Mathematics

• Classes offered by Berlin Mathematical School

  E17iA1.1
  • 19201901 Lecture
    Functional Analysis (Dirk Werner)
    Schedule: Di 10:00-12:00, Do 10:00-12:00 (Class starts on: 2025-10-14)
    Location: A3/019 Seminarraum (Arnimallee 3-5)
    

    Comments

    Content:
    Functional analysis is the branch of mathematics dealing with the study of normed (or general topological) vector spaces and continuous mappings between them. Thus, analysis, topology and algebra are linked.
    The course deals with Banach and Hilbert spaces, linear operators and functionals as well as spectral theory of compact operators.

    Target group: Students from the 3rd/4th semester on.

    Requirements: Good command of the material of the courses Analysis I/II and Linear Algebra I/II.

    Suggested reading

    Literatur:

    • Dirk Werner: Funktionalanalysis, 8. Auflage, Springer-Verlag 2018

  • 19201902 Practice seminar
    Tutorial: Functional Analysis (Dirk Werner)
    Schedule: Do 12:00-14:00 (Class starts on: 2025-10-16)
    Location: KöLu24-26/SR 006 Neuro/Mathe (Königin-Luise-Str. 24 / 26)
    

    Comments

    Inhalt:
    Die Funktionalanalysis ist der Zweig der Mathematik, der sich mit der Untersuchung von normierten (oder allgemeiner topologischen) Vektorräumen und stetigen Abbildungen zwischen ihnen befasst. Hierbei werden Analysis, Topologie und Algebra verknüpft.
    Die Vorlesung behandelt Banach- und Hilberträume, lineare Operatoren und Funktionale sowie Spektraltheorie kompakter Operatoren.
    
    Zielgruppe: Studierende vom 4. Semester an.
    
    Voraussetzungen: Sicheres Beherrschen des Stoffs der Vorlesungen Analysis I/II und Lineare Algebra I/II.
    
    Literatur:

     

    • Dirk Werner: Funktionalanalysis, 6. Auflage, Springer-Verlag 2007, ISBN 978-3-540-72533-6
    • Hans Wilhelm Alt: Lineare Funktionalanalysis : eine anwendungsorientierte Einführung. 5. Auflage. Springer-Verlag, 2006, ISBN 3-540-34186-2
    • Harro Heuser: Funktionalanalysis: Theorie und Anwendung. 3. Auflage. Teubner-Verlag, 1992, ISBN 3-519-22206-X

     

  • 19202001 Lecture
    Discrete Geometrie I (Christian Haase)
    Schedule: Di 10:00-12:00, Mi 12:00-14:00 (Class starts on: 2025-10-14)
    Location: A3/SR 120 (Arnimallee 3-5)
    

    Additional information / Pre-requisites

    Solid background in linear algebra. Knowledge in combinatorics and geometry is advantageous.

    Comments

    This is the first in a series of three courses on discrete geometry. The aim of the course is a skillful handling of discrete geometric structures including analysis and proof techniques. The material will be a selection of the following topics:
    Basic structures in discrete geometry

    • polyhedra and polyhedral complexes
    • configurations of points, hyperplanes, subspaces
    • Subdivisions and triangulations (including Delaunay and Voronoi)
    • Polytope theory
    • Representations and the theorem of Minkowski-Weyl
    • polarity, simple/simplicial polytopes, shellability
    • shellability, face lattices, f-vectors, Euler- and Dehn-Sommerville
    • graphs, diameters, Hirsch (ex-)conjecture
    • Geometry of linear programming
    • linear programs, simplex algorithm, LP-duality
    • Combinatorial geometry / Geometric combinatorics
    • Arrangements of points and lines, Sylvester-Gallai, Erdos-Szekeres
    • Arrangements, zonotopes, zonotopal tilings, oriented matroids
    • Examples, examples, examples
    • regular polytopes, centrally symmetric polytopes
    • extremal polytopes, cyclic/neighborly polytopes, stacked polytopes
    • combinatorial optimization and 0/1-polytopes

     

    For students with an interest in discrete mathematics and geometry, this is the starting point to specialize in discrete geometry. The topics addressed in the course supplement and deepen the understanding for discrete-geometric structures appearing in differential geometry, topology, combinatorics, and algebraic geometry.

