Mathematics

• Classes offered by Berlin Mathematical School

  E17iA1.1
  • 19201901 Lecture
    Functional Analysis (Pavle Blagojevic)
    Schedule: Di 10:00-12:00, Do 10:00-12:00 (Class starts on: 2024-10-15)
    Location: A6/SR 025/026 Seminarraum (Arnimallee 6)
    

    Comments

    Content:
    Functional analysis is the branch of mathematics dealing with the study of normalized (or general topological) vector spaces and continuous images between them. Analysis, topology and algebra are linked.
    The lecture deals with Banach and Hilbert spaces, linear operators and functional 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 lectures Analysis I/II and Linear Algebra I/II.

    Suggested reading

    Literatur:

    • Dirk Werner: Funktionalanalysis, 7. Auflage, Springer-Verlag 2011, ISBN 978-3-642-21016-7

  • 19201902 Practice seminar
    Tutorial: Functional Analysis (Pavle Blagojevic, N.N.)
    Schedule: Do 16:00-18:00 (Class starts on: 2024-10-17)
    Location: A3/019 Seminarraum (Arnimallee 3-5)
    

    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 (Georg Loho)
    Schedule: Di 10:00-12:00, Mi 10:00-12:00 (Class starts on: 2024-10-15)
    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 (Georg Loho)
    Schedule: Mo 16:00-18:00, Fr 10:00-12:00 (Class starts on: 2024-10-14)
    Location: A6/SR 032 Seminarraum (Arnimallee 6)
    
  • 19202101 Lecture
    Basic Module: Numeric II (Volker John)
    Schedule: Mo 10:00-12:00, Mo 14:00-20:00 (Class starts on: 2024-10-14)
    Location: A6/SR 025/026 Seminarraum (Arnimallee 6)
    

    Comments

    Description: Extending basic knowledge on initial value problems with ordinary differential equations from Numerik I, the course presents methods for stiff problems and multistep methods. In the second part of the course iterative methods for solving linear systems of equations are studied.

    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: Do 12:00-14:00 (Class starts on: 2024-10-17)
    Location: A6/SR 025/026 Seminarraum (Arnimallee 6)
    
  • 19202501 Lecture
    Basic Module: Algebra I (Alexandru Constantinescu)
    Schedule: Mo 12:00-14:00, Mi 12:00-14:00, zusätzliche Termine siehe LV-Details (Class starts on: 2024-10-14)
    Location: A3/Hs 001 Hörsaal (Arnimallee 3-5)
    

    Comments

    Content

    
    This is the first part of a three semester course on algebraic geometry. Commutative algebra is the theory of commutative rings and their modules. It formally includes affine algebraic and local analytic geometry. Topics include:

    ? Affine algebraic varieties

    ? Rings, ideals, and modules

    ? Noetherian rings

    ? Local rings and localization

    ? Primary decompositione

    ? Finite and integral extensions

    ? Dimension theory

    ? Regular rings

    Target Group
    Students with the prerequisites mentioned below.

    Prerequisites
    ? Linear Algebra I+II ? Algebra and Number Theory
    
    Literature
    ? Atiyah, M.F.; Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969 ix+128 pp. (This book is probably the best entry to the subject. It is short, concise, and clearly written.)
    ? Further literature will be announced in class.

    
    Homepage: Professor Alexander Schmitt

  • 19202502 Practice seminar
    Practice seminar for Basic Module: Algebra I (Alexandru Constantinescu)
    Schedule: Mo 08:00-10:00 (Class starts on: 2024-10-21)
    Location: A6/SR 031 Seminarraum (Arnimallee 6)
    
  • 19205201 Lecture
    Differential Geometry III (Konrad Polthier)
    Schedule: Di 12:00-14:00 (Class starts on: 2024-10-15)
    Location: T9/046 Seminarraum (Takustr. 9)
    

    Comments

    
    The lecture will introduce selected concepts from differential geometry and their role in solving current application problems.
    
