Lehramt für Mathematik

• Discovering Mathematics I (10 CP)

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  • 19233701 Lecture
    Discovering Mathematics I (Alexandru Constantinescu)
    Schedule: Mi 10:00-12:00, Fr 10:00-12:00 (Class starts on: 2024-10-16)
    Location: A3/Hs 001 Hörsaal (Arnimallee 3-5)
    

    Additional information / Pre-requisites

    This course is aimed at students of teacher training courses.

    Comments

    subject matter

    The focus is on practicing mathematical ways of thinking and working. These are trained on the basis of problems from elementary number theory and elementary geometry.

    compulsory attendance

    Attendance is mandatory for the central exercise on Monday.

  • 19233702 Practice seminar
    Practice seminar for Discovering Mathematics I (Alexandru Constantinescu)
    Schedule: Mo 14:00-16:00, Mo 16:00-18:00, Di 08:00-10:00, Mi 14:00-16:00, Do 12:00-14:00, Do 14:00-16:00 (Class starts on: 2024-10-14)
    Location: A7/SR 031 (Arnimallee 7)
    

• Mathematical Panorama (5 CP)

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  • 19236101 Lecture
    Mathematisches Panorama (Sarah Wolf, Anina Mischau)
    Schedule: Di 12:00-14:00 (Class starts on: 2024-10-15)
    Location: T9/SR 005 Übungsraum (Takustr. 9)
    

    Comments

    This is for a course in German - Short explanation in English:

    Mathematical Panorama is a two-hour overview course for First-Semester students of Mathematics (in particular, but not only, for teacher students) that presents the wide field of modern Mathematics - its history, its topics, its problems, its methods, some basic concepts, applications, etc.

    This could be augmented by another two-hour course Mathematical Panorama II (next summer); both courses together count as the four-hour course Panorama of Mathematics (which is listed in some curricula at FU Berlin). There will also be a seminar in the summer term, where some topics connected to the course are treated actively (by the participants).

    Suggested reading

    • Günter M. Ziegler und Andreas Loos: Panorama der Mathematik, Springer-Spektrum 2018, in Vorbereitung (wird in Auszügen zur Verfügung gestellt)
    • Hans Wußing, 6000 Jahre Mathematik: Eine kulturgeschichtliche Zeitreise, Springer 2009
      • Band 1: Von den Anfängen bis Leibniz und Newton
      • Band 2: Von Euler bis zur Gegenwart
    • Heinz-Wilhelm Alten et al., 4000 Jahre Algebra, Springer 2008
    • Christoph J. Scriba, 5000 Jahre Geometrie, Springer 2009
    • Heinz Niels Jahnke, Geschichte der Analysis: Texte zur Didaktik der Mathematik, Spektrum 1999
    • Richard Courant und Herbert Robbins, What is Mathematics?, Oxford UP 1941 (deutsch: Springer 2010)
    • Phillip J. Davis, Reuben Hersh, The Mathematical Experience, Mariner Books 1999

  • 19236102 Practice seminar
    Übung zu: Mathematisches Panorama (Anina Mischau, Sarah Wolf)
    Schedule: Do 16:00-18:00 (Class starts on: 2024-10-24)
    Location: A7/SR 031 (Arnimallee 7)
    

• Analysis I (10 CP)

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  • 19202801 Lecture
    Analysis I (Marita Thomas)
    Schedule: Di 10:00-12:00, Do 10:00-12:00 (Class starts on: 2024-10-15)
    Location: T9/Gr. Hörsaal (Takustr. 9)
    

    Comments

    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

    Suggested reading

    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.

  • 19202802 Practice seminar
    Tutorial: Analysis I (Marita Thomas)
    Schedule: Mi 12:00-14:00, Mi 14:00-16:00, Mi 16:00-18:00, Fr 08:00-10:00 (Class starts on: 2024-10-16)
    Location: A7/SR 140 Seminarraum (Hinterhaus) (Arnimallee 7)
    

• Linear Algebra I (10 CP)

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  • 19201401 Lecture
    Linear Algebra I (Alexander Schmitt)
    Schedule: Mo 08:00-10:00, Mi 08:00-10:00 (Class starts on: 2024-10-14)
    Location: A3/Hs 001 Hörsaal (Arnimallee 3-5)
    

    Comments

    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.

