Unimodal category : doctoral thesis

english

doktorska dela

2017

approximation   bounded variation   counterexamples   mathematics   monotonicity   persistence module   spectral sequences   unimodal category

Univerza v Ljubljani, Fakulteta za matematiko in fiziko
(Obvezni izvod spletne publikacije)

In this thesis, we completely characterize the unimodal category for functions ?$f: \mathbb{R} \to 0, \infty)$? using a decomposition theorem obtained by generalizing the sweeping algorithm of Baryshnikov and Ghrist. We also give a characterization of the unimodal category for functions ?$f: S^1 \to 0, \infty)$? and provide an algorithm to compute the unimodal category of such a function in the case of finitely many critical points. We then turn to the monotonicity conjecture of Baryshnikov and Ghrist. We show that this conjecture is true for functions on ?$\mathbb{R}$? and ?$S^1$? using the above characterizations and that it is false on certain graphs and on the Euclidean plane by providing explicit counterexamples. We also show that it holds for functions on the Euclidean plane whose Morse-Smale graph is a tree using a result of Hickok, Villatoro and Wang. We then present several open questions indicating promising research directions. After this, we prove an approximate nerve theorem, which is a generalization of the nerve theorem from classical algebraic topology to the context of persistent homology. This is done by introducing the notion of an ?$\varepsilon$?-acyclic cover of a filtered space. We use spectral sequences to relate the persistent homologies of the various spaces involved. The approximation is stated in terms of the interleaving distance between persistence modules. To obtain a tight bound, the technical notions of left and right interleavings are introduced. Finally, examples are provided, which realize the bound and thus prove the tightness of the result.

12.05.2023