered. Topics:

1. fundamentals, elementary logic, ordered pairs, relations, functions, domain and range of a function, inverse functions (injectivity, surjectivity)
2. numbers, induction, calculations in R, C
3. arrangement of R, maximum and minimum, supremum and infimum of real sets, supremum / infimum completeness of R, absolute value of a real number, Q is dense in R
4. sequences and series, limits, cauchy sequences, convergence criteria, series and basic principles of convergence
5. topological aspects of R, open, closed, and compact real sets
6. sequences of functions, series of functions, power series
7. properties of functions, boundedness, monotony, convexity
8. continuity, limits and continuity of functions, uniform continuity, intermediate value theorems, continuity and compactness
9. differentiability, concept of the derivative, differentiation rules, mean value theorem, local and global extrema, curvature, monotony, convexity
10. elementary functions, rational functions, root functions, exponential functions, angular functions, hyperbolic functions, real logarithm, inverse trigonometric functions, curve sketching
11. beginnings of integral calculus

Detailed Information can be found on the Homepage of 19202801 Analysis I.

Literature:

• Bröcker, Theodor: Analysis 1, Spektrum der Wissenschaft-Verlag.
• Forster, Otto: Analysis 1, Vieweg-Verlag.
• Spivak, Michael: Calculus, 4th Edition.

Viele Analysis Bücher sind auch über die Fachbibliothek der FU Berlin elektronisch verfügbar.

Bei Schwierigkeiten mit den Grundbegriffen Menge, Abbildung etc. ist die folgende Ausarbeitung empfehlenswert:

• Scheerer, Hans: Brückenkurs, Skript FU Berlin 2006.

Content:

• Basic terms/concepts: sets, maps, equivalence relations, groups, rings, fields
• Linear equation systems: solvability criteria, Gauss algorithm
• Vector spaces: linear independence, generating systems and bases, dimension, subspaces, quotient spaces, cross products in R^3,
• Linear maps: image and rank, relationship to matrices, behaviour under change of basis
• Dual vector spaces: multilinear forms, alternating and symmetric bilinear forms, relationship to matrices, change of basis
• Determinants: Cramer's rule, eigenvalues and eigenvectors

Prerequisites:

Participation in the preparatory course (Brückenkurs) is highly recommended.

• Siegfried Bosch, Lineare Algebra, 4. Auflage, Springer-Verlag, 2008;
• Gerd Fischer, Lernbuch Lineare Algebra und Analytische Geometrie, Springer-Verlag, 2017;
• Bartel Leendert van der Waerden, Algebra Volume I, 9th Edition, Springer 1993;

Zu den Grundlagen

• Kevin Houston, Wie man mathematisch denkt: Eine Einführung in die mathematische Arbeitstechnik für Studienanfänger, Spektrum Akademischer Verlag, 2012

• Whiteboard: Signing up for the course with the FU Whiteboard system is mandatory for participating. Whiteboard will also serve as a course homepage.
• Active participation: A problem set will be published every week. Each problem set has to be solved within one week. Submissions are allowed by groups of max. two students. For active participation, 50% of the total possible amount of points have to be reached. Moreover, we expect active participation in the tutorials.
• Regular participation: Participating in the lectures is recommended. Participation in the tutorials is required.
• Covid: The modus operandi of the course (online, hybrid, presence) will be determined at the beginning of the semester depending on the Covid situation. Please follow the Whiteboard homepage and announcements.

Inhalt:

Die Auswahl der behandelten numerischen Verfahren enthält Polynominterpolation, Newton-Cotes-Formeln zur numerische Integration und Euler-Verfahren für lineare Differentialgleichungen.

Studierende der Mathematik (Monobachelor und Lehramt) und Bioinformatik, sowie Numerikinteressierte aus Physik, Informatik und anderen Natur- und Geisteswissenschaften.

best approximation of functions, least squares problems, interpolation, numerical quadrature, computation of eigenvalues and eigenvectors, numerical methods for initiatl value problems (part 1)

Stoer, Josef und Roland Bulirsch: Numerische Mathematik - eine Einführung, Band 1. Springer, Berlin, 2005.

Contents

Advanced topcis in algorithm design with a changing focus. The topic is determined newly in each semester. For example, we might consider algorithms for problems on graphs, such as connectivity, shortest paths, or network flows.

Target audience

Masters students in computer science and mathematics.

Recommended prerequisites

"Advanced algorithms" or a similar class.

