For each of the seven teaching areas there are two one-semester basic courses (4 hours/week) with well-defined contents that define the knowledge and skills that any student pursuing advanced work in this area should have. Students will need to pass final exams after each semester that are designed to verify that the students actively master the contents of these courses at a working level. All basic courses will be taught every year. They will be accompanied by tutorial classes (2 hours/week) and homework problems. 

Below we give a brief description of these courses and of their intended contents.

1. Differential geometry, global analysis, and topology.
The two courses can be taken independently from each other. The course on Differential geometry (surface theory) is also recommended for students specializing in visualization.

Outline of the contents:

• Analysis and geometry on manifolds
The course provides an introduction into the basic concepts of differential geometry and the analysis on manifolds:
◦ analysis and geometry of submanifolds in Euclidean space
◦ differentiable manifolds, tangent bundles, tensors
◦ Riemannian manifolds, connections, curvature, geodesics
◦ concepts from nonlinear analysis, topology, differential operators and integration theorems with applications

• Differential geometry (Surface theory)
◦ local and global differential geometry of surfaces and curves
◦ parametrizations
◦ curvature
◦ special classes of surfaces
◦ concepts from complex analysis and topology such as manifolds and Riemann surfaces

2. Algebra and number theory, algebraic and arithmetic geometry.
The two basic courses provide a rigorous introduction to the most important objects and concepts of modern algebraic geometry and number theory. The first semester 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 then provides an introduction to the concepts of modern algebraic geometry.

Prerequisites: group theory, rings and modules, field extensions and Galois theory.

Outline of the contents:

• Commutative algebra
◦ rings, ideals, and modules
◦ prime ideals, maximal ideals, and primary ideals
◦ primary decompositions
◦ Noetherian and Artinian rings
◦ discrete valuation rings and Dedekind domains

• Algebraic geometry
◦ varieties
◦ schemes and sheaves
◦ cohomology theory
◦ curves and surfaces

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.

Prerequistes: Basic notions and models in probability theory, the Law of Large Numbers, Central Limit Theorem and know elements of estimation and test theory.

Outline of the contents:

• Stochastic processes I: discrete time
◦ construction of stochastic processes
◦ conditional expectations and laws
◦ martingales convergence and martingales measures
◦ arbitrage-theory, pricing and hedging
◦ Markov chains
◦ Brownian motion and invariance principles

• Stochastic processes II: continuous time
◦ continuous-time martingales
◦ semimartingales
◦ the Itô stochastic integral
◦ stochastic calculus
◦ stochastic differential equations and PDEs

4. Discrete mathematics and discrete geometry.
The two basic courses in discrete mathematics and in geometry are independent. Each of them is designed to set basic foundations of the field, in view of current research directions that are prominently pursued in Berlin. 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 combinatorics, 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.

Outline of the contents:

• Combinatorics
◦ basic counting problems
◦ enumeration, generating functions
◦ basic graph theory: graphs and hypergraphs
◦ basic structure theory: posets and lattices

• Geometry
◦ affine geometry
◦ projective geometry
◦ Möbius geometry
◦ polyhedra, polyhedral surfaces
◦ polytopes

5. Linear, nonlinear, and combinatorial optimization.
The goal of the two basic courses is to give a solid understanding of the basic role of optimization, models, methods, and consequences. This is taught in view of both the deep theoretical consequences of optimization models (e.g., in terms of duality theory, geometry of convex sets, and polyhedra, etc.), and of the immense theoretical importance of optimization tools in economic and industrial applications. The two basic courses refer to each other, but can be studied independently. It is of great help to know a programming language, e.g. Java, before taking the course "Linear and integer programming"!

Outline of the contents:

• Linear and integer programming
◦ linear programming problems, modeling
◦ simplex algorithm, duality, geometry
◦ polynomiality, ellipsoid method
◦ network problems
◦ integer programming problems, modeling
◦ LP-based methods (cutting plane approaches, branch and bound)
◦ enumeration methods (dynamic programming)

• Nonlinear optimization
◦ nonlinear optimization problems, modeling
◦ optimality conditions
◦ numerical methods (Newton-type, CG-methods, trust-region techniques)
◦ non-differentiable optimization
◦ interior point methods

6. Numerical analysis, scientific computing, and visualization.
The two basic courses provide a rigorous introduction to the most important strategies and concepts of modern numerical mathematics. The courses can be taken independently from each other. 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.

Prerequisites: Non-linear systems of equations, best approximation, linear regression, hermite interpolation, numerical quadrature and initial value problems for ODEs.

Outline of the contents:

• Numerical methods for ODEs and numerical linear algebra
◦ stiff initial value problems and stability
◦ implicit Runge-Kutta and multistep methods
◦ numerical methods for DAEs
◦ iterative solution of linear equations and eigenvalue problems

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.

Outline of the contents:

• Dynamical systems
◦ flow properties of dynamical systems
◦ omega limit sets
◦ stability of fixed points and periodic orbits
◦ invariant manifolds
◦ examples of local and global bifurcations
◦ chaotic behavior

In addition to the above 14 basic courses, the BMS offers three one-semester courses that will provide students the opportunity to fill potential gaps in their general mathematical background. These courses will not be exclusively aimed at BMS students, but will be part of the master programs of the universities.

Outline of the contents:

• Complex analysis
◦ holomorphic functions, power series
◦ curve integrals and Cauchy theorem
◦ singularities
◦ analytical continuation and homotopy, residue calculus
◦ Riemann mapping theorem
◦ elliptic functions

 • Topology
◦ topological spaces, continuous maps
◦ simplicial complexes
◦ homotopy groups
◦ (co)homology groups
◦ Euler and Lefschetz numbers, fixed point theorems
◦ manifolds, classification of surfaces
◦ exact sequences, Mayer–Vietoris theorem, Poincaré duality