     

     

     

     

     

     

     

     

     

    Suggested reading

    • G.M. Ziegler "Lectures in Polytopes"
    • J. Matousek "Lectures on Discrete Geometry"
    • Further literature will be announced in class.

  • 19202002 Practice seminar
    Practice seminar for Discrete Geometrie I (Sofia Garzón Mora, Christian Haase)
    Schedule: Mi 14:00-16:00 (Class starts on: 2025-10-15)
    Location: A6/SR 031 Seminarraum (Arnimallee 6)
    
  • 19202101 Lecture
    Basic Module: Numeric II (Robert Gruhlke)
    Schedule: Mo 12:00-14:00, Mi 12:00-14:00 (Class starts on: 2025-10-15)
    Location: A3/Hs 001 Hörsaal (Arnimallee 3-5)
    

    Comments

    Description: Extending basic knowledge on odes from Numerik I, we first concentrate on one-step methods for stiff and differential-algebraic systems and then discuss Hamiltonian systems. In the second part of the lecture we consider the iterative solution of large linear systems.

    Target Audience: Students of Bachelor and Master courses in Mathematics and of BMS

    Prerequisites: Basics of calculus (Analysis I, II) linear algebra (Lineare Algebra I, II) and numerical analysis (Numerik I)

  • 19202102 Practice seminar
    Practice seminar for Basic Module: Numeric II (André-Alexander Zepernick)
    Schedule: Mi 10:00-12:00, Fr 08:00-10:00 (Class starts on: 2025-10-15)
    Location: A6/SR 025/026 Seminarraum (Arnimallee 6)
    
  • 19202601 Lecture
    Differential Geometry I (Konrad Polthier)
    Schedule: Mo 10:00-12:00, Mi 10:00-12:00 (Class starts on: 2025-10-15)
    Location: KöLu24-26/SR 006 Neuro/Mathe (Königin-Luise-Str. 24 / 26)
    

    Additional information / Pre-requisites

    For further information, see Lecture Homepage.

    Comments

    Topics of the lecture will be:

    • curves and surfaces in Euclidean space,
    • metrics and (Riemannian) manifolds,
    • surface tension, notions of curvature,
    • vector fields, tensors, covariant derivative
    • geodesic curves, exponential map,
    • Gauß-Bonnet theorem, topology,
    • connection to discrete differential geometry.

    This course is a BMS course and will be held in English on request.

    Prerequisits:

    Analysis I, II, III and Linear Algebra I, II

     

     

    Suggested reading

    Literature

    • W. Kühnel: Differentialgeometrie:Kurven - Flächen - Mannigfaltigkeiten, Springer, 2012
    • M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall
    • J.-H. Eschenburg, J. Jost: Differentialgeometrie und Minimalflächen, Springer, 2014
    • C. Bär: Elementare Differentialgeometrie, de Gruyter, 2001

  • 19202602 Practice seminar
    Practice seminar for Differential Geometry I (Tillmann Kleiner, Konrad Polthier)
    Schedule: Mi 08:00-10:00 (Class starts on: 2025-10-15)
    Location: A6/SR 031 Seminarraum (Arnimallee 6)
    
  • 19205901 Lecture
    Advanced Module: Discrete Geometry III (N.N.)
    Schedule: Di 10:00-12:00 (Class starts on: 2025-10-14)
    Location: A6/SR 025/026 Seminarraum (Arnimallee 6)
    

    Additional information / Pre-requisites

    The target audience are students with a solid background in discrete geometry and/or convex geometry (en par with Discrete Geometry I & II). The topic of this course is a state-of-art of advanced topics in discrete geometry that find applications and incarnations in differential geometry, topology, combinatorics, and algebraic geometry.
    