Spezialliteratur aus Zeitschriften

Proseminar/ Seminar in Group Theory

Content: In this seminar we will start with an introduction to the concepts about profinite groups, namely, inverse and direct limits, pro-C- groups for various group theoretical classes C, Sylow theory for profinite groups, finitely generated profinite groups, profinite groups as Galois groups, free profinite groups, profinite topologies, some special profinite groups  like profinite Frobenius groups, automorphism group of a profinite group, profinite completion, profinite torsion groups.

http://page.mi.fu-berlin.de/ersoy/ProfiniteGroups

Literature:
Luis Ribes and Pavel Zalesski: Profinite Groups

Prerequisite: Linear Algebra, a course in Algebra or Group Theory.

Inhalt

Diese Vorlesung für das Bachelorstudium soll als natürliche Fortsetzung von Lineare Algebra I und II Fundamente legen für Vorlesungen/Zyklen wie Diskrete Geometrie, Algebraische Geometrie und Differenzialgeometrie.

Sie behandelt grundlegende Modelle der Geometrie, insbesondere

euklidische, affine, sphärische, projektive und hyperbolische Geometrie,Möbiusgeometrie, Polarität und Dualität Strukturgruppen, Messen (Längen, Winkel, Volumina), explizite Berechnungen und Anwendungen, Beispiele sowie Illustrationsthemen;

Dabei werden weitere Bezüge hergestellt, zum Beispiel zur Funktionentheorie und zur Numerik.

    • Apéry, François: Models of the real projective plane. Computer
graphics of Steiner and Boy surfaces. With a preface by Egbert
Brieskorn. Friedr. Vieweg & Sohn, Braunschweig, 1987. xii+156 pp.
    • Berchtold, Florian: Geometrie. Von Euklid bis zur hyperbolischen
Geometrie mit Ausblick auf angrenzende Gebiete. Springer Spektrum,
Heidelberg, 2017. x+210 pp.
    • [Geo] Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J.:
Geometry. 2nd ed. Cambridge University Press, Cambridge, 2012. xiv+587
pp.
    • Lüneburg, Heinz: Die euklidische Ebene und ihre Verwandten.
Birkhäuser, Basel, 1999. viii+207 S.
    • Scheid, Harald; Schwarz, Wolfgang: Elemente der Geometrie. 5th rev.
ed. Springer Spektrum, Heidelberg, 2017. ix+373 pp.
    • Skript

Function theory is a classical field of mathematics, which deals with the properties of complex-differentiable functions on the complex number plane and has connections to algebra, analysis, number theory and geometry.

The concept of complex differentiability restricts real-differentiable functions from R2 to R2 to angle-preserving images. We will discover that complex-differentiable functions are quite rigid objects, but they are endowed with many amazing analytical, geometric, and visual properties.

A major result discussed in this lecture is Cauchy's integral theorem which states that the integral of any complexly differentiable function along a closed path in the complex plane is zero. We will see many nice consequences of this result, e.g. Cauchy's integral formula, the residual theorem and a proof of the fundamental theorem of algebra, as well as modern graphical representation methods.

Wolfgang Fischer, Ingo Lieb: Funktionentheorie. 8. Auflage. Vieweg, Braunschweig 2003, ISBN 3-528-77247-6.
T. Needham: Anschauliche Funktionentheorie, Oldenbourg Wissenschaftsverlag, (2001).
Reinhold Remmert, Georg Schumacher: Funktionentheorie 1. 5. Auflage. Springer, Berlin 2002, ISBN 3-540-59075-7

Contents:

• Construction of stochastic processes;
• conditional expectation;
• martingales in discrete time: convergence, stopping theorems, inequalities;
• convergence types in stochastics;
• uniform integrability;
• Markov chains in discrete and continuous time: recurrence and transience, invariant measures;
• convergence in distribution of stochatic processes;
• Brownian motion and invariance principle.

Detailed Information can be found on the Homepage of 19212901 Basismodul: Stochastics II.

Literature:

• Klenke: Wahrscheinlichkeitstheorie
• Durrett: Probability. Theory and Examples.

Weitere Literatur wird im Lauf der Vorlesung bekannt gegeben.
Further literature will be given during the lecture.

Prerequisite: Stochastics I  and  Analysis I — III.

Inhalt

• This course is the sequel of the course of Stochastics I. The main objective is to go beyond the first principles in probability theory by introducing the general language of measure theory, and the application of this framework in a wide variety of probabilistic scenarios.
  More precisely, the course will cover the following aspects of probability theory:
• Measure theory and the Lebesgue integral
• Convergence of random variables and 0-1 laws
• Generating functions: branching processes and characteristic functions
• Markov chains
• Introduction to martingales

Content:

Selection from the following topics:

• Enumeration (twelvefold way, inclusion-exclusion, double counting, recursions, generating functions, inversion, Ramsey's Theorem, asymptotic counting)
• Discrete Structures (graphs, set systems, designs, posets, matroids)
• Graph Theory (trees, matchings, connectivity, planarity, colorings)

