Linear Algebraic Group and Finite Groups of Lie Type

Let k be an algebraically closed field. A linear algebraic group over k is a closed subgroup of the General Linear Group over k. In this course we will first cover the basic concepts about linear algebraic groups and their morphisms, examples of algebraic groups, connectedness, dimension, Jordan decomposition, unipotent subgroups. We will classify commutative linear algebraic groups via their Jordan decomposition. Then we will cover tori, characters and cocharacters. Then we will go on with the structure of connected solvable groups and Lie-Kolchin Theorem, actions of linear algebraic groups, existence of rational representations, properties of the Borel subgroup and Borel fixed point theorem. We will define the Lie algebra of a linear algebraic group and the adjoint representation. In the second chapter we will introduce root systems and the classification theorem of Chevalley to study the structure of reductive and semisimple linear algebraic groups. Then we will study BN pairs and Bruhat decomposition, parabolic subgroups and Levi decomposition, subgroups of maximal rank, Borel-de Siebenthal theorem. We will prove some results about centralizers and conjugacy classes in simple linear algebraic groups. In the third chapter we will deal with endomorphisms of linear algebraic groups and then finite groups of Lie type, as fixed points of Steinberg endomorphisms. We will classify simple groups of Lie type. We will cover Weyl groups, root systems and root subgroups. We will end the course with a discussion on maximal subgroups of finite classical groups, and theorems of Liebeck, Seitz and Aschbacher.

Linear Algebraic Groups and Finite Groups of Lie Type, Donna Testerman- Gunther Malle

Supplementary reading:

Endomorphisms of Linear Algebraic Groups, R. Steinberg

Simple Groups of Lie Type, R. Carter

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.

Literatur

1. Marcel Berger. Geometry I
2. David A. Brannan, Matthew F. Esplen, and Jeremy J. Gray. Geometry
3. Gerd Fischer. Analytische Geometrie
4. V.V. Prasolov und V.M. Tikhomirov. Geometry

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.

Literatur:

E. Freitag and R. Busam 'Complex analysis', (Springer) 2nd Edition 2009 (the original German version is called 'Funktionentheorie')

Contents:

• Construction of stochastic processes;
• conditional expectation;
• martingales and Markov chains in discrete time;
• weak convergence;
• first time-continuous processes (in particular Brownian motion)

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)

Online algorithms are used in scenarios where the input is revealed gradually, for instance in learning tasks, or when there is interaction between agents, or between an agent and its environment. The primary concern is dealing with uncertainty and to find good solutions even with partial information; in this setting computational efficiency often takes a secondary role.

The field of online algorithms is one of the most active areas of algorithmic research with a rich set of established techniques developed during the past decades, as well as recent exciting breakthroughs and open questions.

The seminar is aimed at students interested in algorithms and theoretical computer science.  Requirement: ALP3/HA or similar algorithmic background, mathematical maturity.

We can cover some subset of the following topics: caching, paging, load balancing, routing, navigation, data structuring, k-server, scheduling, ski rental, online learning, packing and covering, coloring, primal-dual methods, online algorithms with advice, and others.

Presentations will be in English, topics can be based on articles or selected material from the books:

• Borodin and El-Yaniv: Online Computation and Competitive Analysis, Cambridge University Press, 1998
• Buchbinder and Naor: The Design of Competitive Online Algorithms via a Primal-Dual Approach, Foundations and Trends in TCS, 2009
• Fiat and Woeginger: Online Algorithms, The State of the Art, Springer, 1998

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.

Literatur

1. Marcel Berger. Geometry I
2. David A. Brannan, Matthew F. Esplen, and Jeremy J. Gray. Geometry
3. Gerd Fischer. Analytische Geometrie
4. V.V. Prasolov und V.M. Tikhomirov. Geometry

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

Passende Literatur wird in der Vorlesung bekanntgegeben.

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.

Content

1. Additions to Analysis I. Non-authentic integrals
2. Uniform convergence of function sequences. Power series. Sentence of Taylor.
3. Elements of topology. Standardized and metric spaces. Open quantities. Convergence. Completed quantities. Consistency. Compactness
4. Differential calculus of several variables. Partial, total and continuous differentiability. Block via the inverse function. Block of implicit functions.
5. Iterated integrals.
6. Ordinary differential equations. Basic terms, elementary solvable differential equations, existential and unambiguous results for systems.

