matthias beck

FU Berlin Winter 2020/21

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

Online musings. The FU currently does not allow our course to be in person, so I will stream (and record) every lecture via FU WebEx. Please check the FU guidelines dealing with the Corona virus frequently. The exercise discusssions will also be on FU WebEx; see the FU Whiteboard for more information and a link to our class sessions.

Exercises & final exam. We will post (on the FU Whiteboard sites) weekly homework problems, due on Wednesdays. To obtain the Aktive Teilnahme stamp, you will need to reach at least a 50% total score on the homework. The final exam is be a 48-hour take-home exam starting at 12:00 on 9 March 2021, posted here and on the FU Whiteboard sites. The Nachklausur is also a 48-hour take-home exam starting at 12:00 on 6 April 2021.

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.