     

    Suggested reading

    • 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

  • 19201402 Practice seminar
    Practice seminar for Linear Algebra I (Alexander Schmitt)
    Schedule: Mo 10:00-12:00, Mo 16:00-18:00, Mi 10:00-12:00, Fr 14:00-16:00 (Class starts on: 2024-10-14)
    Location: A3/SR 119 (Arnimallee 3-5)
    

• Analysis II (10 CP)

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  • 19211601 Lecture
    Analysis II (Isabelle Schneider)
    Schedule: Di 10:00-12:00, Do 10:00-12:00 (Class starts on: 2024-10-15)
    Location: A3/Hs 001 Hörsaal (Arnimallee 3-5)
    

    Suggested reading

    • O. Forster: Analysis 1 und 2. Vieweg/Springer.
    • Königsberger, K: Analysis 1,2, Springer.
    • E. Behrends: Analysis Band 1 und 2, Vieweg/Springer.
    • H. Heuser: Lehrbuch der Analysis 1 und 2, Teubner/Springer.

  • 19211602 Practice seminar
    Practice seminar for Analysis II (Isabelle Schneider)
    Schedule: Mo 10:00-12:00, Mi 10:00-12:00, Fr 10:00-12:00 (Class starts on: 2024-10-14)
    Location: A6/SR 009 Seminarraum (Arnimallee 6)
    

• Linear Algebra II (10 CP)

  0082fA2.2
  • 19211701 Lecture
    Linear Algebra II (N.N.)
    Schedule: Mi 12:00-14:00, Fr 08:00-10:00 (Class starts on: 2024-10-16)
    Location: A3/Hs 001 Hörsaal (Arnimallee 3-5)
    

    Comments

    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.

  • 19211702 Practice seminar
    Practice seminar for Linear Algebra II (N.N.)
    Schedule: Mo 10:00-16:00 (Class starts on: 2024-10-14)
    Location: A6/SR 025/026 Seminarraum (Arnimallee 6)
    

• Numbers, Equations, Algebraic Structures (10 CP)

  0082fA2.3
  • 19200701 Lecture
    Algebra and Theory of Numbers (Kivanc Ersoy)
    Schedule: Mo 08:00-10:00, Mi 08:00-10:00 (Class starts on: 2024-10-14)
    Location: A3/SR 120 (Arnimallee 3-5)
    

    Comments

    Subject matter:
    Selected topics from:

        Divisibility into rings (especially Z- and polynomial rings); residual classes and congruencies; modules and ideals
        Euclidean, principal ideal and factorial rings
        The quadratic law of reciprocity
        Primality tests and cryptography
        The structure of abel groups (or modules about main ideal rings)
        Symmetric function set
        Body extensions, Galois correspondence; constructions with compasses and rulers
        Non-Label groups (set of Lagrange, normal dividers, dissolvability, sylow groups)

  • 19200702 Practice seminar
    Practice seminar for Algebra and Theory of Numbers (N.N.)
    Schedule: Mi 12:00-18:00 (Class starts on: 2024-10-16)
    Location: A3/SR 119 (Arnimallee 3-5)
    

• Probability and Statistics (10 CP)

  0082fA3.1
  • 19220901 Lecture
    Probability and Statistics (Olaf Parczyk)
    Schedule: Di 08:00-10:00, Do 08:00-10:00 (Class starts on: 2024-10-15)
    Location: A3/Hs 001 Hörsaal (Arnimallee 3-5)
    

    Comments

    Es werden insbesondere folgende Inhalte vermittelt.
    –  Diskrete Wahrscheinlichkeitsräume und -maße
    –  Diskrete und stetige Zufallsvariablen und ihre Verteilungen, wichtige Beispiele
    –  Erwartungswert, (Ko-)Varianz, Korrelation
    –  Bedingte Wahrscheinlichkeit, Unabhängigkeit
    –  Schwaches Gesetz der großen Zahl
    –  Zentraler Grenzwertsatz
    –  Datenanalyse und deskriptive Statistik: Histogramme; empirische Verteilung; Kenngrößen von Stichprobenver-teilungen; Beispiele irreführender deskriptiver Statistiken; lineare Regression
    –  Elementare Begriffe und Techniken des Testens und Schätzens: Maximum-Likelihood-Prinzip; Konfidenzinter-valle; Hypothesentests; Fehler erster und zweiter Art 

     

    Suggested reading

    E. Behrends: Elementary Stochastics, Springer, 2013
        H.-O. Georgii: Stochastics: Introduction to Probability Theory and Statistics, De Gruyter, 2007
        U. Krengel: Introduction to probability theory and statistics, Vieweg, 2005
        D. Meintrup, S. Schäffler, Stochastics: Theory and Applications, Springer, 2005.
        Most of the books listed below are available online at the UB. For this purpose, there is an extensive hand apparatus for stochastics in the mathematic library.

