Summary
We study integer points in polyhedra, their structure (with respect to the polytope), classifications, their applications in algebra, linear optimization, and number theory, and algorithms.
We meet twice a week, with additional tutorials every week.
Topics
This course is about lattice points in polyhedra and applications in other areas, where these structures naturally appear. A lattice is a discrete subgroup of \(\mathbb{R}^n\), e.g. the lattice \(\mathbb{Z}^n\) of points with integral coordinates. A polytope is the convex hull of finitely many points. it is a lattice polytope if these points are lattice points.
We discuss the structure of these polytopes with respect to the lattice, restrictions this imposes, their characteristic properties and and their applications to other areas of mathematics.
We plan to cover:
- lattice polytopes in small dimension
- Ehrhart Theory
- Classifications
- Hilbert bases
- Algorithm of Barvinok
- Flatness Theorem and integer linear programming
- triangulations
You should be familiar with linear algebra. Knowledge in polyhedral theory is useful, but we can also briefly cover relevant parts in the lecture and discuss this in the tutorials.
Some Literature
- Barvinok, Alexander: A course in convexity. Graduate Studies in Mathematics. 54. Providence, RI: American Mathematical Society (AMS). x, 366 p. (2002).
- Barvinok, Alexander: Integer Points in Polyhedra. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), 2008.
- Bruns, Winfried; Gubeladze Joseph: Polytopes, Rings, and K-Theory. Springer 2009.
- Beck, Matthias; Robins, Sinai: Computing the continuous discretely. Integer-point enumeration in polyhedra. Springer Undergraduate Texts in Mathematics
- De Loera, Jesus; Hemmecke, Raymond, Köppe, Matthias: Algebraic and Geometric Ideas in the Theory of Discrete Optimization
- Schrijver, Alexander: Theory of linear and integer programming. Repr. Chichester: Wiley. xi, 471 p. (1998)
- Sturmfels, Bernd: Gröbner bases and convex polytopes. University Lecture Series. 8. Providece, RI: American Mathematical Society (AMS). xi, 162 p. (1996).
- Ziegler, Günter M.: Lectures on polytopes. Graduate Texts in Mathematics, 152. Springer-Verlag, New York, 1995.