• J. Matousek, J. Nesetril (2002/2007): An Invitation to Discrete Mathematics, Oxford University Press, Oxford/Diskrete Mathematik, Springer Verlag, Berlin, Heidelberg.
• L. Lovasz, J. Pelikan, K. Vesztergombi (2003): Discrete Mathemtics - Elementary and Beyond/Diskrete Mathematik, Springer Verlag, New York.
• N. Biggs (2004): Discrete Mathematics. Oxford University Press, Oxford.
• M. Aigner (2004/2007): Diskrete Mathematik, Vieweg Verlag, Wiesbaden/Discrete Mathemattics, American Mathematical Society, USA.
• D. West (2011): Introduction to Graph Theory. Pearson Education, New York.

Target group:

BMS students, Master and Bachelor students

Content:

Selection from the following topics:

• Counting (basics, double counting, Pigeonhole Principle, recursions, generating functions, Inclusion-Exclusion, inversion, Polya theory)
• Discrete Structures (graphs, set systems, designs, posets, matroids)
• Graph Theory (trees, matchings, connectivity, planarity, colorings)
• Algorithms (asymptotic running time, BFS, DFS, Dijkstra, Greedy, Kruskal, Hungarian, Ford-Fulkerson)

Contents

Advanced topcis in algorithm design with a changing focus. The topic is determined newly in each semester. For example, we might consider algorithms for problems on graphs, such as connectivity, shortest paths, or network flows.

Target audience

Masters students in computer science and mathematics.

Recommended prerequisites

"Advanced algorithms" or a similar class.

Spezialliteratur aus Zeitschriften

Inhalt

Diese Vorlesung für das Bachelorstudium soll als natürliche Fortsetzung von Lineare Algebra I und II Fundamente legen für Vorlesungen/Zyklen wie Diskrete Geometrie, Algebraische Geometrie und Differenzialgeometrie.

Sie behandelt grundlegende Modelle der Geometrie, insbesondere

euklidische, affine, sphärische, projektive und hyperbolische Geometrie,Möbiusgeometrie, Polarität und Dualität Strukturgruppen, Messen (Längen, Winkel, Volumina), explizite Berechnungen und Anwendungen, Beispiele sowie Illustrationsthemen;

Dabei werden weitere Bezüge hergestellt, zum Beispiel zur Funktionentheorie und zur Numerik.

    • Apéry, François: Models of the real projective plane. Computer
graphics of Steiner and Boy surfaces. With a preface by Egbert
Brieskorn. Friedr. Vieweg & Sohn, Braunschweig, 1987. xii+156 pp.
    • Berchtold, Florian: Geometrie. Von Euklid bis zur hyperbolischen
Geometrie mit Ausblick auf angrenzende Gebiete. Springer Spektrum,
Heidelberg, 2017. x+210 pp.
    • [Geo] Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J.:
Geometry. 2nd ed. Cambridge University Press, Cambridge, 2012. xiv+587
pp.
    • Lüneburg, Heinz: Die euklidische Ebene und ihre Verwandten.
Birkhäuser, Basel, 1999. viii+207 S.
    • Scheid, Harald; Schwarz, Wolfgang: Elemente der Geometrie. 5th rev.
ed. Springer Spektrum, Heidelberg, 2017. ix+373 pp.
    • Skript

The course provides an introduction to fundamental methods and principles in visualization of mathematics, data sciences, and industrial applications. We discuss mathematical foundations and data structures. We present several core algorithms and tools.

The students get practical hands-on experience with at least one visualization system during a larger project work and develop strategies and visualizations for appropriate problems.

Topics include:

• Data structures and key algorithms
• Visualization of curves, surfaces and volumes
• Applications to scientific and industrial problems
• 3D modelling and animation
• Concepts from dynamic geometry
• 3DScanning and 3D-printing basics

Literature: will be announced in the course.

Previous knowledge: Analysis I + II, Linear Algebra I

Previous knowledge: Analysis I + II, Linear Algebra I

Content:
This is the first part of a three semester introduction into the basic mathematical field of Analysis. Differential and integral calculus in a real variable will be covered. Topics:

1. fundamentals, elementary logic, ordered pairs, relations, functions, domain and range of a function, inverse functions (injectivity, surjectivity)
2. numbers, induction, calculations in R, C
3. arrangement of R, maximum and minimum, supremum and infimum of real sets, supremum / infimum completeness of R, absolute value of a real number, Q is dense in R
4. sequences and series, limits, cauchy sequences, convergence criteria, series and basic principles of convergence
5. topological aspects of R, open, closed, and compact real sets
6. sequences of functions, series of functions, power series
7. properties of functions, boundedness, monotony, convexity
8. continuity, limits and continuity of functions, uniform continuity, intermediate value theorems, continuity and compactness
9. differentiability, concept of the derivative, differentiation rules, mean value theorem, local and global extrema, curvature, monotony, convexity
10. elementary functions, rational functions, root functions, exponential functions, angular functions, hyperbolic functions, real logarithm, inverse trigonometric functions, curve sketching
11. beginnings of integral calculus

Detailed Information can be found on the Homepage of 19202801 Analysis I.