  • 19220902 Practice seminar
    Practice seminar for Probability and Statistics (Olaf Parczyk)
    Schedule: Mo 10:00-12:00, Mo 12:00-14:00, Do 10:00-12:00 (Class starts on: 2024-10-14)
    Location: 1.3.48 Seminarraum T3 (Arnimallee 14)
    

• Proseminar Mathematics - Teacher Training

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  • 19203311 Seminar
    Proseminar/Seminar Gruppentheorie (N.N.)
    Schedule: -
    Location: keine Angabe
    

    Additional information / Pre-requisites

    Participants should feel comfortable with the contents of 'Linear algebra 1', and perhaps 'Linear algebra 2'. If there are interested students who have already done the 'Algebra und Zahlentheorie' module, we will also be able to find interesting topics for them.

    Comments

    In this (pro)seminar we will recapitulate the basics of group theory and study a few deeper properties and theorems. Possible topics are: solvable groups, nilpotent groups, representations of finite groups, the theorem of Schur-Zassenhaus, the theorems of P. Hall.

  • 19213417 Seminar / Undergraduate Course
    Undergraduate Seminar: Analysis (Ehrhard Behrends)
    Schedule: Mo 14:00-16:00 (Class starts on: 2024-10-14)
    Location: A7/SR 140 Seminarraum (Hinterhaus) (Arnimallee 7)
    

    Comments

    Basics of non-standard analysis (according to Nelson)

    In the analysis, calculating with in?nitesimalen sizes is formalized by the Grenzwertbegri?. Robinson has extended the body of real numbers so that it contains in?nitesimale elements, i.e. positive numbers that are smaller than all positive real numbers. Nelson later observed that the axioms of set theory can be expanded so that the "usual body" of the real numbers "standard" and "non-standard elements". Among the non-standard elements ?nden are again those that are positive and smaller than all positive standard elements.

     

    The proseminar will discuss how the basic statements of the analysis can be reformulated and proven with the help of the in?nitesimalen elements. The proseminar is divided into the following sections:

    I. Logical basis of NSA

    II. Analysis I in the language of the NSA

    III. advanced applications of NSA

    IV. A look at Robinson's NSA

    Homepage Prof. Schmitt

     

    Suggested reading

    Behrends: Analysis I und II

  • 19214210 Proseminar
    Science Communication on Mathematics (Anna Maria Hartkopf)
    Schedule: -
    Location: keine Angabe
    

    Comments

    The goal of this seminar is the development of a science communication format on a mathematical topic. 

  • 19229917 Seminar / Undergraduate Course
    Proseminar/Seminar Geometrie / Optimierung / KI / Spieltheorie (Georg Loho)
    Schedule: Di 16:00-18:00 (Class starts on: 2024-10-15)
    Location: A7/SR 031 (Arnimallee 7)
    

    Comments

    Je nach Bedarf: 

    "KI als Werkzeug in der Mathematik" oder "Geometrie & Optimierung" oder etwas wie "Spieltheorie & Gesellschaft"

    Suggested reading

    G. D. James "The representation theory of the symmetric group" Springer, Lecture Notes in Mathemtaics vol 682, 1978

    B. E. Sagan "The Symmetric Group - Representations, Combinatorial Algorithms, and Symmetric Functions" 2nd Edition, 2000

  • 19234810 Proseminar
    Women in the History of Mathematics and Computer Science (N.N.)
    Schedule: -
    Location: keine Angabe
    

    Additional information / Pre-requisites

    For mathematicians and computer scientists in a monobachelor's degree, creditable as ABV!

    Comments

    The seminar focuses on the development and rediscovery of the life stories and the work of some important mathematicians and computer scientists in the 19th and 20th centuries. The life and work of Sophie Germaine (1776-1831), Ada Lovelace (1815-1852), Sonja Kovalevskaya (1850-1891), Emmy Noether (1882-1935), Ruth Moufang (1905-1977), Grace Murray Hopper (1906-1992) and other female scientists are examined.

    The seminar is not about highlighting these women as an exception, because it would only set them on their exotic status. Rather, it is about a historical contextualization of their life and work. This not only enables an exemplary examination of social and cultural inclusion and exclusion processes along the gender category, but also the development of new perspectives on the traditional cultural history of both disciplines. The seminar is based on the approach of researching or discovering learning, i.e. the students will independently prepare and present individual seminar topics in group work. These presentations will then be discussed in the seminar. Through the use of observation sheets, a feedback culture is also to be tested that will be helpful in dealing with pupils and/or colleagues in later professional life.