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/.

Numerical methods for: iterative solution of nonlinear systems of equations (fixpoint and Newton methods), curve fitting, interpolation, numerical quadrature, and numerics of ODE systems.

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

Linear Algebraic Group and Finite Groups of Lie Type

Let k be an algebraically closed field. A linear algebraic group over k is a closed subgroup of the General Linear Group over k. In this course we will first cover the basic concepts about linear algebraic groups and their morphisms, examples of algebraic groups, connectedness, dimension, Jordan decomposition, unipotent subgroups. We will classify commutative linear algebraic groups via their Jordan decomposition. Then we will cover tori, characters and cocharacters. Then we will go on with the structure of connected solvable groups and Lie-Kolchin Theorem, actions of linear algebraic groups, existence of rational representations, properties of the Borel subgroup and Borel fixed point theorem. We will define the Lie algebra of a linear algebraic group and the adjoint representation. In the second chapter we will introduce root systems and the classification theorem of Chevalley to study the structure of reductive and semisimple linear algebraic groups. Then we will study BN pairs and Bruhat decomposition, parabolic subgroups and Levi decomposition, subgroups of maximal rank, Borel-de Siebenthal theorem. We will prove some results about centralizers and conjugacy classes in simple linear algebraic groups. In the third chapter we will deal with endomorphisms of linear algebraic groups and then finite groups of Lie type, as fixed points of Steinberg endomorphisms. We will classify simple groups of Lie type. We will cover Weyl groups, root systems and root subgroups. We will end the course with a discussion on maximal subgroups of finite classical groups, and theorems of Liebeck, Seitz and Aschbacher.

Linear Algebraic Groups and Finite Groups of Lie Type, Donna Testerman- Gunther Malle

Supplementary reading:

Endomorphisms of Linear Algebraic Groups, R. Steinberg

Simple Groups of Lie Type, R. Carter

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.

Students present recent papers on topics in delay equations.

You can find the schedule here.

Students present recent papers on topics in dynamical systems.

You can find the schedule here.

Convex Analysis deals with the properties of convex functions, convex sets and their impact on minimization problems. This seminar addresses bachelor students who have already attended Analysis I-III. Some knowledge in Functional Analysis is also helpful.

Content:  The goal of the course is the practical implementation and realization of stochastic models and methods. We will learn algorithms and statistical methods with applications in the "real world" and students will implement them and try them out on datasets if necessary. Concrete contents will be mentioned beforehand on the seminar website.

The seminar will be held as a block course. In the first three weeks there will be three lectures explaining basics of the implementation of stochastic methods (random numbers, Monte Carlo simulation, descriptive statistics, simulation of stochastic processes). At the beginning of the semester, students will be given several projects to work on and implement by the end of the semester. In the last weeks of the semester, there will be several dates with short student presentations in which the project results will be presented.

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

Literatur wir in der Vorbesprechung bekanntgegeben.

Literature will be announced in the preliminary discussion

Target Group: The seminar is aimed at Bachelor's and Master's students (level adjustment by choosing appropriate projects).
Prerequisites: Stochastics I. Parallel attendance of Stochastics II is recommended.

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.

Literatur:

E. Freitag and R. Busam 'Complex analysis', (Springer) 2nd Edition 2009 (the original German version is called 'Funktionentheorie')

Contents:

• Construction of stochastic processes;
• conditional expectation;
• martingales and Markov chains in discrete time;
• weak convergence;
• first time-continuous processes (in particular Brownian motion)

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.

Literatur

1. Marcel Berger. Geometry I
2. David A. Brannan, Matthew F. Esplen, and Jeremy J. Gray. Geometry
3. Gerd Fischer. Analytische Geometrie
4. V.V. Prasolov und V.M. Tikhomirov. Geometry

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; dabei werden Decision Theatres und School Lab Workshops kombiniert. 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 Modelle 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 fachwissenschaftliche Grundlagen des Kommunikationsformats (bspw. mathematische und agenten-basierte Modellierung oder die Arbeit mit empirischen Daten) 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 und analysieren zudem in Gruppenarbeit selbst kleine agentenbasierte Modelle.

