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The étale fundamental group, étale homotopy and anabelian geometry
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2017 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesisAlternative title
Den étala fundamentalgruppen, étalehomotopi och anabelsk geometri (Swedish)
Abstract [en]

In 1983 Grothendieck wrote a letter to Faltings, [Gro83], outlining what is today known as the anabelian conjectures. These conjectures concern the possibility to reconstruct curves and schemes from their étale fundamental group. Although Faltings never replied to the letter, his student Mochizuki began working on it. A major achievement by Mochizuki and Tamagawa was to prove several important versions of these conjectures.

In this thesis we will first introduce Grothendieck’s Galois theory with the aim to define the étale fundamental group and formulate Mochizuki’s result. After recalling some necessary homotopy theory, we will introduce the étale homotopy type, which is an extension of the étale fundamental group developed by Artin, Mazur and Friedlander. This is done in order to describe some recentwork by Schmidt and Stix that improves on the results of Mochizuki and Tamagawa by extending them from étale fundamental groups to étale homotopy types of certain (possibly higher-dimensional) schemes.

 

Abstract [sv]

 I ett brev till Faltings 1983, [Gro83], lade Grothendieck grunden till det som idag kallas anabelsk geometri. I brevet presenterar han ett antal förmodningar som handlar om möjligheten att återskapa kurvor och scheman från deras étala fundamentalgrupper. Faltings svarade aldrig på brevet, men hans student Mochizuki bevisade ett antal viktiga specialfall.

I den här uppsatsen introducerar vi först Grothendiecks Galoisteori för att definera den étala fundamentalgruppen och formulera Mochizukis resultat. Sedan går vi igenom grundläggande homotopiteori, som behövs för att introducera étalehomotopi, som utvecklats av Artin, Mazur och Friedlander. Med dessa hjälpmedel tittar vi närmare på ett resultat av Stix och Schmidt som bygger på Mochizukis resultat och utvidgar det från fundamentalgrupper till homotopityper

Place, publisher, year, edition, pages
2017.
Series
TRITA-MAT-E ; 2017:17
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-219869OAI: oai:DiVA.org:kth-219869DiVA, id: diva2:1165816
External cooperation
LMU Ludwig-Maximilians-Universität, München
Subject / course
Mathematics
Educational program
Master of Science - Mathematics
Supervisors
Examiners
Available from: 2017-12-13 Created: 2017-12-13 Last updated: 2017-12-13Bibliographically approved

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