Master of Science Arne Lien will Wednesday September 18th, 2024, at 12:15 hold his disputas for the PhD degree in Science. The title of his thesis is:
"Slicing orbit Spaces, Geometry and combinatorics of hyperbolic and even-hyperbolic slices"
Motivated by a connection to Timofte’s degree and halfdegree principle we study canonical hyperbolic slices, that is, sets of univariate hyperbolic polynomials that share the same first few coefficients. We study the geometric and combinatorial properties of a natural stratification of these slices and use these properties to improve upon the degree principle. Amongst the geometric properties we establish is a description of the dimension and relative interior of the strata along with a characterisation of some natural points of “escapes” from these strata. And on the combinatorial side we show that the lattice of strata is determined by the zero-dimensional strata and that the boundary complex of the dual lattice is generically a combinatorial sphere. We finish by showing that a similar story can be told about a natural stratification of even-hyperbolic slices. These are the subsets of hyperbolic slices consisting of the polynomials with only nonnegative roots and such sets arise in the context of the degree principle for the hyperoctahedral group.
The disputas and trial lecture will be streamed from these sites:
Disputas (12:15 - 15:00)
Trial Lecture (10:15 - 11:00)
The thesis is available at Munin here.