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

• Agenten-basierte Modellierung: 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
• im Kurs wird die Modellierungsumgebung NetLogo verwendet, die Bereitschaft, sich hier einzuarbeiten ist also erforderlich

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.

Literature:

1. Tammo tom Dieck: Topologie, De Gruyter Lehrbuch
2. Allen Hatcher: Algebraic Topology, Chapter I. Also available online from the author's website
3. Klaus Jänich: Topologie, Springer-Verlag
4. Gerd Laures, Markus Szymik: Grundkurs Topologie, Spektrum Akademischer Verlag
5. James R. Munkres: Topology, Prentice Hall

The lecture serves as an introduction to fundamental ideas and concepts for scientific visualization with applications to mathematics, computer graphics and natural sciences. Hereby, it will focused on a selection of the following topics:

• Basic data structures
• Representation of geometries as discrete meshes
• Discrete surface theory
• Subdivision curves and surfaces
• Wavelets
• Modelling
• Vector field visualization
• Visualization software and typical examples

Knowledge of beginning mathematics courses required.

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

Passende Literatur wird in der Vorlesung bekanntgegeben.

Content

1. Additions to Analysis I. Non-authentic integrals
2. Uniform convergence of function sequences. Power series. Sentence of Taylor.
3. Elements of topology. Standardized and metric spaces. Open quantities. Convergence. Completed quantities. Consistency. Compactness
4. Differential calculus of several variables. Partial, total and continuous differentiability. Block via the inverse function. Block of implicit functions.
5. Iterated integrals.
6. Ordinary differential equations. Basic terms, elementary solvable differential equations, existential and unambiguous results for systems.

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 R3
• 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 matices, 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

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.

Numerical methods for: iterative solution of nonlinear systems of equations (fixpoint and Newton methods), curve fitting, interpolation, numerical quadrature, and numerics of ODE systems.

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)

Inhalt:

This is the second in a series of three courses on discrete geometry. The aim of the course is a skillful handling of discrete geometric structures with an emphasis on metric and convex geometric properties. In the course we will develop central themes in metric and convex geometry including proof techniques and applications to other areas in mathematics.

The material will be a selection of the following topics:
Linear programming and some applications

• Linear programming and duality
• Pivot rules and the diameter of polytopes

Subdivisions and triangulations

• Delaunay and Voronoi
• Delaunay triangulations and inscribable polytopes
• Weighted Voronoi diagrams and optimal transport

Basic structures in convex geometry

• convexity and separation theorems
• convex bodies and polytopes/polyhedra
• polarity
• Mahler’s conjecture
• approximation by polytopes

Volumes and roundness

• Hilbert’s third problem
• volumes and mixed volumes
• volume computations and estimates
• Löwner-John ellipsoids and roundness
• valuations

Geometric inequalities

• Brunn-Minkowski and Alexandrov-Fenchel inequality
• isoperimetric inequalities
• measure concentration and phenomena in high-dimensions

Geometry of numbers

• lattices
• Minkowski's (first) theorem
• successive minima
• lattice points in convex bodies and Ehrhart's theorem
• Ehrhart-Macdonald reciprocity

Sphere packings

• lattice packings and coverings
• the Theorem of Minkowski-Hlawka
• analytic methods

Applications in optimization, number theory, algebra, algebraic geometry, and functional analysis

The course will use material from P. M. Gruber, " Convex and Discrete Geometry" (Springer 2007) and various other sources. There will be brief lecture notes available for course participants with detailed pointers to the literature.

Solid background in linear algebra and some analysis. Basic knowledge and experience with polytopes and/or convexity (as from the course "Discrete Geometry I") will be helpful. .

Positional Games is a rapidly developing field of combinatorics, whose aim is to systematically develop an extensive mathematical basis for a variety of two-player perfect information games. These range from such popular games as Tic-Tac-Toe and Hex to purely abstract games played on graphs and hypergraphs. The subject of positional games is strongly related to several other branches of combinatorics such as Ramsey theory, extremal graph and set theory, and the probabilistic method. In the course we also develop its strong connection to random graphs and the theory of algorithms, in particular hypergraph coloring, satisfiability, and the Lovász Local Lemma.          