    The topics include curvature measures, geometric flows, minimal surfaces, harmonic mapping, parallel transport, branched coverings, as well as their discretization and implementation.
    Practical problems come for example from the fields of geometric design, geometry processing, visualization, materials science, medicine, architecture.

    Prerequisites: Differential geometry I

  • 19205202 Practice seminar
    Practice seminar for Differential Geometry III (Tillmann Kleiner)
    Schedule: Fr 08:00-10:00 (Class starts on: 2024-10-18)
    Location: A6/SR 025/026 Seminarraum (Arnimallee 6)
    

    Comments

    The first tute will take place in semester week 2.

  • 19205801 Lecture
    Discrete Mathematics II - Algorithmic Comb. (Tibor Szabo)
    Schedule: Di 14:00-16:00, Do 14:00-16:00 (Class starts on: 2024-10-15)
    Location: A3/Hs 001 Hörsaal (Arnimallee 3-5)
    

    Comments

    Topics of the course

    • Algorithms and complexity (sorting, Dijkstra, TSP, approximation algorithms, matchings vs Hamiltonicity, P vs NP, certificates (Hall, Tutte), Hungarian algorithm, network flows and its applications (Menger, Baranyai), (list)-coloring, stable matching (Gale-Shapley Algorithm) and its application (Galvin))
    • Linear Programming (Simplex Algorithm), Duality and its applications in Combinatorics and Algorithms
    • Randomized algorithms (randomized matching algorithms, hypergraph-coloring, derandomization, Erdos-Selfridge Criterion, algorithmic Local Lemma)

     

    Suggested reading

    • L. Lovász, J. Pelikán, K. Vesztergombi, Discrete Mathematics
    • J. Matousek - B. Gaertner, Understanding and Using Linear Programming
    • D. West, Introduction to Graph Theory

    Further reading:

    • V. Chvátal, Linear Programming.
    • Schrijver, Theory of Linear and Integer Programming
    • Schrijver, Combinatorial Optimization

  • 19205802 Practice seminar
    Practice seminar for Discrete Mathematics II - Algorithmic Comb. (Tibor Szabo)
    Schedule: Di 12:00-14:00 (Class starts on: 2024-10-15)
    Location: A3/SR 120 (Arnimallee 3-5)
    

    Comments

    October 6th at 8:15-8:45 there will be an information question/answer opportunity about the rules and requirements of the course. (The first lecture starts at 9:00.)
    The first three weeks of the course will be given in a block course format during the week preceding the semester, October 6-10, hence there are no regular lectures during the period October 13-31. During the week of October 6-10 there will be lectures on four days, 9:15-12:00, about the fundamentals of Additive Combinatorics. These lectures will be accompanied by exercise sessions in the afternoon. In order to gain points towards their exercise credit, participants of Discrete Math II will be required to submit written solutions to some of these exercises during the first three weeks of the semester. The regular lecture and exercise hours will resume from November 3. For further details please check the course website: Discrete Mathematics II

  • 19205901 Lecture
    Advanced Module: Discrete Geometry III (Pavle Blagojevic)
    Schedule: Di 14:00-16:00 (Class starts on: 2024-10-15)
    Location: A7/SR 031 (Arnimallee 7)
    

    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 (Pavle Blagojevic)
    Schedule: Do 10:00-12:00 (Class starts on: 2024-10-17)
    Location: A3/SR 115 (Arnimallee 3-5)
    
  • 19206401 Lecture
    Numerics IV: Modeling, Simulation, and Optimization (Christof Schütte)
    Schedule: Termine siehe LV-Details (Class starts on: 2024-10-18)
    Location: A6/SR 007/008 Seminarraum (Arnimallee 6)
    

    Comments

    Inhalt:

    Abstract:

    Modeling, Simulation, and Optimization (MSO) is one of the cornerstones of application-oriented mathematics.

