Discrete Geometry, WS 04/05

Michael Joswig, Fachbereich Mathematik, TU Darmstadt.

VL 2+1: Tuesday, 16:15-17:55, S3 06/052. Exercise (irregularly, as announced in the lecture): Wednesday, 8:00-9:40, S2 15/301.

Some General References

1. Martin Aigner and Günter M. Ziegler: Proofs from the Book, 3rd edition, Springer, 2003.
2. Branko Grünbaum: Convex Polytopes, (2nd edition with comments by Volker Kaibel, Victor Klee, and Günter M. Ziegler), Springer, 2003.
3. Michael Joswig and Nobuki Takayama (eds.): Algebra, Geometry, and Software Systems, Springer, 2002.
4. Jiri Matousek: Lectures on Discrete Geometry, Springer, 2002.
5. Günter M. Ziegler: Lectures on Polytopes, Springer, 1995.

Selected Tropical Geometry References

1. Mike Develin, Francisco Santos, and Bernd Sturmfels: On the rank of a tropical matrix, arxiv.org/math.CO/0312114, MSRI Proceedings, to appear.
2. Mike Develin and Bernd Sturmfels: Tropical Convexity, Documenta Math. 9 (2004), 1-27.
3. Michael Joswig: Tropical Halfspaces, arxiv.org/math.CO/0312068, MSRI Proceedings, to appear.
4. Ki Hang Kim and Fred W. Roush: Factorization of polynomials in one variable over the tropical semiring, arxiv.org/math.CO/0501167.
5. Grigory Mikhalkin: Enumerative tropical algebraic geometry in R2, arxiv.org/math.AG/0312530, J. AMS, to appear.
6. Jürgen Richter-Gebert, Bernd Sturmfels, and Thorsten Theobald: First steps in tropical geometry, arxiv.org/math.AG/0306366, Contemp. Math., AMS, to appear.
7. Lior Pachter and Bernd Sturmfels: Tropical Geometry of Statistical Models, arxiv.org/q-bio.QM/0311009.
8. Frank Sottile: Tropical Geometry, arxiv.org/math.AG/0501146.
9. David Speyer: Tropical Linear Spaces, arxiv.org/math.CO/0410455.
10. David Speyer and Bernd Sturmfels: Tropical Mathematics, arxiv.org/math.CO/0408099.
11. Thorsten Theobald: On the frontiers of polynomial computations in tropical geometry, arxiv.org/math.CO/0411012

... and a list of open problems maintained by Mike Develin.

Additional Remarks

Cauchy's Rigidity Theorem says that 3-dimensional convex polytopes are not flexible. For a proof see "Proofs from the Book", Chapter 12.

polymake

polymake, version 2.1.0, is installed on the machines in the pool. In order to be able to use it you should follow these instructions:

• Add /home/polymake/bin to your PATH environment variable, i.e., if you are using bash, e.g., then do:
  PATH=/home/polymake/bin:$PATH
• Make the shared libraries of the proper gcc version 3.4.3 available:
  export LD_LIBRARY_PATH=/opt/gnu/lib
• For your convenience it is recommended that you add both the above command lines to your .bashrc (or equivalent) file.
• Please, do polymake --reconfigure once (even if you did it before).
• External packages installed: geomview, JavaView, nauty.
• Documentation in HTML locally available at this local link.

Visit polymake's homepage for updates etc.

Polytope examples and more to be found on the Electronic Geometry Models server.