Research & Projects

My research is about sections of polytopes, in particular lattice polytopes. I'm interested in:

Algorithms: Can we describe algorithms to compute a section with special properties (maximal volume, maximal number of lattice points, maximal number of vertices)?
Computational Complexity: What is the computational complexity of computing such sections?
Ehrhart Theory: Classical Ehrhart Theory studies "How does the number of lattice points change in a polytope under dilation of the polytope?". We extend this theory to counting subsets of lattice points: How does the set of sections with the maximal number of lattice points behave under dilation of the polytope?
Combinatorial Reconstruction: What does the combinatorial type (or f-vector) of a section of a polytope say about the combinatorial type (or f-vector) of the polytope?

The lattice diameter (segment) of a lattice polygon.

Lattice Diameters

This project is joint work with Gennadiy Averkov, Jesús De Loera, Gyivan Lopez-Campos and Antonio Torres. The lattice diameter of a polytope P measures the maximum number of collinear lattice points in P. This invariant is closely related to the shortest vector problem, Minkowski's successive minima in the geometry of numbers, and the lattice width, and it also helps in the classification of polytopes.

Our Results: We have polynomial-time algorithms to compute the lattice diameter of a rational polytope, for fixed dimension d, and show that computation is NP-hard in non-fixed dimension. As an extension of Ehrhart Theory we show that the counting function of lattice diameter segments in dilates of a polytope eventually agrees with a quasi-polynomial. Lastly we solve the discrete variant of Borsuk's partition problem with respect to the lattice diameter.

Our paper will appear in IPCO proceedings 2026. Here are slides.

Lower-Dimensional Sections of Polytopes

This is ongoing work, joint with Marie-Charlotte Brandenburg, Chiara Meroni and Jesús De Loera. We are parametrizing lower-dimensional sections of polytopes, and building an algorithm in Julia to compute "all" lower-dimensional sections of a polytope.

Poster.

Equivalent codimension 2 sections of a polytope.

A polytope with O(n^2) faces, but slices with only O(n) vertices.

f-vectors of Sections of Polytopes

This project is ongoing work with Anna Birkemeyer, Marie-Charlotte Brandenburg and Niklas Prün, and developed out of an REU Dive into Research at Ruhr University Bochum, in the summer of 2025. We study reconstruction type problems between f-vectors of slices of polytopes and f-vectors of the polytope itself.

Some Open Problems

Conjecture: Let L(P, k, i) denote the number of i-dimensional sections of kP with maximally many lattice points. We conjecture that the function k -> L(P, k, i) eventually agrees with a quasi-polynomial of degree dim(P) - i.
Corzatt's Covering Problem: In the 1970s Corzatt conjectured that every convex set of lattice points in the plane has a minimal cover by lines with at most four directions. It is still unsolved! This is related to the number of lattice diameter directions of lattice polygons.
Which polytopes P, with an interior lattice point, satisfy vol(P) > #(P cap Z^d)? Can we characterize them in dimension three? This question arose during my Bachelor's Thesis which was expository work on a theorem by Hensley showing that the volume can be bounded above by the number of interior lattice points of a polytope. Can we also bound the volume above by the total number of lattice points?

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