    It covers a broad spectrum of research activities, ranging from the design of mathematical models for real-world processes, via efficient numerical simulation algorithms, to the solution of optimization problems for finding optimal scenarios or controls for the process under consideration. This lecture will give an overview over the techniques used in MSO and its application in different areas (life science, mobility, energy, sustainability, …). The lecture will be complemented by several pilot projects in which student groups will develop MSO solutions for realistic (but not too complex) application problems.

    Zielpublikum:
    Master Mathematik

    Suggested reading

    • Brokate and J. Sprekels: Hysteresis and Phase Transitions. Springer (1996)K.
    • Deckelnick, G. Dziuk, and Ch.M. Elliott: Computation of geometric partial differential equations and mean curvature flow. Acta Numerica, p. 1-94 (2005)
    • G. Dziuk and Ch.M. Elliott: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27, p. 262-292 (2007)
    • J.A. Sethian: Level Set Methods and Fast Marching Methods, CambridgeUniversity Press (1999)
    • T.J. Willmore: Riemannian Geometry, Clarendon, Oxford (1993)

  • 19206402 Practice seminar
    Practice seminar for Advanced Module: Numerics IV (Christof Schütte)
    Schedule: Di 10:00-12:00 (Class starts on: 2024-10-15)
    Location: A3/SR 115 (Arnimallee 3-5)
    
  • 19208001 Lecture
    Probability Theory III (Nicolas Perkowski, N.N.)
    Schedule: Mi 08:00-10:00, Do 10:00-12:00 (Class starts on: 2024-10-16)
    Location: T9/046 Seminarraum (Takustr. 9)
    

    Comments

    Stochastic analysis is the study of stochastic processes that evolve in continuous time. We will treat the following subjects, among others:

    Gaussian processes; Brownian motion, construction and properties; filtrations and stopping times; continuous time martingales; continuous semimartingales; quadratic variation; stochastic integration; Itô’s formula; Girsanov’s theorem and change of measure; time change; martingale representation; stochastic differential equations and diffusion processes, connections with partial differential equations.

    Detailed Information can be found on the Homepage of 19208001 Stochastics III.

  • 19208002 Practice seminar
    Tutorial: Probability Theory III (N.N.)
    Schedule: Do 16:00-18:00 (Class starts on: 2024-10-17)
    Location: A6/SR 032 Seminarraum (Arnimallee 6)
    
  • 19214611 Seminar
    Reseach Module: Algebra (Alexander Schmitt)
    Schedule: Termine siehe LV-Details (Class starts on: 2024-10-15)
    Location: A3/SR 115 (Arnimallee 3-5)
    

    Additional information / Pre-requisites

    Target group:

    Math students

    Prerequisites

    Algebra I and II

    Homepage: Prof. Altmann

    Comments

    Contents:

    Seminar page

    Suggested reading

    Literatur

    Wird bekanntgegeben.

  • 19222301 Lecture
    Advanced Module: Algebra III (Alexander Schmitt)
    Schedule: Termine siehe LV-Details (Class starts on: 2024-10-16)
    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

    Suggested reading

    G.R. Kempf: Algebraic varieties. London Mathematical Society Lecture Note
    Series. 172. Cambridge: Cambridge University Press. 1993. x, 163 p.

    A. Schmitt: Homological algebra, lecture notes.

    J.L. Taylor: Several complex variables with connections to algebraic
    geometry and Lie groups. Graduate Studies in Mathematics. 46. Providence,
    RI: American Mathematical Society. 2002. xvi, 507 p.