  • 19241710 Proseminar
    Proseminar Mathematics Panorama (Anna Maria Hartkopf)
    Schedule: -
    Location: keine Angabe
    

    Comments

    Science Communication on Mathematics

    Suggested reading

    1. Hans Wußing, 6000 Jahre Mathematik: Eine kulturgeschichtliche Zeitreise;
    2. Band 1: Von den Anfängen bis Leibniz und Newton, Band 2: Von Euler bis zur Gegenwart, Springer 2009
    3. Heinz-Wilhelm Alten et al., 4000 Jahre Algebra, Springer 2008
    4. Christoph J. Scriba, 5000 Jahre Geometrie, Springer 2009
    5. Heinz-Niels Jahnke, Geschichte der Analysis: Texte zur Didaktik der Mathematik, Spektrum 1999
    6. Richard Courant und Herbert Robbins, Was ist Mathematik?, Springer 2010
    7. Phillip J. Davis, Reuben Hersh, The Mathematical Experience, Mariner Books 1999
    8. Knoebel, Arthur; Laubenbacher, Reinhard; Lodder, Jerry; Pengelley, David
    9. Mathematical masterpieces, Springer 2007
    10. Laubenbacher, Reinhard; Pengelley, David, Mathematical expeditions. Chronicles by the explorers, Springer 1999
    11. sowie abhängig vom Thema

• Computer-Oriented Mathematics I (5 CP)

  0082fA4.1
  • 19200501 Lecture
    Computerorientated Mathematics I (5 LP) (Ralf Kornhuber, Claudia Schillings)
    Schedule: Fr 12:00-14:00 (Class starts on: 2024-10-18)
    Location: T9/Gr. Hörsaal (Takustr. 9)
    

    Comments

    Contents:
    Computers play an important role in (almost) all situations in life today. Computer-oriented mathematics provides basic knowledge in dealing with computers for solving mathematical problems and an introduction to algorithmic thinking. At the same time, typical mathematical software such as Matlab and Mathematica will be introduced. The motivation for the questions under consideration is provided by simple application examples from the aforementioned areas. The content of the first part includes fundamental terms of numerical calculation: number representation and rounding errors, condition, efficiency and stability.

    Homepage: All current information on lectures and lectures

    Suggested reading

    Literatur: R. Kornhuber, C. Schuette, A. Fest: Mit Zahlen Rechnen (Skript zur Vorlesung)

  • 19200502 Practice seminar
    Practice seminar for Computerorientated Mathematics I (5 LP) (André-Alexander Zepernick)
    Schedule: Mo 08:00-16:00 (Class starts on: 2024-10-14)
    Location: A3/SR 119 (Arnimallee 3-5)
    

• Computer Algebra

  0082fA4.3
  • 19203419 Seminar with practice
    Computer Algebra (Sofia Garzón Mora)
    Schedule: Termine siehe LV-Details (Class starts on: 2025-02-24)
    Location: A3/ 024 Seminarraum (Arnimallee 3-5)
    

    Comments

    1) Prime number tests, factorization in Z

    2) LLL-algorithums

    3) Polynomial factorization over finite fields over Z, Q or in K [x1,...,xn]

    4) Gröbner bases, resultatants, eliminations

    5) Primary decompostion, radical ideals, Syzygies and free resolutions

    6) Practical applications, such as the examination of processors, states of balance in economic models, the description of configuration spaces in molecules, robotics or Sudoku

    For all topics the emphasis is on practical work using a concrete computer-algebra system (such as Singular).

     

    • Prerequisite Lineare Algebra I

       

  • 19207219 Seminar with practice
    Formal Proof Vetification (Tibor Szabo)
    Schedule: -
    Location: keine Angabe
    

    Comments

    Selected topics from:

    • Installation of LEAN, using the proof assistant and setting up your own project
    • Fundamentals of Dependent Type Theory and Propositions as Types 
    • Functional proofs and tactics 
    • Basics of logic in LEAN 
    • Inductive types and proofs by induction 
    • Selection of well-known mathematical concepts in LEAN (set theory, integers, vector spaces, convergence, ...)
    • Selection of simple proofs and proof strategies (infinitely many prime numbers, stable sets in hypercube, ...)
    • The mathlib library 

    For all topics, the focus is on practical work with a concrete proof assistant (e.g., LEAN).

    Prerquisites: Linaer Algebra I and Analysis I

    Suggested reading

    Literatur:

    • The Hitchhiker’s Guide to Logical Verification von Anne Baanen, Alexander Bentkamp, Jasmin Blanchette, Johannes Hölzl und Jannis Limperg
    • The Mechanics of Proof by Heather Macbeth 
    • Functional Programming in Lean von David Thrane Christiansen 
    • Theorem Proving in Lean 4 von Jeremy Avigad, Leonardo de Moura, Soonho Kong und Sebastian Ullrich
    • Mathematics in Lean

• Modules w/o courses in this semester

  2 Modules
  • Discovering Mathematics II (5 CP) 0082fA1.2
  • Computer-Oriented Mathematics II (5 CP) 0082fA4.2