# 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 16 one-semester basic courses, at least 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 mathematical physics

The two Basic Courses "Analysis and geometry on manifolds" and "Riemannian geometry" give an introduction to the most important concepts of differential geometry; these are fundamental for Riemannian and symplectic geometry as well as for geometric analysis and mathematical physics. The first course provides an introduction to the basic notions of geometry and analysis on manifolds, while the second one imparts fundamental knowledge in global Riemannian geometry.

## 2. Algebraic and arithmetic geometry, number theory

The two Basic Courses "Commutative algebra" and "Algebraic geometry" 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 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.

## 3. Probability, statistics, and financial mathematics

The two Basic Courses "Stochastic processes I: discrete time" and "Stochastic processes II: continuous timee" give an introduction to the most important concepts of modern probability theory. These are fundamental to the theory of stochastic processes and their stochastic and statistical analysis, as well as to mathematical finance. The first course focuses mainly on stochastic processes in discrete time. The second gives a solid introduction to continuous-time stochastic processes and the basics of stochastic calculus.

## 4. Discrete mathematics and combinatorial optimization

The three Basic Courses in discrete mathematics, discrete optimization, and nonlinear optimization are independent. Each of them is designed to cover basic foundations of the field, in view of current research directions pursued in Berlin. The discrete combinatorics course treats basic structures and methods from the core areas of discrete mathematics, in particular enumerative combinatorics, algebraic combinatorics, and graph theory, topics which are also of great importance in nearly all other parts of mathematics. The discrete optimization course gives a solid understanding of the basic role of discrete optimization, models, methods, and consequences. The course gives a view both of the deep theoretical consequences of discrete optimization models (in terms of duality theory, geometry, and polyhedra, for instance) and of the immense practical importance of optimization tools in economic and industrial applications. Nonlinear optimization is an indispensable tool for dealing with realworld problems, e.g., for identifying system parameters and/or for optimizing the performance of a technical or economical process. In the course we consider nonlinear differentiable optimization problems in finite dimensions. In a mixture of theoretical analysis and numerics we discuss necessary and sufficient optimality conditions for unconstrained and constrained problems. We develop algorithms for the numerical solution of these problems, study their convergence properties and use MATLAB to implement some of them.

## 5. Geometry, topology, and visualization

This two-semester sequence gives a graduate-level introduction to classical geometries, discrete differential geometry and mathematical visualization and algebraic topology. While noneuclidean and projective geometry are often taught at the undergraduate level at other universities, it is rare to find them at this advanced level. Discrete differential geometry is an area of special research interest in Berlin.

## 6. Numerical analysis and scientific computing

The two Basic Courses "Numerical methods for ODEs" and "Numerical methods for PDEs" provide a rigorous introduction to the most important strategies and concepts of modern numerical mathematics, and are independent of each other. The first course focuses mainly on numerical methods for ordinary differential equations, but also on deepening 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 and differential equations

The two Basic Courses "Dynamical systems" and "Partial differential equations" 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 16 basic courses, the BMS offers two 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:

- Complex analysis
- Functional analysis