Literature:

• Bröcker, Theodor: Analysis 1, Spektrum der Wissenschaft-Verlag.
• Forster, Otto: Analysis 1, Vieweg-Verlag.
• Spivak, Michael: Calculus, 4th Edition.

Viele Analysis Bücher sind auch über die Fachbibliothek der FU Berlin elektronisch verfügbar.

Bei Schwierigkeiten mit den Grundbegriffen Menge, Abbildung etc. ist die folgende Ausarbeitung empfehlenswert:

• Scheerer, Hans: Brückenkurs, Skript FU Berlin 2006.

Content:

• Basic terms/concepts: sets, maps, equivalence relations, groups, rings, fields
• Linear equation systems: solvability criteria, Gauss algorithm
• Vector spaces: linear independence, generating systems and bases, dimension, subspaces, quotient spaces, cross products in R^3,
• Linear maps: image and rank, relationship to matrices, behaviour under change of basis
• Dual vector spaces: multilinear forms, alternating and symmetric bilinear forms, relationship to matrices, change of basis
• Determinants: Cramer's rule, eigenvalues and eigenvectors

Prerequisites:

Participation in the preparatory course (Brückenkurs) is highly recommended.

• Siegfried Bosch, Lineare Algebra, 4. Auflage, Springer-Verlag, 2008;
• Gerd Fischer, Lernbuch Lineare Algebra und Analytische Geometrie, Springer-Verlag, 2017;
• Bartel Leendert van der Waerden, Algebra Volume I, 9th Edition, Springer 1993;

Zu den Grundlagen

• Kevin Houston, Wie man mathematisch denkt: Eine Einführung in die mathematische Arbeitstechnik für Studienanfänger, Spektrum Akademischer Verlag, 2012

• Whiteboard: Signing up for the course with the FU Whiteboard system is mandatory for participating. Whiteboard will also serve as a course homepage.
• Active participation: A problem set will be published every week. Each problem set has to be solved within one week. Submissions are allowed by groups of max. two students. For active participation, 50% of the total possible amount of points have to be reached. Moreover, we expect active participation in the tutorials.
• Regular participation: Participating in the lectures is recommended. Participation in the tutorials is required.
• Covid: The modus operandi of the course (online, hybrid, presence) will be determined at the beginning of the semester depending on the Covid situation. Please follow the Whiteboard homepage and announcements.

Inhalt:

Die Auswahl der behandelten numerischen Verfahren enthält Polynominterpolation, Newton-Cotes-Formeln zur numerische Integration und Euler-Verfahren für lineare Differentialgleichungen.

Studierende der Mathematik (Monobachelor und Lehramt) und Bioinformatik, sowie Numerikinteressierte aus Physik, Informatik und anderen Natur- und Geisteswissenschaften.

Analysis in one of the two fundamental courses in mathematics. In the lectures Analysis I - III we treat mathematical induction, convergence, sequences and series, compactness, differentiation and integration, transformation theorems, the implicit function theorem, and more. These are the tools of the trade and no mathematician gets along without them.

The trade is learned by solving the exercises and participating in the exercise groups.

Homepage: http://dynamics.mi.fu-berlin.de/lectures/22SS-Fiedler-Analysis2/index.php

Contents:

• Determinants
• Eigenvalues and eigenvectors: diagonalizability, trigonalizability, set of Cayley-Hamilton, Jordanian normal form
• Bilinear forms
• Vectorräume with scalar product: Euclidean, unitary vectorräume, orthogonal projection, isometries, self-adjusted images, Gram-Schmidt orthonormalization methods, major axis transformation

Prerequisites:
Linear Algebra I
Literature:

Will be mentioned in the lecture.

See http://page.mi.fu-berlin.de/werner99/.

best approximation of functions, least squares problems, interpolation, numerical quadrature, computation of eigenvalues and eigenvectors, numerical methods for initiatl value problems (part 1)

Stoer, Josef und Roland Bulirsch: Numerische Mathematik - eine Einführung, Band 1. Springer, Berlin, 2005.