    Requirements: Preferably Discrete Geometry I and II.

    Comments

    This is the third in a series of three courses on discrete geometry. This advanced course will cover a selection of the following topics (depending on the interests of the audience):   1. Oriented Matroids along the lines of the book Oriented Matroids by Björner, Las Vergnas, Sturmfels, White, and Ziegler; and/or   2. Triangulations along the lines of the book Triangulations by de Loera, Rambau, and Santos; and/or   3. Discriminants and tropical geometry along the lines of the book Discriminants, Resultants, and multidimensional determinants by Gelfand, Kapranov, and Zelevinsky; and/or   4. Combinatorics and commutative algebra along the lines of the book Combinatorics and commutative algebra by Stanley.

    Suggested reading

    Will be announced in class.

  • 19205902 Practice seminar
    Practice seminar for Advanced Module: Discrete Geometry III (N.N.)
    Schedule: Mi 14:00-16:00 (Class starts on: 2025-10-15)
    Location: A3/SR 120 (Arnimallee 3-5)
    
  • 19206201 Lecture
    Basic Module: Topology II (Pavle Blagojevic)
    Schedule: Di 14:00-16:00, Do 14:00-16:00 (Class starts on: 2025-10-14)
    Location: A6/SR 032 Seminarraum (Arnimallee 6)
    

    Comments

    Content: Homology, cohomology and applications, CW-complexes, basic notions of homotopy theory

    Suggested reading

    Literatur

    • Hatcher, Allen: Algebraic Topology; Cambridge University Press.
    • http://www.math.cornell.edu/~hatcher/AT/ATpage.html
    • Lück, Wolfgang: Algebraische Topologie, Homologie und Mannigfaltigkeiten; Vieweg.

  • 19212901 Lecture
    Stochastics II (Felix Höfling)
    Schedule: Di 12:00-14:00, Do 08:00-10:00 (Class starts on: 2025-10-14)
    Location: A3/Hs 001 Hörsaal (Arnimallee 3-5)
    

    Additional information / Pre-requisites

    Prerequisite: Stochastics I  and  Analysis I — III.

    Comments

    Contents:

    • Basics: conditional expectation, characteristic function, convergence types, uniform integrability;
    • Construction of stochastic processes and examples: Gaussian processes, Lévy processes, Brownian motion;
    • martingales in discrete time: convergence, stopping theorems, inequalities;
    • Markov chains in discrete and continuous time: recurrence and transience, invariant measures.

    Suggested reading

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

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

  • 19212902 Practice seminar
    Practice seminar for Stochastics II (Felix Höfling)
    Schedule: Di 14:00-16:00 (Class starts on: 2025-10-14)
    Location: A6/SR 031 Seminarraum (Arnimallee 6)
    

    Comments

    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

     

     

  • 19217311 Seminar
    PhD Seminar "What is...?" (Holger Reich)
    Schedule: Fr 12:00-14:00 (Class starts on: 2025-10-17)
    Location: A6/SR 031 Seminarraum (Arnimallee 6)
    

    Additional information / Pre-requisites

    The "What is ...?" seminars are usually held before the BMS Friday seminar to complement the topic of the talk.

    Audience: Anybody interested in mathematics is invited to attend the "What is ...?" seminars. This includes Bachelors, Masters, Diplom, and PhD students from any field, as well as researchers like Post-Docs.
    Requirements: The speakers assume that the audience has at least a general knowledge of graduate-level mathematics.

