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Deutsch Intern
    Institute of Mathematics

    Annual Master Focus

    Every academic year a new focus is offered: lectures and seminars prepare the way for current research topics in a select field. Mathematics International Master students may choose such a focus during their studies but are not obliged to do so.

    Prepare your way for current research topics in COMPUTATIONAL NUMBER THEORY such as advanced methods for primality testing and factoring integers, cryptosystems, computational aspects in algebraic number theory, the computation of L-functions, and local methods and their various applications.

    Organizers: Prof. Dr. Jörn Steuding, Prof. Dr. Stephan Elsenhans.

    Winter semester 2020/2021

    • Lecture series with exercises: Number Theoretical Algorithms (Prof. Dr. S. Elsenhans, 10 ECTS)
      In this class we will study methods for testing primality, factoring integers and computing discrete logarithms in finite fields. Further, we will introduce the continued fraction method for rational approximation. This can be applied to some Diophantine equations. (Literature: Cohen: A Course in Computational Algebraic Number Theory, Forster: Algorithmische Zahlentheorie, Eric Bach: Algorithmic Number Theory)
       
    • Lecture Series with exercises: Computational Aspects of Algebraic and Analytic Number Theory (Prof. Dr. J. Steuding, 10 ECTS)
      We begin with a brief introduction to algebraic number theory and study in particular the splitting of primes and ideals in number fields. A second focus is on computational aspects of zeta- and L-functions as they appear in algebraic number theory and the theory of modular forms; here we are concerned with special values, functional equations, the analytic class number formula (related to the ideal class group), and the localization of zeros (with respect to the two unsolved Millennium problems from number theory, namely the Riemann hypothesis and the conjecture of Birch & Swinnerton-Dyer). Excellent literature on these topics: "Algebraic Number Theory"  by Fröhlich & Taylor, "Zetafunktionen und quadratische Körper"  by Zagier, "Riemann's Zeta-Function"  by Edwards.
       
    • Seminars or research groups on related or different topics from the regular master program (5-10 ECTS). It is also possible to choose a further lecture series with exercises from our master programs Mathematics, Computational Mathematics, Mathematical Physics and Economathematics.

    Summer semester 2021

    • Lecture series with exercises: Bernoulli Polynomials and Their Applications (Prof. Dr. K. Dilcher, 10 ECTS)
      The first part of this course will be a thorough introduction to Bernoulli numbers and polynomials, as well as higher-order analogues and generalizations based on Dirichlet characters. This is followed by applications to combinatorics, asymptotic analysis, and some other fields of mathematics, but the main focus will be on various different and important applications in several areas of number theory. Modern computational issues related to Bernoulli numbers will also be discussed. (Literature: Arakawa et al., "Bernoulli numbers and zeta functions"; Temme, "Special Functions")
       
    • Lecture series with exercises: Local Methods and Their Applications (Prof. Dr. S. Elsenhans, 10 ECTS)
      The first part of the class will be an Introduction to p-adic numbers and local fields. The second half will present the following applications: Factoring polynomials over the rationals, computing subfields of numberfields and computing Galois groups of number fields. (Literature: Jürgen Neukich: Algebraic Number Theory)
       
    • Seminars or research groups on related or different topics from the regular master program (5-10 ECTS). It is also possible to choose a further lecture series with exercises from our master programs Mathematics, Computational Mathematics, Mathematical Physics and Economathematics.

    Winter semester 2021/2022 and summer semester 2022

    In the following two semesters further classes, seminars or research in groups on special topics will be offered that allow to deepen the knowledge in computational number theory and its applications, and that can lead to a master thesis (15-20 ECTS + 30 ECTS for the master thesis).

    It is also possible to choose lecture series with exercises, seminars or research groups on different topics from our  master programs Mathematics, Computational Mathematics, Mathematical Physics and Economathematics.

    Organizers: Prof. Dr. Komla Domelevo, Prof. Dr. Christian Klingenberg and Prof. Dr. Anja Schlömerkemper

    Winter semester 2021/2022

    • Lecture series with exercises: Partial differential equations of mathematical physics (Prof. Dr. C. Klingenberg, 10 ECTS)
      We will elucidate the physical origins of the main partial differential equations (PDEs) of classical mathematical physics, including the fundamental equations of fluid and solid mechanics, thermodynamics, and classical electrodynamics. These PDEs form the backbone of modern applications in physics and other natural sciences.
       
    • Lecture series with exercises: Applied Analysis (Prof. Dr. A. Schlömerkemper, 10 ECTS)
      In this course we focus on linear partial differential equations and study their properties with methods from functional analysis. To this end we will consider various function spaces, embedding theorems and compactness properties and investigate second-order elliptic equations as well as linear evolution equations.
       
    • Seminars or research groups on related or different topics from the regular master program (5-10 ECTS). It is also possible to choose a further lecture series with exercises from our master programs Mathematics, Computational Mathematics, Mathematical Physics and Economathematics.

    Summer semester 2022

    • Lecture series with exercises: Navier-Stokes equations (Prof. Dr. A. Schlömerkemper, 10 ECTS)
      The Navier-Stokes equations model the motion of viscous fluids. They are applied in the modeling of weather, air flow around a wing, ocean currents, flow of viscoelastic materials and many more. Despite their wide range of application, there are still major open mathematical problems. For instance, it is not known in three dimensions whether smooth solutions to the incompressible Navier-Stokes equations exist at all times for smooth initial data, which is part of the so-called Millennium problem of the Clay Mathematical Institute. Still there are important results in the mathematical theory of the three-dimensional Navier-Stokes equations, which we will address in this lecture series.
       
    • Lecture series with exercises: Hyperbolic equations (Prof. Dr. C. Klingenberg, 10 ECTS)
      We will consider an important class of mathematical models coming from science and technology that are described by partial differential equations based on the balance of the physical quantities mass, momentum, and energy. Examples that will be presented are the compressible Euler equations, the equations of magnetohydrodynamics, and kinetic equations like the Vlasov or Boltzmann equations.
       
    • Lecture series with exercises: Numerics of partial differential equations (Prof. Dr. K. Domelevo, 10 ECTS)
      The partial differential equations introduced in the first semester of this Master Focus can be connected to their applications via their numerical discretization. In this course these methods will be introduced, for example finite elements method, discontinuous Galerkin method, finite differences and finite volume methods. Applications will be kept in mind in the sections of topics.
       
    • Seminars or research groups on related or different topics from the regular master program (5-10 ECTS). It is also possible to choose a further lecture series with exercises from our master programs Mathematics, Computational Mathematics, Mathematical Physics and Economathematics to replace one of the above.

    Winter semester 2022/2023 and summer semester 2023

    In the following two semesters further classes, seminars or research in groups on special topics will be offered that allow to deepen the knowledge in partial differential equations and their applications, and that can lead to a master thesis (15-20 ECTS + 30 ECTS for the master thesis).

    It is also possible to choose lecture series with exercises, seminars or research groups on different topics from our  master programs Mathematics, Computational Mathematics, Mathematical Physics and Economathematics.