Mathematics

• Classes offered by Berlin Mathematical School

  E17iA1.1
  • 19205401 Lecture
    Basic module: Topology I (Christian Haase)
    Schedule: Mo 12:00-14:00, Mi 12:00-14:00, zusätzliche Termine siehe LV-Details (Class starts on: 2025-04-16)
    Location: 1.3.14 Hörsaal A (Arnimallee 14)
    

    Comments

    
    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, simplicial complexes
    4. Homotopies between continuous maps, degree of a map, fundamental group.
    5. Seifert-van Kampen Theorem.
    6. Covering spaces.
    7. Simplicial homology
    8. Combinatorial applications

    Suggested reading

    Literature:

    1. M. A. Armstron: Basic Topology, Springer UTM
    2. Allen Hatcher: Algebraic Topology, Chapter I. Also available online from the author's website
    3. Jirí Matoušek: Using the Borsuk-Ulam Theorem, Springer UTX
    4. Mark de Longueville: A Course in Topological Combinatorics, Springer UTX
    5. Tammo tom Dieck: Topologie, De Gruyter Lehrbuch
    6. Klaus Jänich: Topologie, Springer-Verlag
    7. Gerd Laures, Markus Szymik: Grundkurs Topologie, Spektrum Akademischer Verlag
    8. James R. Munkres: Topology, Prentice Hall

  • 19205402 Practice seminar
    Exercise for Basic Module: Topology I (Sofia Garzón Mora)
    Schedule: Mo 16:00-18:00 (Class starts on: 2025-04-28)
    Location: A3/Hs 001 Hörsaal (Arnimallee 3-5)
    
  • 19206011 Seminar
    Discrete Mathematics Masterseminar (Tibor Szabo)
    Schedule: Fr 10:00-12:00, zusätzliche Termine siehe LV-Details (Class starts on: 2025-04-11)
    Location: T9/SR 005 Übungsraum (Takustr. 9)
    

    Additional information / Pre-requisites

     

     

    Comments

    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).

     

  • 19212801 Lecture
    Theory of Functions (Nicolas Perkowski)
    Schedule: Di 14:00-16:00, Do 12:00-14:00, zusätzliche Termine siehe LV-Details (Class starts on: 2025-04-15)
    Location: A3/Hs 001 Hörsaal (Arnimallee 3-5)
    

    Comments

    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.

    Suggested reading

    Literatur:

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

  • 19212802 Practice seminar
    Practice seminar for Theory of Functions (Julian Kern)
    Schedule: Di 16:00-18:00 (Class starts on: 2025-04-22)
    Location: A3/Hs 001 Hörsaal (Arnimallee 3-5)
    
  • 19213101 Lecture
    Geometry (Giulia Codenotti)
    Schedule: Di 12:00-14:00, Mi 12:00-14:00 (Class starts on: 2025-04-15)
    Location: A6/SR 032 Seminarraum (Arnimallee 6)
    

    Comments

    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.

    Suggested reading

    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

  • 19213102 Practice seminar
    Practice seminar for Geometry (Giulia Codenotti)
    Schedule: Mo 10:00-12:00, Mo 16:00-18:00 (Class starts on: 2025-04-14)
    Location: A3/SR 119 (Arnimallee 3-5)
    
  • 19214701 Lecture
    Discrete Mathematics I (Ralf Borndörfer)
    Schedule: Di 14:00-16:00, Do 12:00-14:00 (Class starts on: 2025-04-15)
    Location: T9/SR 005 Übungsraum (Takustr. 9)
    

    Additional information / Pre-requisites

    Target group:

    BMS students, Master and Bachelor students

    Whiteboard:

    You need access to the whiteboard in order to receive information and participate in the exercises.

    Large tutorial:

    Participation is recommended, but non-mandatory.