    Comments

    Content: The "What is ...?" seminar is a 30-minute weekly seminar that concisely introduces terms and ideas that are fundamental to certain fields of mathematics but may not be familiar in others.
    The vast mathematical landscape in Berlin welcomes mathematicians with diverse backgrounds to work side by side, yet their paths often only cross within their individual research groups. To encourage interdisciplinary cooperation and collaboration, the "What is ...?" seminar attempts to initiate contact by introducing essential vocabulary and foundational concepts of the numerous fields represented in Berlin. The casual atmosphere of the seminar invites the audience to ask many questions and the speakers to experiment with their presentation styles.
    The location of the seminar rotates among the Urania, FU, TU, and HU. On the weeks when a BMS Friday takes place, the "What is ...?" seminar topic is arranged to coincide with the Friday talk acting as an introductory talk for the BMS Friday Colloquium. For a schedule of the talks and their locations, check the website. The website is updated frequently throughout the semester.

    Talks and more detailed information can be found here
    Homepage: http://www.math.fu-berlin.de/w/Math/WhatIsSeminar

  • 19222301 Lecture
    Advanced Module: Algebra III (Holger Reich)
    Schedule: Mo 12:00-14:00 (Class starts on: 2025-10-20)
    Location: A3/SR 120 (Arnimallee 3-5)
    

    Comments

    Course contents: a selection of the following topics

    • properties of morphisms (proper, projective, smooth)
    • divisors
    • (quasi-)coherent sheaves
    • cohomology
    • Hilbert functions

    further properties of morphisms (proper, integral, regular, smooth, étale, ...)

    • Grothendieck topologies
    • cohomology (Cech, étale, ...)

    Suggested reading

    For example: Introduction to Schemes, Geir Ellingsrud and John Christian Otten

  • 19222302 Practice seminar
    Practice seminar for Advanced Module: Algebra III (Holger Reich)
    Schedule: Do 10:00-12:00 (Class starts on: 2025-10-16)
    Location: A7/SR 031 (Arnimallee 7)
    
  • 19223111 Seminar
    BMS Fridays (Holger Reich)
    Schedule: Fr 14:00-18:00 (Class starts on: 2025-10-17)
    Location: T9/Gr. Hörsaal (Takustr. 9)
    

    Comments

    The Friday colloquia of BMS represent a common meeting point for Berlin mathematics at Urania Berlin: a colloquium with broad emanation that permits an overview of large-scale connections and insights. In thematic series, the conversation is about “mathematics as a whole,” and we hope to be able to witness some breakthroughs.

    Typically, there is a BMS colloquium every other Friday afternoon in the BMS Loft at Urania during term time. BMS Friday colloquia usually start at 2:15 pm. Tea and cookies are served before each talk at 1:00 pm.

    More details: https://www.math-berlin.de/academics/bms-fridays

  • 19226511 Seminar
    Seminar Multiscale Methods in Molecular Simulations (Luigi Delle Site)
    Schedule: Fr 12:00-14:00 (Class starts on: 2025-10-17)
    Location: SR A9
    

    Additional information / Pre-requisites

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

    Comments

    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.

    Suggested reading

    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

  • 19234401 Lecture
    Discrete Mathematics II - Optimization (Ralf Borndörfer)
    Schedule: Mo 14:00-16:00, Do 12:00-14:00 (Class starts on: 2025-10-16)
    Location: Mo A6/SR 032 Seminarraum (Arnimallee 6), Do A7/SR 031 (Arnimallee 7)
    

    Additional information / Pre-requisites

    Credit This course can be chosen as Discrete Mathematics II (DM II). If Discrete Mathematics II - Extreme Combinatorics is taken at the same time, one of the two courses can be chosen as DM II and the other as a supplementary module. Language This lecture is in English.

    Exam

    The exam takes place at the last lecture. The 2nd exam takes place in the last week of the summer holidays before the lectures resume.

    Comments

    This lecture starts the optimization branch of discrete mathematics. It deals with algorithmic graph theory and linear optimization.

