matthias beck

FU Berlin Summer 2021

Course description. This is the second in a series of three courses on discrete geometry. This advanced course will cover a selection of topics from enumerative geometric combinatorics, centered around volume computation for polyhedra. In the discrete setting, this computation is based on Ehrhart's theory of counting lattice points in polytopes. We will witness an interplay of methods from combinatorics, geometry, and number theory, with an emphasis on actually computing discrete volume functions.

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

References. We will mostly follow

• M. Beck and S. Robins, Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra, Springer 2007 & 2015

but there are several other good references:

• A. Barvinok, Integer Points in Polyhedra, European Mathematical Society 2008
• M. Beck and R. Sanyal, Combinatorial Reciprocity Theorems, American Mathematical Society 2018
• T. Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw 1992
• R. Stanley, Enumerative Combinatorics I, Cambridge 2012

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; they will be due each Friday. To obtain the Aktive Teilnahme stamp, you will need to reach at least a 50% total score on the homework. The final exam will be a 48-hour take-home exam starting at 12:00 on 20 July 2021, posted here and on the FU Whiteboard sites. The Nachklausur will also be a 48-hour take-home exam starting at 12:00 on 21 September 2021; anyone wishing to take the Nachklausur needs to register (email to the instructor) by 1 September 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, whether in person or online.