    Exams:

    1st exam: Thurday July 17, 14:00-16:00, room tba, i.e., in the last lecture
    2nd exam: Thursday October 09, 10:00-12:00, room tba, i.e., in the last week before the lectures of the winter semester start

    Comments

    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)

    Suggested reading

    • 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.

  • 19214702 Practice seminar
    Practice seminar for Discrete Mathematics I (Silas Rathke)
    Schedule: Di 16:00-18:00, Do 14:00-16:00 (Class starts on: 2025-04-22)
    Location: A3/SR 119 (Arnimallee 3-5)
    

    Comments

    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)

  • 19214741 Zentralübung
    Large tutorial for Discrete Mathematics I (Ralf Borndörfer)
    Schedule: Mi 08:00-10:00 (Class starts on: 2025-04-16)
    Location: T9/SR 005 Übungsraum (Takustr. 9)
    
  • 19215001 Lecture
    Constructive Combinatorics (Tibor Szabo)
    Schedule: Di 14:00-16:00 (Class starts on: 2025-04-15)
    Location: A3/SR 119 (Arnimallee 3-5)
    

    Additional information / Pre-requisites

    Basic Bachelor Algebra, Probability, and Disrete Mathematics.

    Comments

    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.

    Suggested reading

    A script will be provided.

  • 19215002 Practice seminar
    Constructive Combinatorics exercises (Tibor Szabo)
    Schedule: Do 12:00-14:00 (Class starts on: 2025-04-17)
    Location: A6/SR 009 Seminarraum (Arnimallee 6)
    

    Comments

    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.

  • 19219701 Lecture
    Algebra with Probability in Combinatorics (Tibor Szabo)
    Schedule: Do 08:00-10:00 (Class starts on: 2025-04-17)
    Location: T9/049 Seminarraum (Takustr. 9)
    

    Comments

    In this lecture specialized topics of Combinatorics and Graph Theory are presented.

  • 19223901 Lecture
    Uncertainty Quantification and Quasi-Monte Carlo (Claudia Schillings)
    Schedule: Mo 10:00-12:00 (Class starts on: 2025-04-14)
    Location: A6/SR 032 Seminarraum (Arnimallee 6)
    

    Comments

    High-dimensional numerical integration plays a central role in contemporary study of uncertainty quantification. The analysis of how uncertainties associated with material parameters or the measurement configuration propagate within mathematical models leads to challenging high-dimensional integration problems, fueling the need to develop efficient numerical methods for this task. Modern quasi-Monte Carlo (QMC) methods are based on tailoring specially designed cubature rules for high-dimensional integration problems. By leveraging the smoothness and anisotropy of an integrand, it is possible to achieve faster-than-Monte Carlo convergence rates. QMC methods have become a popular tool for solving partial differential equations (PDEs) involving random coefficients, a central topic within the field of uncertainty quantification. This course provides an introduction to uncertainty quantification and how QMC methods can be applied to solve problems arising within this field.

    Suggested reading

    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.

  • 19223902 Practice seminar
    Übung zu UQ and QMC (Claudia Schillings)
    Schedule: Di 10:00-12:00 (Class starts on: 2025-04-15)
    Location: A3/SR 120 (Arnimallee 3-5)
    
  • 19226511 Seminar
    Seminar Multiscale Methods in Molecular Simulations (Luigi Delle Site)
    Schedule: Fr 12:00-14:00 (Class starts on: 2025-04-25)
    Location: Die Veranstaltung findet in der Arnimallee 9 statt (Seminarraum).
    

    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

  • 19241301 Lecture
    Partial Differential Equations I (André Erhardt)
    Schedule: Di 12:00-14:00, Do 10:00-12:00 (Class starts on: 2025-04-15)
    Location: T9/SR 005 Übungsraum (Takustr. 9)
    

    Comments

    • Basic differential equations (Laplace,- heat and wave equations)
    • Representation formulas
    • Solution methods
    • Introduction to Hilbert space methods

    This can serve as the basis for a BSc and/or MSc project.  