    Contents

    Complexity: complexity measures, run time of algorithms, the classes P and NP, NP-completeness
    Matroids and Independence Systems: independence systems, matroids, trees, forests, oracles, optimization over independence systems
    Shortest Paths: nonnegative weights, general weights, all pairs
    Network Flows: Max-Flow-Min-Cut Theorem, augmenting paths, minimum cost flows, transport and allocation problems
    Polyhedra: faces, dimension formula, projections of polyhedra, transformation, polarity, representation theorems
    Foundations of linear optimization: Farkas Lemma, Duality Theorem
    Simplex algorithm: basis, degeneration, basis exchange, revised simplex algorithm, bounds, dual simplex algorithm, post-optimization, numerics
    Interior point and ellipsoid method: basics

    Audience

    This course is aimed at mathematics students with knowledge of discrete mathematics I, linear algebra and analysis. Some exercises require the use of a computer.

    Suggested reading

    M. Grötschel, Lineare Optimierung, eines der Vorlesungsskripte

    V. Chvátal, Linear Programming, Freeman 1983

    Additional

    Garey & Johnson, Computers and Intractability,  1979 (Complexity Theory)

    Bertsimas & Tsitsiklis, Introduction to Linear Optimization, 97 (Linear Programming)

    Korte & Vygen, Combinatorial Optimization, 2006 (Flows, Shortest Paths, Matchings)

  • 19234441 Zentralübung
    Large tutorial for Discrete Mathematics II - Optimization (Ralf Borndörfer)
    Schedule: Mi 08:00-10:00 (Class starts on: 2025-10-15)
    Location: A6/SR 007/008 Seminarraum (Arnimallee 6)
    
  • 19235101 Lecture
    Function and distribution spaces (Willem Van Zuijlen)
    Schedule: Di 14:00-16:00 (Class starts on: 2025-10-14)
    Location: A6/SR 009 Seminarraum (Arnimallee 6)
    

    Additional information / Pre-requisites

    Prerequisits: Analysis I — III, Linear Algebra I, II. 
    Recommended: Functional Analysis.

    Comments

    In this course we consider function spaces and spaces of distributions, also called generalised functions. Distributions play an important role in the theory of partial differential equations, as in contrast to functions, they are always differentiable. Hence, during the course we motivate the context via PDEs here and there. We will discuss: 

    Distribution spaces and their notion of convergence (on general domains)
    Sobolev spaces (on general domains)
    Tempered distributions and the Fourier transform (on R^d)
    Besov spaces (on R^d)
    Bony's para- and resonance products
     

    Suggested reading

    There will be lecture notes.

  • 19235102 Practice seminar
    Exercise: Function and distribution spaces (Willem Van Zuijlen)
    Schedule: Di 16:00-18:00 (Class starts on: 2025-10-14)
    Location: A3/ 024 Seminarraum (Arnimallee 3-5)
    
  • 19242001 Lecture
    Partial Differential Equations II (Erica Ipocoana)
    Schedule: Di 08:00-10:00, Do 10:00-12:00 (Class starts on: 2025-10-14)
    Location: A6/SR 031 Seminarraum (Arnimallee 6)
    

    Comments

    This course builds on the material of the course Partial Differential Equations I as taught in the previous summer term. Methods for linear partial differential equations will be deepened and extended to nonlinear partial differential equations. A central topic of the course is the theory of monotone and maximal monotone operators. 

     

  • 19242002 Practice seminar
    Tutorial Partial Differential Equations II (Erica Ipocoana)
    Schedule: Do 12:00-14:00 (Class starts on: 2025-10-16)
    Location: A6/SR 031 Seminarraum (Arnimallee 6)
    
  • 19247111 Seminar
    Ordinary Differential Equations (Marita Thomas)
    Schedule: Di 16:00-18:00 (Class starts on: 2025-10-14)
    Location: A6/SR 009 Seminarraum (Arnimallee 6)
    

    Comments

    Ordinary differential equations arise in many applications from physics, chemistry, biology or economics. This seminar extends the topics that were covered in the Analysis III course, e.g., eigenvalue problems and stability theory will be addressed.