Masters course in Triangulations: Semester I 2009

Triangulations of point configurations arise naturally within combinatorics, linear programming and algebra but are also interesting for their own sake. For example, counting the number of distinct (complex) solutions to a system of polynomial equations can be understood in terms of computing a triangulation of a point configuration coming from the monomials that appear in your system. Triangulations can also be used to find the optimal solutions to infinite families of linear programs and many classical polyhedra, like the permutahedron and the Stasheff polytope, can be succinctly understood and studied in terms of triangulations.

If you choose to participate in this course you should learn how to think rigourously in both a geometric and combinatorial framework. In particular, you should start to learn how to become very good at counting (that's not a joke -- "bijecting" might be a better word) and how to carefully write definitions that fit with the good geometric intuition that you will have plenty of opportunities to develop.

The course text and plan:

We will use the soon-to-be-published Triangulations of Point Configurations of DeLoera, Rambau and Santos. The text will be made available in the first week. The topics to be covered are as follows:
• Triangulations in Mathematics: introduction and examples.
• Defining the main players: regular triangulations and flips.
• Life in low dimensions.
• The general tool box.
• Regular triangulations and the secondary polytope.
• Special configurations: Gale duality, the permutahedron.
• Non-regular triangulations and disconnected flip graphs.
• Further topics, if time permits:
  • The Cayley trick and tropical algebra
  • Non-faces of triangulations and toric initial ideals
  • Hilbert bases, unimodular triangulations and Ehrhart theory
  • possibly more

Meeting times and office hours:

We meet for class on Mondays 11-12 and Wednedays 9-10, both in C219. There will be a tutorial each Friday 4 -5, in C402. My office door is open (most of the time) for questions and/or a chat.

Grading and Prerequisites:

If you expect a respectable final grade, regular and active participation in this class is mandatory and your grade will be split between assigned homework (45%), a final exam(40%) and a paper presentation(15%). There will be a weekly assigned homework which may be assessed both in writing and orally.

You will need a thorough understanding of linear algebra. Some polynomial ring theory is desirable but not neccessary. Although the class is primarily a masters course, students from the H.Dip. in Mathematics stream are more than welcome, as are enthusiastic undergraduates. For the purposes of assessments, H.Dip. students and undergraduates would not be expected to give a paper presentation. Please email or drop by if you have any questions.