Proseminar/ Seminar in Group Theory

Content: In this seminar we will start with an introduction to the concepts about profinite groups, namely, inverse and direct limits, pro-C- groups for various group theoretical classes C, Sylow theory for profinite groups, finitely generated profinite groups, profinite groups as Galois groups, free profinite groups, profinite topologies, some special profinite groups  like profinite Frobenius groups, automorphism group of a profinite group, profinite completion, profinite torsion groups.

http://page.mi.fu-berlin.de/ersoy/ProfiniteGroups

Literature:
Luis Ribes and Pavel Zalesski: Profinite Groups

Prerequisite: Linear Algebra, a course in Algebra or Group Theory.

Content: The seminar will concern the discussion of state-of-art techniques in molecular simulation which allow for a simulation of several space (especially) and time scale within one computational approach.

The discussion will concerns both, specific computational coding and conceptual developments.

Related Basic Literature:

(1) M.Praprotnik, L.Delle Site and K.Kremer, Ann.Rev.Phys.Chem.59, 545-571 (2008)

(2) C.Peter, L.Delle Site and K.Kremer, Soft Matter 4, 859-869 (2008).

(3) M.Praprotnik and L.Delle Site, in "Biomolecular Simulations: Methods and Protocols" L.Monticelli and E.Salonen Eds. Vol.924, 567-583 (2012) Methods Mol. Biol. Springer-Science

Audience: At least 6th semester with a background in statistical and quantum mechanics, Master students and PhD students (even postdocs) are welcome.

The seminar follows (parts of) the book Invitation to Nonlinear Algebra by Michalek und Sturmfels.

It centers around the solution (sets) of systems of polynomial equations as they appear in many other branches of mathematics, such as combinatorics, algebraic topology, commutative algebra, convex and discrete geometry, tensors and multilinear algebra, number theory, representation theory, and symbolic and numerical computation.

Invitation to Nonlinear Algebra by Michalek und Sturmfels.
AMS GSM 211

Vorbesprechung Dienstag 22. Februar um 12.

Preliminary meeting Tuesday, February 22 at 12.

Students present recent papers on topics in delay equations.

Students present recent papers on topics in dynamical systems.

Many real-world processes can naturally be modeled by systems of interacting agents. Approaches reach from informal descriptions of interaction dynamics and their computational implementations to more formal, mathematical models. On the basis of recent journal publications, this seminar will focus on state-of-the art approaches and main challenges for the mathematical modeling of complex social systems. Example applications shall include: Epidemics spreading, opinion dynamics, processes in economic and socio-ecological systems.

The course will cover a selection of the following topics:

• Dynamics on and of social networks

• Agent-based models and their formulation by means of Markov processes (possibly with a connection to chemical reaction systems)

• Population scaling and approximation by differential equations (ODEs, SDEs, PDEs…)

• Numerical simulation techniques

Will be announced in the first session.

The target group of this seminar are BMS students, Master students, and advanced bachelor students.

Prerequisites: Stochastics I-II (Stochastic processes) and Numerics I

Language: English

Function theory is a classical field of mathematics, which deals with the properties of complex-differentiable functions on the complex number plane and has connections to algebra, analysis, number theory and geometry.

The concept of complex differentiability restricts real-differentiable functions from R2 to R2 to angle-preserving images. We will discover that complex-differentiable functions are quite rigid objects, but they are endowed with many amazing analytical, geometric, and visual properties.

A major result discussed in this lecture is Cauchy's integral theorem which states that the integral of any complexly differentiable function along a closed path in the complex plane is zero. We will see many nice consequences of this result, e.g. Cauchy's integral formula, the residual theorem and a proof of the fundamental theorem of algebra, as well as modern graphical representation methods.

Wolfgang Fischer, Ingo Lieb: Funktionentheorie. 8. Auflage. Vieweg, Braunschweig 2003, ISBN 3-528-77247-6.
T. Needham: Anschauliche Funktionentheorie, Oldenbourg Wissenschaftsverlag, (2001).
Reinhold Remmert, Georg Schumacher: Funktionentheorie 1. 5. Auflage. Springer, Berlin 2002, ISBN 3-540-59075-7

Contents:

• Construction of stochastic processes;
• conditional expectation;
• martingales in discrete time: convergence, stopping theorems, inequalities;
• convergence types in stochastics;
• uniform integrability;
• Markov chains in discrete and continuous time: recurrence and transience, invariant measures;
• convergence in distribution of stochatic processes;
• Brownian motion and invariance principle.

Detailed Information can be found on the Homepage of 19212901 Basismodul: Stochastics II.

Literature:

• Klenke: Wahrscheinlichkeitstheorie
• Durrett: Probability. Theory and Examples.

Weitere Literatur wird im Lauf der Vorlesung bekannt gegeben.
Further literature will be given during the lecture.

Prerequisite: Stochastics I  and  Analysis I — III.