Prerequisite is the succesful completion of the modul Discrete Mathematics I (or equivalent, please contact the instructor). 

J. Beck: Combinatorial Games, Tic-Tac-Toe Theory, Cambridge University Press, 2008.

D. Hefetz, M. Krivelevich, M. Stojakovic, T. Szabo: Positional Games, Birkhäuser, 2014. 

Prerequisites: Basic Bachelor Algebra, Probability, and Disrete Mathematics.

Abstract:
Despite the effectiveness of the probabilistic method in extremal combinatorics, explicit constructive approaches remain of paramount importance. On the one hand, they are often superior to purely existential arguments, and, even when they are not, the search for the most efficient deterministic combinatorial structure is naturally motivated by questions of complexity.
The course discusses classic Turan- and Ramsay-type problems of extremal combinatorics from this constructive perspective.
Besides combinatorics, the methods often involve algebraic and probabilistic techniques (affine and projective geometries over finite fields, eigenvalues and quasirandom graphs, the discrete Fourier transform).
For further details please check Prof. Szabó's homepage.

Content:
The seminar covers advanced topics in Extremal and Probabilistic Combinatorics.
Target audience:
BMS students, Master students, or advanced Bachelor students.
Prerequisites:
Prerequisite is the successful completion of the modul Discrete Mathematics II or III (or equivalent background: please contact the instructor).

This seminar will be about polytopes.

Themenvergabe und speziellere Literaturangaben in der Vorbesprechung zum Seminar.

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.

Literature:

1. Tammo tom Dieck: Topologie, De Gruyter Lehrbuch
2. Allen Hatcher: Algebraic Topology, Chapter I. Also available online from the author's website
3. Klaus Jänich: Topologie, Springer-Verlag
4. Gerd Laures, Markus Szymik: Grundkurs Topologie, Spektrum Akademischer Verlag
5. James R. Munkres: Topology, Prentice Hall

The lecture serves as an introduction to fundamental ideas and concepts for scientific visualization with applications to mathematics, computer graphics and natural sciences. Hereby, it will focused on a selection of the following topics:

• Basic data structures
• Representation of geometries as discrete meshes
• Discrete surface theory
• Subdivision curves and surfaces
• Wavelets
• Modelling
• Vector field visualization
• Visualization software and typical examples

Knowledge of beginning mathematics courses required.

This lecture will continue the introduction to algebraic topology. The emphasis will be on homotopy theory: homotopy sets and groups, Puppe sequences, cofibrations and fibrations, the theorem of Blakers-Massey, colimits and limits, the theorem of Hurewicz, introduction to spectral sequences, ...

Detailed Information can be found on the Homepage of 19215101 Aufbaumodul: Topologie III.

Tammo tom Dieck: Algebraic topology, EMS Textbooks in Mathematics 2008

We strongly recommend to also take the course >Homotopy Theory<.

The Seminar will deal with advanced topics of topology and homotopy theory. We expect from the participants familiarity with basic concepts of topology like the ones covered in the lectures Topologie I and II.

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

Literatur wird auf der Homepage angegeben.
Literature will be given on the homepage.

The mathematical modeling of many processes in nature and industry leads  
to partial differential equations. Generally, such equations cannot be solved
analytically. It is only possible to compute numerical approximations to the  
solution on the basis of discretized equations. This course studies  
discretizations of elliptic partial differential equations. Major topics are  
finite difference methods and finite element methods.

 

• F. John: Partial Differential Equations. Springer (1982)
• M. Renardy, R. C. Rogers: An introduction to partial differential equations, Springer, 2. Auflage (2004)
• A. Quarteroni, R. Sacco, F. Saleri: Numerische Mathematik 2. Springer (2002)
• D. Braess: Finite Elemente. Springer, 3. Auflage (2002)
• P. A. Raviart, J. M. Thomas: Introduction à l'analyse numérique des équations aux dérivées partielles, Dunod (1998)

Prerequisites

Prerequisites for this course are basic knowledge in calculus (Analysis I-III) and Numerical Analysis (Numerik I). Some knowledge in Functional Analysis will help a lot.

