Public defence in the field of Mathematics and Statistics, MSc Teemu Lundström

Title of the thesis: Combinatorics of poset polytopes

Thesis defender: Teemu Lundström
Opponent: Professor Raman Sanyal, Goethe-Universität Frankfurt, Germany
Custos: Associate Professor Kaie Kubjas, Aalto University School of Science

This thesis is in the area of combinatorics and discrete geometry. 

The thesis studies the interplay between a finite partially ordered set (a poset) and two polytopes constructed from it: order polytopes and chain polytopes. We study how the combinatorics of the poset affect the geometric and combinatorial properties of the corresponding polytopes and compare these two classes of polytopes to each other. 

A particular focus has been on a conjecture that relates the number of faces of these two polytopes to each other in the case when the two polytopes are constructed from the same underlying poset. We make new progress on this conjecture with novel techniques, proving the conjecture for a large class of posets, as well as proving it for all posets in the case of two-dimensional faces. 

In addition, for a particular class of order polytopes, we study two polynomials associated to these polytopes, and for one of these polynomials, we show how their coefficients relate to certain permutation statistics. 

The results of this thesis increase our understanding of these two classes of polytopes, and provide new techniques to approach open problems related to geometric properties of partially ordered sets.

Keywords: Polytope, partially ordered set, combinatorics, discrete geometry

Thesis available for public display 7 days prior to the defence at Aalto University's public display page.