Inhalt

• This course is the sequel of the course of Stochastics I. The main objective is to go beyond the first principles in probability theory by introducing the general language of measure theory, and the application of this framework in a wide variety of probabilistic scenarios.
  More precisely, the course will cover the following aspects of probability theory:
• Measure theory and the Lebesgue integral
• Convergence of random variables and 0-1 laws
• Generating functions: branching processes and characteristic functions
• Markov chains
• Introduction to martingales

Inhalt

Diese Vorlesung für das Bachelorstudium soll als natürliche Fortsetzung von Lineare Algebra I und II Fundamente legen für Vorlesungen/Zyklen wie Diskrete Geometrie, Algebraische Geometrie und Differenzialgeometrie.

Sie behandelt grundlegende Modelle der Geometrie, insbesondere

euklidische, affine, sphärische, projektive und hyperbolische Geometrie,Möbiusgeometrie, Polarität und Dualität Strukturgruppen, Messen (Längen, Winkel, Volumina), explizite Berechnungen und Anwendungen, Beispiele sowie Illustrationsthemen;

Dabei werden weitere Bezüge hergestellt, zum Beispiel zur Funktionentheorie und zur Numerik.

    • Apéry, François: Models of the real projective plane. Computer
graphics of Steiner and Boy surfaces. With a preface by Egbert
Brieskorn. Friedr. Vieweg & Sohn, Braunschweig, 1987. xii+156 pp.
    • Berchtold, Florian: Geometrie. Von Euklid bis zur hyperbolischen
Geometrie mit Ausblick auf angrenzende Gebiete. Springer Spektrum,
Heidelberg, 2017. x+210 pp.
    • [Geo] Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J.:
Geometry. 2nd ed. Cambridge University Press, Cambridge, 2012. xiv+587
pp.
    • Lüneburg, Heinz: Die euklidische Ebene und ihre Verwandten.
Birkhäuser, Basel, 1999. viii+207 S.
    • Scheid, Harald; Schwarz, Wolfgang: Elemente der Geometrie. 5th rev.
ed. Springer Spektrum, Heidelberg, 2017. ix+373 pp.
    • Skript

Dieses Projektseminar steht in Verbindung mit „Schule@DecisionTheatreLab“, einem Experimentallabor für Wissenschaftskommunikation gefördert von der Berlin University Alliance und dem Excellenzcluster MATH+. Das Projekt entwickelt ein innovatives Kommunikationsformat basierend auf mathematischen Modellen und führt dieses mit Gruppen von Schüler*innen durch. Decision Theatres sind Diskussionsveranstaltungen, in denen Teilnehmende eine gesellschaftliche Herausforderung mit Wissenschaftler*innen diskutieren und dabei mit einem mathematischen Modell experimentieren können. In School Lab Workshops geben Mathematiker*innen in einer Mischung aus Workshops und Vorträgen an Schulen erste Erklärungen und Anleitungen für die Entwicklung mathematischer Modellierungen und deren Anwendung auf relevante Fragen der Gesellschaft.

Das Projektseminar ist interdisziplinär ausgerichtet und verbindet mathematische Forschung mit didaktischen und sozialwissenschaftlichen Perspektiven. So werden sowohl Grundlagen des Kommunikationsformats (bspw. mathematische und agenten-basierte Modellierung oder die Arbeit mit empirischen Informationen) als auch ein Bezug zum Mathematikunterricht an Schulen und damit zur Vermittlung von Mathematik erarbeitet. Die Studierenden arbeiten direkt an der Vorbereitung, Durchführung, Beobachtung und Auswertung von Decision Theatre Veranstaltungen mit und entwickeln in Gruppenarbeit einfachere agenten-basierte Modelle, die wiederum in die School Lab Workshops des Projekts einfließen können.

In dem Projektseminar ist ein intensiver Austausch zwischen Studierenden aus dem Monostudiengang und aus dem Lehramtsstudiengang der Mathematik intendiert. Über das Kennenlernen von und die Mitwirkung in einem aktuellen mathematischen wie didaktischen Forschungsprojekt und dessen Abläufe wie Methoden erhalten die Studierenden die Chance jeweils ihren Blick über den Tellerand ihres Studiengangs hinaus zu erweitern. Es ergeben sich folgende Schwerpunkte:

• Agenten-basierte Modelle: Definition, Implementierung, Sensitivitätsanalyse und Kalibrierung, Politikszenarien, mathematische Formulierung (z.B. als Markovkette)
• synthetische Populationen: Daten, Algorithmen, Software Tools
• Weiterentwicklung von mathematischen Modellen im Dialog mit Nicht-Wissenschaftler*innen (hier insbesondere Schüler*innen)
• Chancen der Einbettung des Kommunikationsformates und neue Perspektiven auf Modellieren in der Vermittlung von Mathematik (bspw. im Unterricht)
• Unterrichtsbeobachtung und Interaktion mit Schüler*innengruppen

Wird in der ersten Sitzung bekannt gegeben.