Homepage:Wiki der Numerik II

Content:
Geometric partial differential equations are describing the evolution ofand processes on surfaces. Geometric flows such as the now classical mean curvature flow, Willmore flow, and pdes on moving surfaces are typical examples. In this lecture, we will consider various formulations including phase field models of Allen-Cahn and Cahn-Hilliard type and concentrate on basic numerical techniques such as surface finite element methods, adaptivity,unfitted finite element methods, and efficient numerical solvers. Zielgruppe Advanced students in the Master Program Mathematics.Various possible topics for a Master thesis will come up during this course. Voraussetzungen Basic knowledge on partial differential equations and their numericalsolution (e.g. Numerik III).

Target audience:
This lecture is a continuation of the preceding course on "Numerical methods for partial differential equations (Numerik III)". It is intended to broaden the way towards a master thesis in the field of computational PDEs.

Prerequisites:
Participants should have some knowledge about PDEs and their numerical approximation by finite elements as provided, e.g., by the preceing course on "Numerical methods for partial differential equations (Numerik III)".

Homepage: More Information can be found on the Homepage

•  Winkelmann and Schütte (2020): Stochastic Dynamics in Computational Biology, in: Frontiers in Applied Dynamical Systems: Reviews and Tutorials, Volume 8 (Springer)
• Gardiner (2004): Stochastic methods: a handbook for the natural and social sciences (Springer)
• Riley, Hobson and Bence (2002): Mathematical Methods for Physics and Engineering (Cambridge)
• Deuflhard and Roeblitz (2015): A guide to numerical modelling in systems biology Texts in computational science and engineering, Volume 12 (Springer) 

The following books will be relevant:

• O. P. Le Maître and O. M. Knio. Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics. Scientific Computation. Springer, New York, 2010.
• R. C. Smith. Uncertainty Quantification: Theory, Implementation, and Applications, volume 12 of Computational Science & Engineering. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2014.
• T. J. Sullivan. Introduction to Uncertainty Quantification. Springer, New York, in press.
• D. Xiu. Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton, NJ, 2010.

Climate and Climate Impact Research must, much more than the classical natural sciences (except astrophysics), rely heavily computational simulation and observational data to gain insights into their objects of study. The reason is that laboratory experiments for the Earth system or for significant parts of it are impossible to realize. As a consequence, tools of mathematical analysis, computational science, and data science play a central role in these research areas.

In this seminar, we will discuss state-of-the-art research papers as well as selected passages from textbooks to provide participating students with some first insights into the broad methodological spectrum adopted routinely in climate and climate impact research.

The lecture serves as an introduction to fundamental ideas and concepts for scientific visualization with applications to mathematics, computer graphics and natural sciences. Hereby, it will focused on a selection of the following topics:

• Basic data structures
• Representation of geometries as discrete meshes
• Discrete surface theory
• Subdivision curves and surfaces
• Wavelets
• Modelling
• Vector field visualization
• Visualization software and typical examples

Knowledge of beginning mathematics courses required.

Content: "Stochastics IV: Malliavin calculus and applications"
I will give an introduction to Malliavin calculus, a fundamental tool in modern stochastic analysis that allows differentiation of random variables defined on a Gaussian probability space, such as Wiener space. It serves as an infinite-dimensional generalization of analytical concepts like Fourier transforms and Sobolev spaces. The calculus enables the development of an „analysis on Wiener space“ and it provides a framework for studying properties of stochastic processes. We will also discuss applications of Malliavin calculus, including:

• existence of transition densities for stochastic differential equations under Hörmander’s condition („the noise allows movement in each direction from each point“)
• hypercontractivity („all moments of polynomials of Gaussians are comparable“, this is crucial for the pathwise approach to SPDEs)
• the fourth moment theorem (a universality result akin to the central limit theorem, but it applies in very different situations than the CLT and strikingly we only need to check convergence of the second and fourth moment to get weak convergence)
• construction of two-dimensional Euclidean quantum field theories.

We will start with a crash course on (finite-dimensional) distributions, Fourier transforms and Sobolev spaces. Then we will mostly follow Hairer’s lecture notes.

Detailed Information can be found on the Homepage of 19242101 Stochastics IV.

