Discrete Geometry III:

Enumerative Geometric Combinatorics

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FU Berlin Winter 2019/20

Lecture & Exercises | Tue/Thu 12-14, Seminarraum @ Villa |

Instructor | Prof. Matthias Beck |

Office | Villa |

Office hours | Wed 11-12, 14-15 & by appointment |

**Course description.**
This is the last in a series of three courses on discrete geometry. This advanced course will cover a selection of topics
from enumerative geometric combinatorics:

- Lattice-point structure of rational polyhedra
- Rational generating functions
- Hyperplane arrangements
- Applications to combinatorial number theory, poset theory, and partition analysis

**Prerequisites.**
Linear algebra, basic combinatorics, polyhedral geometry (convex polytopes, faces, polarity, etc.) as covered in Discrete
Geometry I-II.

**References.**

- A. Barvinok, A Course in Convexity, American Mathematical Society 2002
- A. Barvinok, Integer Points in Polyhedra, European Mathematical Society 2008
- M. Beck and S. Robins, Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra, Springer 2007 & 2015
- M. Beck and R. Sanyal, Combinatorial Reciprocity Theorems, American Mathematical Society 2018
- J. De Loera, J. Rambau, and F. Santos, Triangulations: Structures for Algorithms and Applications, Springer 2010
- T. Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw 1992
- R. Stanley, Enumerative Combinatorics I, Cambridge 2012
- G. Ziegler, Lectures on Polytopes, Springer 1995

**Exercises & final project.**
I will post (below) weekly homework problems, due on Thursdays before class.
To obtain the *Aktive Teilnahme* stamp, you will need to reach at least a 50% total score on the homework.
The final project is a written report on a research topic related to the material covered in our course; I will mention numerous possible topics during class
and collect them here.
A first draft of your report is due on 1 March; the final draft is due on 1 April (no joke).

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I want to ensure that each of you accomplishes the goals of this course as comfortably and successfully as possible. At any time you feel overwhelmed or lost, please come and talk with me. Always ask lots of questions in class; my courses are interactive. You are always encouraged to see me in my office.