Prostori konfiguracija i operade malih kocaka

master's thesis

• Author: Vlatko Crnković
• Mentor: Matija Bašić (mentor)
• Committee member: Matija Bašić (predsjednik povjerenstva)
• Committee member: Tomislav Berić (član povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Committee member: Andrej Dujella (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu promatramo pojam operade, preciznije topološke operade, i neke njene primjene. Pojam je u današnjem obliku uveden u radu Petera Maya, [7], za potrebe karakterizacije višestrukih prostora petlji, no krajem dvadesetog stoljeća nalazi primjene i u drugim granama matematike. Rad je organiziran u tri poglavlja s dodatkom na kraju koji navodi osnovne rezultate teorije kategorija i topologije potrebne za praćenje sadržaja. U prvom poglavlju definiramo pojam operade i... More $\mathcal{P}$-prostora te pojam monade i algebre nad monadom. Navodimo neke osnovne primjere ovakvih objekata te opisujemo dvije temeljne konstrukcije, jednu koja operadi $\mathcal{P}$ pridružuje monadu čije algebre klasificiraju $\mathcal{P}$-prostore, te drugu koja operadi $\mathcal{P}$pridružuje pripadnu rezolventu $W (\mathcal{P})$. Drugo poglavlje bavi se operadama malih kocaka, ključnim objektom u Mayevom radu karakterizacije višestrukih prostora petlji. Opisujemo kako operada malih kocaka prirodno djeluje na višestruke prostore petlji te navodimo dva ključna rezultata Mayevog rada. Na kraju poglavlja pokazujemo da je operada malih kocaka homotopski ekvivalentna prostorima konfiguracija $F(\mathbb{R}^n, \cdot)$. Opisujemo kompaktifikaciju prostora konfiguracija te na njoj definiranu strukturu operade, takozvanu Kontsevichevu operadu. Treće poglavlje prati rad Ryana Budneya, [3], koji opisuje slabi homotopski tip prostora dugih uzlova $\mathcal{K}$. Budneyev rad se zasniva na Shubertovom radu, [11], koji pokazuje da operacije sume inducira strukturu komutativnog monoida na $\pi_0\mathcal{K}$ te daje do na poredak i izotopiju jedinstvenu dekompoziciju uzla na proste. Definiramo djelovanje operade malih 2-kocaka na određenom modelu za $\mathcal{K}$ te pratimo Budneyev dokaz fokusirajući se na spomenuto djelovanje. Less

Abstract (english)

In this paper we observe the notion of an operad, more precisely a topological operad, and some of its applications. The term operad, as it is known today, was first introduced in the work of Peter May, [7], for the purposes of characterizing iterated loop spaces, though, towards the end of the twentieth century, it finds its applications in other branches of mathematics. This paper is organized into three chapters, with an appendix that contains basic results of... More category theory and topology required for its understanding. In the first chapter we define operads and $\mathcal{P}-$spaces along with monads and algebras over a monad. We go over some basic examples of such objects and describe two basic constructions, one that associates to an operad $\mathcal{P}$ its corresponding monad $P$, such that algebras over the monad $P$ characterize $\mathcal{P}$-spaces, and another that gives a certain resolution $W (\mathcal{P})$ of the monad $P$. The second chapter deals with the little cube operad, the key object in May’s work on the loopspace recognition principle. We describe the natural action of the little cube operad on iterated loop spaces and briefly go over May’s two main resutls. In the last section of the chapter we show that the little cube operad is homotopy equivalent to configuration spaces $F(\mathbb{R}^n, \cdot)$. We also describe a compactification of those configuration spaces and its associated operad structure, the Kontsevich operad. The third chapter follows the work of Ryan Budney, [3], which describes the weak homotopy type of the spaces of long knots $\mathcal{K}$. Budney’s work is loosely based on the work of Schubert, [11], in which it is shown that the connected sum operation induces a commutative monoid structure on $\pi_0\mathcal{K}$. We define the little 2-cube action on a certain model for $\mathcal{K}$ and follow Budeny’s proof mainly focusing on the mentioned action. Less

• Keywords: operada; topološka operada; monada; algebra nad monadom; operade malih kocaka; Kontsevicheva operada
• Keywords (english): operad; topological operad; monad; algebra over monad; little cube operad; Kontsevich operad
• Language: croatian
• URN:NBN: urn:nbn:hr:217:210709
• Study programme: Title: Pure Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-01-23 08:54:17