Präsenzunterricht vorbehaltlich der pandemischen Lage.

Die Veranstaltungen mit Schüler*innen können ggf außerhalb der üblichen Veranstaltungszeit stattfinden.

Voraussetzungen:

• mindestens ein Interesse an Programmieren, grundlegende Programmierkenntnisse wären wünschenswert
• Interesse an mathematischer Modellierung und gesellschaftlichen Diskursen

Given a square matrix A, the first order system d/dt u(t) = A u(t) can be solved by the matrix exponential: u(t) = exp(t A) u(0). In particular, as a function of t this gives rise to a one-parameter group.

Similarly, partial differential equations (e.g., the heat equation) fit within this framework if the matrix A is replaced by a suitable operator. For the heat equation, it is the Laplace operator. However, since the Laplace operator is not bounded, the series for the exponential will not converge and this raises the question how the exponential should be defined in this case. Moreover, solutions of the heat equation exist in general only for positive times and the solution operator can form at best a semigroup.

This course aims at developing a solid mathematical foundation to this approach. As a byproduct, we obtain a very general solution concept for certain evolution equations and a framework to characterize long-term behavior of the solutions.

• K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations. Springer. 2000.
• A. Pazy. Semigroups of linear operators and applications to partial differential equations. Springer. 1983.

Given a square matrix A, the first order system d/dt u(t) = A u(t) can be solved by the matrix exponential: u(t) = exp(t A) u(0). In particular, as a function of t this gives rise to a one-parameter group.

Similarly, partial differential equations (e.g., the heat equation) fit within this framework if the matrix A is replaced by a suitable operator. For the heat equation, it is the Laplace operator. However, since the Laplace operator is not bounded, the series for the exponential will not converge and this raises the question how the exponential should be defined in this case. Moreover, solutions of the heat equation exist in general only for positive times and the solution operator can form at best a semigroup.

This course aims at developing a solid mathematical foundation to this approach. As a byproduct, we obtain a very general solution concept for certain evolution equations and a framework to characterize long-term behavior of the solutions.

• K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations. Springer. 2000.
• A. Pazy. Semigroups of linear operators and applications to partial differential equations. Springer. 1983.

Content:

Selection from the following topics:

• Enumeration (twelvefold way, inclusion-exclusion, double counting, recursions, generating functions, inversion, Ramsey's Theorem, asymptotic counting)
• Discrete Structures (graphs, set systems, designs, posets, matroids)
• Graph Theory (trees, matchings, connectivity, planarity, colorings)

• J. Matousek, J. Nesetril (2002/2007): An Invitation to Discrete Mathematics, Oxford University Press, Oxford/Diskrete Mathematik, Springer Verlag, Berlin, Heidelberg.
• L. Lovasz, J. Pelikan, K. Vesztergombi (2003): Discrete Mathemtics - Elementary and Beyond/Diskrete Mathematik, Springer Verlag, New York.
• N. Biggs (2004): Discrete Mathematics. Oxford University Press, Oxford.
• M. Aigner (2004/2007): Diskrete Mathematik, Vieweg Verlag, Wiesbaden/Discrete Mathemattics, American Mathematical Society, USA.
• D. West (2011): Introduction to Graph Theory. Pearson Education, New York.

Target group:

BMS students, Master and Bachelor students

Content:

Selection from the following topics:

• Counting (basics, double counting, Pigeonhole Principle, recursions, generating functions, Inclusion-Exclusion, inversion, Polya theory)
• Discrete Structures (graphs, set systems, designs, posets, matroids)
• Graph Theory (trees, matchings, connectivity, planarity, colorings)
• Algorithms (asymptotic running time, BFS, DFS, Dijkstra, Greedy, Kruskal, Hungarian, Ford-Fulkerson)

Course Overview
This is a beginning course from the series of three courses Topology I—III:

1. Basic notions: topological spaces, continuous maps, connectedness, compactness, products, coproducts, quotients.
2. Groups acting on topological spaces
3. Gluing constructions.
4. Homotopies between continuous maps, degree of a map, fundamental group.
5. Seifert-van Kampen Theorem.
6. Covering spaces.

Detailed Information can be found on the Homepage of the lecture.