Literature:

• Hairer’s lecture notes and further literature:
• Nualart, D. (2006): The Malliavin calculus and related topics. Springer.
• Janson, S. (1997). Gaussian Hilbert spaces. Cambridge University Press.

Prerequisite: Stochastics I -- III.

The lecture aims at a deeper understanding of cryptographic algorithms, especially which design criteria have to be considered for the development of secure encryption algorithms. For that purpose we will get to know and evaluate different cryptanalytic methods for symmetrical and asymmetrical encryption techniques – e.g. linear and differential cryptanalysis on block ciphers, correlation attacks on stream ciphers and algorithms to solve the factorization problem and the discrete logarithm problem. Weaknesses in the implementation, e.g. to exploit side-channel attacks, will be discussed only peripherally.

Positional Games is a rapidly developing field of combinatorics, whose aim is to systematically develop an extensive mathematical basis for a variety of two-player perfect information games. These range from such popular games as Tic-Tac-Toe and Hex to purely abstract games played on graphs and hypergraphs. The subject of positional games is strongly related to several other branches of combinatorics such as Ramsey theory, extremal graph and set theory, and the probabilistic method. In the course we also develop its strong connection to random graphs and the theory of algorithms, in particular hypergraph coloring, satisfiability, and the Lovász Local Lemma.          

Prerequisite is the succesful completion of the modul Discrete Mathematics I (or equivalent, please contact the instructor). 

J. Beck: Combinatorial Games, Tic-Tac-Toe Theory, Cambridge University Press, 2008.

D. Hefetz, M. Krivelevich, M. Stojakovic, T. Szabo: Positional Games, Birkhäuser, 2014. 

Prerequisites: Basic Bachelor Algebra, Probability, and Disrete Mathematics.

Abstract:
Despite the effectiveness of the probabilistic method in extremal combinatorics, explicit constructive approaches remain of paramount importance. On the one hand, they are often superior to purely existential arguments, and, even when they are not, the search for the most efficient deterministic combinatorial structure is naturally motivated by questions of complexity.
The course discusses classic Turan- and Ramsay-type problems of extremal combinatorics from this constructive perspective.
Besides combinatorics, the methods often involve algebraic and probabilistic techniques (affine and projective geometries over finite fields, eigenvalues and quasirandom graphs, the discrete Fourier transform).
For further details please check Prof. Szabó's homepage.

This lecture will continue the introduction to algebraic topology. The emphasis will be on homotopy theory: homotopy sets and groups, Puppe sequences, cofibrations and fibrations, the theorem of Blakers-Massey, colimits and limits, the theorem of Hurewicz, introduction to spectral sequences, ...

Detailed Information can be found on the Homepage of 19215101 Aufbaumodul: Topologie III.

Tammo tom Dieck: Algebraic topology, EMS Textbooks in Mathematics 2008

We strongly recommend to also take the course >Homotopy Theory<.

The course will cover advanced topics from homotopy theory.

Detailed Information can be found on the Homepage of 19234201 Homotopy Theory.

Tammo tom Dieck: Algebraic topology, EMS Textbooks in Mathematics 2008

The course complements and continues topics from Topology III.

Content:
The seminar covers advanced topics in Extremal and Probabilistic Combinatorics.
Target audience:
BMS students, Master students, or advanced Bachelor students.
Prerequisites:
Prerequisite is the successful completion of the modul Discrete Mathematics II or III (or equivalent background: please contact the instructor).

This seminar will be about polytopes.

Themenvergabe und speziellere Literaturangaben in der Vorbesprechung zum Seminar.

Climate and Climate Impact Research must, much more than the classical natural sciences (except astrophysics), rely heavily computational simulation and observational data to gain insights into their objects of study. The reason is that laboratory experiments for the Earth system or for significant parts of it are impossible to realize. As a consequence, tools of mathematical analysis, computational science, and data science play a central role in these research areas.

In this seminar, we will discuss state-of-the-art research papers as well as selected passages from textbooks to provide participating students with some first insights into the broad methodological spectrum adopted routinely in climate and climate impact research.

The Seminar will deal with advanced topics of topology and homotopy theory. We expect from the participants familiarity with basic concepts of topology like the ones covered in the lectures Topologie I and II.

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

Literatur wird auf der Homepage angegeben.
Literature will be given on the homepage.

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)