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Interior Lattice Points of Lattice Polytopes, Hensley’s Approach
Student-Run Research Seminar| Speaker: | Anouk Brose, UC Davis |
| Location: | 3106 MSB |
| Start time: | Wed, Jan 31 2024, 1:10PM |
In dimension two, the relation of interior lattice points, boundary lattice points, and the volume of a lattice polytope are fully characterized – this follows from Pick’s Theorem. In higher dimensions less is known. Hensley’s paper Lattice Vertex Polytopes with Interior Lattice Points was the first to bound the volume of a lattice polytope (with at least one interior lattice point), by its dimension and the number of interior lattice points. That is, Hensley found a bound B(d, k) depending only on the dimension $d \ge 1$ and interior lattice points $k \ge 1$ such that $\text{vol}(P ) \le B(d, k)$. I will explain methods used in the proof; which together with Egyptian fractions, showcase how to search for extremal polytopes of the inequality (and these correspond to Fano toric varieties).
Free pizzas as always!:)