  • 19222302 Practice seminar
    Practice seminar for Advanced Module: Algebra III (Jan Sevenster)
    Schedule: Mi 12:00-14:00 (Class starts on: 2024-10-16)
    Location: T9/SR 005 Übungsraum (Takustr. 9)
    
  • 19226511 Seminar
    Seminar Multiscale Methods in Molecular Simulations (Luigi Delle Site)
    Schedule: Fr 12:00-14:00 (Class starts on: 2024-10-18)
    Location: A7/SR 031 (Arnimallee 7)
    

    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

  • 19234201 Lecture
    Topology and Topoi (Georg Lehner)
    Schedule: Mo 10:00-12:00 (Class starts on: 2024-10-14)
    Location: A6/SR 009 Seminarraum (Arnimallee 6)
    

    Comments

    There are various dualities in mathematics that share formal similarities. One is given by Galois theory: For a given field, the poset of Galois extensions and the poset of subgroups of its absolute Galois group are dual to another. Another example is covering space theory: For a given topological space, there is a duality between coverings and subgroups of its fundamental group. We will also discuss Stone dualities, as well as various incarnations of these.

    These dualities are special cases of a very general phenomenon that can be expressed by looking at Grothendieck Topoi. These Topoi are categories of sheaves on a site, and can also be thought of as generalized topological spaces. Any topos has a pro-finite homotopy group and there is an abstract Galois theory that generalizes both classical Galois theory and covering space theory.

    Towards the end of the lecture series we will look at shape theory. Shape theory allows one to do homotopy theory even with wild topological spaces. To any higher topos, one can associate its shape. We will attempt to prove the result that for a locally contractible topos, its sub-topos of locally contractible objects is equivalent to the category of local systems over its shape.

    Suggested reading

    Johnstone - Stone Spaces
    MacLane, Moerdijk - Sheaves in Geometry and Logic
    Hoyois - Higher Galois Theory

  • 19235701 Lecture
    Introduction to mathematical modeling with partial differential equations (Marita Thomas)
    Schedule: Di 10:00-12:00 (Class starts on: 2024-10-15)
    Location: A6/SR 009 Seminarraum (Arnimallee 6)
    

    Comments

    The course gives an introduction to mathematical modeling with partial differential equations. It discusses a selection of the following topics:  

    • General principles of continuum mechanics and thermodynamics

    • Symmetries and conservation laws

    • Variational principles

    • Derivation and discussion of models from hydrodynamics, solid mechanics, thermoelasticity, geodynamics, climate research or quantum mechanics

    This course can be attended as the first part of the BMS Basic Course "Mathematical Modeling with PDEs", which stretches over two semesters at FU Berlin. The second part will be covered in the subsequent summer term by the course 19215301 + 19215302 "Mathematical Modelling in Climate Research". 

     

  • 19235702 Practice seminar
    Introduction to mathematical modeling with partial differential equations (Marita Thomas)
    Schedule: Di 14:00-16:00 (Class starts on: 2024-10-15)
    Location: A6/SR 009 Seminarraum (Arnimallee 6)
    
  • 19242001 Lecture
    Partial Differential Equations II (Erica Ipocoana)
    Schedule: Di 08:00-10:00, Do 08:00-10:00 (Class starts on: 2024-10-15)
    Location: A6/SR 007/008 Seminarraum (Arnimallee 6)
    

    Comments

    Diese Veranstaltung baut auf dem Kursmaterial von Partielle Differentialgleichungen I des vorangegangenen Sommersemesters auf. Methoden für lineare partielle Differentialgleichungen werden vertieft und erweitert auf nichtlineare partielle Differentialgleichungen. Im Mittelpunkt der Vorlesung steht die Theorie monotoner und maximal monotoner Operatoren. 

    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: Di 16:00-18:00 (Class starts on: 2024-10-15)
    Location: A3/SR 120 (Arnimallee 3-5)
    
  • 19247111 Seminar
    Topics in measure and integration theory (Marita Thomas)
    Schedule: Di 16:00-18:00 (Class starts on: 2024-10-15)
    Location: A3/SR 119 (Arnimallee 3-5)
    

    Comments

    This seminar builds upon the Analysis III course to deepen topics in measure and integration theory. Topics are, for example: covering theorems, Lebesgue-, Hausdorff- and Radon measures, Radon Nikodym derivatives.