| Popis: |
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. V tej disertaciji popolnoma karakteriziramo unimodalno kategorijo funkcij ▫$f: mathbb{R} to [0, infty)$▫ s pomočjo izreka o dekompoziciji, ki ga dobimo kot posplošitev algoritma s pometanjem, ki sta ga vpeljala Baryshnikov in Ghrist. Podamo tudi karakterizacijo unimodalne kategorije za funkcije ▫$f: S^1 to [0, infty)$▫ in od tod dobimo algoritem za izračun unimodalne kategorije take funkcije v primeru, ko ima le končno mnogo kritičnih točk. Nato obravnavamo domnevo Baryshnikova in Ghrista o monotonosti. Pokažemo, da ta domneva drži za funkcije na ▫$mathbb{R}$▫ in ▫$S^1$▫ s pomočjo zgornjih karakterizacij, in da ne drži za funkcije na določenih grafih in na evklidski ravnini, tako da konstruiramo eksplicitne protiprimere. Poleg tega pokažemo, da drži za funkcije na evklidski ravnini, katerih Morse-Smaleov graf je drevo, z uporabo rezultata, ki so ga dokazali Hickok, Villatoro in Wang. Nato predstavimo nekaj odprtih vprašanj, ki nakazujejo obetavne smeri raziskovanja. Potem dokažemo še aproksimativni izrek o živcu, ki je posplošitev izreka o živcu iz klasične algebraične topologije v kontekst vztrajne homologije. To storimo z vpeljavo pojma ▫$varepsilon$▫-acikličnega pokritja filtriranega prostora. Z uporabo spektralnih zaporedij povežemo vztrajne homologije raznih prostorov, na katere pri tem naletimo. Aproksimacija je podana v jeziku prepletne razdalje med vztrajnostnimi moduli. Da dobimo optimalne meje, vpeljemo tehnična pojma levih in desnih prepletanj. Nazadnje podamo še primere, kjer so meje realizirane in s tem dokažemo optimalnost rezultata. |