Literature:

1. Tammo tom Dieck: Topologie, De Gruyter Lehrbuch
2. Allen Hatcher: Algebraic Topology, Chapter I. Also available online from the author's website (english)
3. James R. Munkres: Topology, Prentice Hall (english)

Content:
This is the first part of a three semester introduction into the basic mathematical field of Analysis. Differential and integral calculus in a real variable will be covered. Topics:

1. fundamentals, elementary logic, ordered pairs, relations, functions, domain and range of a function, inverse functions (injectivity, surjectivity)
2. numbers, induction, calculations in R, C
3. arrangement of R, maximum and minimum, supremum and infimum of real sets, supremum / infimum completeness of R, absolute value of a real number, Q is dense in R
4. sequences and series, limits, cauchy sequences, convergence criteria, series and basic principles of convergence
5. topological aspects of R, open, closed, and compact real sets
6. sequences of functions, series of functions, power series
7. properties of functions, boundedness, monotony, convexity
8. continuity, limits and continuity of functions, uniform continuity, intermediate value theorems, continuity and compactness
9. differentiability, concept of the derivative, differentiation rules, mean value theorem, local and global extrema, curvature, monotony, convexity
10. elementary functions, rational functions, root functions, exponential functions, angular functions, hyperbolic functions, real logarithm, inverse trigonometric functions, curve sketching
11. beginnings of integral calculus

Detailed Information can be found on the Homepage of 19202801 Analysis I.

Literature:

• Bröcker, Theodor: Analysis 1, Spektrum der Wissenschaft-Verlag.
• Forster, Otto: Analysis 1, Vieweg-Verlag.
• Spivak, Michael: Calculus, 4th Edition.

Viele Analysis Bücher sind auch über die Fachbibliothek der FU Berlin elektronisch verfügbar.

Bei Schwierigkeiten mit den Grundbegriffen Menge, Abbildung etc. ist die folgende Ausarbeitung empfehlenswert:

• Scheerer, Hans: Brückenkurs, Skript FU Berlin 2006.

Analysis in one of the two fundamental courses in mathematics. In the lectures Analysis I - III we treat mathematical induction, convergence, sequences and series, compactness, differentiation and integration, transformation theorems, the implicit function theorem, and more. These are the tools of the trade and no mathematician gets along without them.

The trade is learned by solving the exercises and participating in the exercise groups.

Homepage: http://dynamics.mi.fu-berlin.de/lectures/22SS-Fiedler-Analysis2/index.php

Content:

• Basic terms/concepts: sets, maps, equivalence relations, groups, rings, fields
• Linear equation systems: solvability criteria, Gauss algorithm
• Vector spaces: linear independence, generating systems and bases, dimension, subspaces, quotient spaces, cross products in R^3,
• Linear maps: image and rank, relationship to matrices, behaviour under change of basis
• Dual vector spaces: multilinear forms, alternating and symmetric bilinear forms, relationship to matrices, change of basis
• Determinants: Cramer's rule, eigenvalues and eigenvectors

Prerequisites:

Participation in the preparatory course (Brückenkurs) is highly recommended.

• Siegfried Bosch, Lineare Algebra, 4. Auflage, Springer-Verlag, 2008;
• Gerd Fischer, Lernbuch Lineare Algebra und Analytische Geometrie, Springer-Verlag, 2017;
• Bartel Leendert van der Waerden, Algebra Volume I, 9th Edition, Springer 1993;

Zu den Grundlagen

• Kevin Houston, Wie man mathematisch denkt: Eine Einführung in die mathematische Arbeitstechnik für Studienanfänger, Spektrum Akademischer Verlag, 2012

• Whiteboard: Signing up for the course with the FU Whiteboard system is mandatory for participating. Whiteboard will also serve as a course homepage.
• Active participation: A problem set will be published every week. Each problem set has to be solved within one week. Submissions are allowed by groups of max. two students. For active participation, 50% of the total possible amount of points have to be reached. Moreover, we expect active participation in the tutorials.
• Regular participation: Participating in the lectures is recommended. Participation in the tutorials is required.
• Covid: The modus operandi of the course (online, hybrid, presence) will be determined at the beginning of the semester depending on the Covid situation. Please follow the Whiteboard homepage and announcements.

Contents:

• Determinants
• Eigenvalues and eigenvectors: diagonalizability, trigonalizability, set of Cayley-Hamilton, Jordanian normal form
• Bilinear forms
• Vectorräume with scalar product: Euclidean, unitary vectorräume, orthogonal projection, isometries, self-adjusted images, Gram-Schmidt orthonormalization methods, major axis transformation

Prerequisites:
Linear Algebra I
Literature:

Will be mentioned in the lecture.

See http://page.mi.fu-berlin.de/werner99/.

Inhalt:

Die Auswahl der behandelten numerischen Verfahren enthält Polynominterpolation, Newton-Cotes-Formeln zur numerische Integration und Euler-Verfahren für lineare Differentialgleichungen.

Studierende der Mathematik (Monobachelor und Lehramt) und Bioinformatik, sowie Numerikinteressierte aus Physik, Informatik und anderen Natur- und Geisteswissenschaften.