Summer Workshop on Lattice Polytopes
July 23 -- August 10, 2018, Osaka, Japan
Schedule
| 9:00-10:30 |
First Lecture |
| 11:00-12:30 |
Second Lecture |
| 12:30-14:30 |
Lunch |
| 14:30-15:30 |
First Exercise |
| 16:00-17:00 |
Second Exercise |
Lecture 1
Teacher: Johannes Hofscheier (McMaster University, Canada)
Title: Toric geometry with a view towards lattice polytopes
Asbstract:
Toric varieties are algebraic varieties defined by combinatorial data, and there is a rich interplay between algebra, combinatorics and geometry. The goal of this lecture is to give an introduction to this wonderful theory. We will cover affine toric varieties, projective toric varieties, divisors, line bundles and ampleness. If time permits, we will then look into the exciting problems concerned with projective normality.
Lecture 2
Teacher: Katharina Jochemko (KTH Royal Institute of Techonolgy, Sweden)
Title: Valuations on lattice polytopes
Asbstract:
Valuations are a classical topic in convex geometry. In the continuous setting, valuations are well-studied and the volume plays a prominent role in many structural results. A foundational result is Hadwiger's famous theorem characterizing continuous, rigid-motion invariant valuations on convex bodies. Valuations on lattice polytopes are less studied. The Betke-Kneser Theorem establishes a fascinating discrete analog of Hadwiger's theorem for lattice-invariant valuations on lattice polytopes in which the number of lattice points - the discrete volume - takes a fundamental role. Starting from there, in this series of lectures we explore striking parallels, analogies and also differences between the continuous and the discrete world of valuations with a focus on positivity questions and links to Ehrhart theory.