VL: Discrete Geometry: Polytopes and Polynomials, WS 14/15

Michael Joswig, Institut für Mathematik, TU Berlin.

Contents

Assuming a basic background in polytope theory, this course covers topics in polytopal combinatorics with a view towards applications to solving systems of polynomial equations.

Subject overview:

• graphs of polytopes: simple polytopes, Balinski's theorem
• lattice points and Ehrhart polynomials
• triangulations, regular subdivisions, mixed subdivisions and mixed volume
• secondary fans
• Theorems of Bernstein, Kushnirenko and Khovanskii
• a tiny bit of toric varieties and tropical geometry

References (more to be added)

1. Beck and Robins: Computing the continuous discretely. UTM. Springer, 2007.
2. Cox, Little, O'Shea: Ideals, varieties, and algorithms. Third edition. UTM. Springer, 2007.
3. Cox, Little, O'Shea: Using algebraic geometry. Second edition. GTM, Springer, 2005.
4. De Loera, Rambau and Santos: Triangulations. Springer, 2010.
5. Joswig and Theobald: Polyhedral and algebraic methods in computational geometry. Springer, 2013.
6. Thomas: Lectures in geometric combinatorics. Student Mathematical Library, 33. IAS/Park City Mathematical Subseries. AMS, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 2006.
7. Ziegler: Lectures on polytopes. GTM 152. Springer, 1995.