Pure and Applied Mathematics in Seven Areas

The subject of the Berlin Mathematical School is Mathematics, which encompasses many fields that are traditionally termed either "pure" or "applied" mathematics.

The BMS prefers, however, not to make that distinction; instead, the teaching areas covered by the BMS are grouped into seven parts, each of which covers a quite broad, but coherent, part of mathematics. The core offering of the BMS Phase I study program consists of 14 one-semester basic courses, two for each of the seven teaching areas. These courses are modern introductions to research in the respective areas, stressing interdisciplinary and trans-disciplinary connections and applications, modern trends and current questions. Their purpose is to provide solid foundations in the field, geared towards ambitious students who after the BMS Phase I will head towards mathematics PhD research work.

1. Differential geometry, global analysis and topology

The basic course "Analysis and geometry on manifolds" provides an introduction into the basic concepts of differential geometry and the analysis on manifolds, while the other "Differential geometry (surface theory)" deepens the understanding of the concepts of differential geometry and develops the connection with complex analysis.

2. Algebra and number theory, algebraic and arithmetic geometry

The first semester course "Commutative algebra" focuses mainly on deepening the knowledge in algebra, namely in commutative algebra, which is the basic prerequisite for algebraic geometry and number theory. The second semester course "Algebraic geometry" then provides an introduction to the concepts of modern algebraic geometry.

3. Probability theory and financial mathematics

The two basic courses provide a rigorous introduction to the most important objects and concepts of modern probability theory. The first semester focuses mainly on stochastic processes in discrete time, while the second semester provides a sound introduction to continuous time stochastic processes and the foundations of stochastic calculus.

4. Discrete mathematics and geometry

The "Combinatorics" course treats basic structures and methods from discrete mathematics that are also of great importance in nearly all other parts of mathematics; it covers the core of the main branches of discrete mathematics, namely enumerative and algebraic combinatorics, and graph theory. The "Geometry" course treats the fundamental geometries in view of their role for current research throughout mathematics, which encompasses discrete geometry (polyhedral theory), differential geometry, visualization, and mathematical physics.

5. Linear, nonlinear, and combinatorial optimization

The goal of the two basic courses, "Linear and integer programming" and "Nonlinear optimization", is to give a solid understanding of the basic role of optimization, models, methods, and consequences. This is taught in view of both the theoretical consequences of optimization models (e.g., in terms of duality theory, geometry of convex sets, and polyhedra, etc.), and of the theoretical importance of optimization tools in economic and industrial applications.

6. Numerical analysis, scientific computing, and visualization

The first semester focuses mainly on numerical methods for ordinary differential equations, but also on deepening the knowledge in numerical linear algebra, especially regarding iterative methods for large systems. The second semester gives an introduction to partial differential equations from fundamental theory to modern numerical concepts.

7. Applied analysis, mathematical physics, and dynamical systems

The two basic courses here provide a thorough introduction to the theory of ordinary differential equations and dynamical systems, and to that of partial differential equations.

In addition to the above 14 basic courses, the BMS will offer three one-semester courses that will provide students the opportunity to fill potential gaps in their general mathematical background:

• Complex analysis (Funktionentheorie)
• Functional analysis
• Topology