    Suggested reading

    L.C. Evans, Partial Differential Equations
     

  • 19241302 Practice seminar
    Exercises to Partial Differential Equations I (Piotr Pawel Wozniak)
    Schedule: Di 16:00-18:00, Do 12:00-14:00 (Class starts on: 2025-04-15)
    Location: 0.1.01 Hörsaal B (Arnimallee 14)
    
  • 19242101 Lecture
    Stochastics IV: (Guilherme de Lima Feltes, Nicolas Perkowski)
    Schedule: Mi 10:00-12:00 (Class starts on: 2025-04-16)
    Location: A6/SR 009 Seminarraum (Arnimallee 6)
    

    Additional information / Pre-requisites

    Prerequisite: Stochastics I, II, III. 
    Recommended: Functional Analysis.

    Comments

    Content: We will learn two different methods for solving stochastic partial differential equations. The classical method is based on Ito calculus, and we will use it to solve semilinear SPDEs with space-time white noise in one space dimension. But we will see that in higher dimensions this theory only works for linear equations, and motivated by that we will introduce "paracontrolled distributions“, which we developed in the last years based on ideas from harmonic analysis and rough paths, and which allow us to solve some interesting semilinear equations in higher dimensions. Along the way we will learn about regularity theory for semilinear PDEs, Gaussian Hilbert spaces, and much more.

    • Ito calculus for Gaussian random measures;
    • semilinear stochastic PDEs in one dimension;
    • Schauder estimates;
    • Gaussian hypercontractivity;
    • paraproducts and paracontrolled distributions;
    • local existence and uniqueness for semilinear SPDEs in higher dimensions;
    • properties of solutions

    Detailed Information can be found on the Homepage of 19246301 SPDEs: Classical and New.

    Suggested reading

    Literature
    There will be lecture notes.

  • 19242102 Practice seminar
    Exercise: Stochastics IV (Guilherme de Lima Feltes)
    Schedule: Mi 12:00-14:00, zusätzliche Termine siehe LV-Details (Class starts on: 2025-04-16)
    Location: A6/SR 009 Seminarraum (Arnimallee 6)
    
  • 19243001 Lecture
    Partial Differential Equations III (Erica Ipocoana)
    Schedule: Di 08:00-10:00 (Class starts on: 2025-04-15)
    Location: A6/SR 025/026 Seminarraum (Arnimallee 6)
    

    Additional information / Pre-requisites

    Voraussetzungen: Partielle Differentialgleichungen I und II

    Comments

    The course builds upon the the PDE II course offered in the previous winter term. Methods for boundary value problems of elliptic PDEs are deepend. A central aspect of the course are variational methods, in particular multi-dimensional calculus of variations. 

     

    Suggested reading

    Wird in der Vorlesung bekannt gegeben / to be announced.

  • 19243002 Practice seminar
    Tutorial Partial Differential Equations III (Erica Ipocoana)
    Schedule: Do 12:00-14:00 (Class starts on: 2025-04-24)
    Location: A3/SR 119 (Arnimallee 3-5)
    
  • 19247111 Seminar
    Variational methods & Gamma-convergence (Marita Thomas)
    Schedule: Termine siehe LV-Details (Class starts on: 2025-04-15)
    Location: A6/SR 025/026 Seminarraum (Arnimallee 6)
    

    Comments

    This seminar addresses bachelor and master students interested in the analysis of partial differential equations (PDEs). It focuses on elliptic PDEs, where the direct method of the calculus of variations provides a powerful tool to handle linear as well as nonlinear problems by investigating the minimality properties of the functional associated with the PDE. Closely related to this is the method of Gamma-convergence, which allows it to study sequences of functionals and minimization problems. A background with courses in analysis, functional analysis, and introduction to PDEs is useful to attend the seminar, but the topics for the presentations will be adapted to the background of the participants.   The main part of the seminar will be held en block in the teaching-free period.