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