Investigations in Hadamard spaces

by Arian Bërdëllima
Doctoral thesis
Date of Examination:2020-08-27
Date of issue:2021-08-27
Advisor:Prof. Dr. Russell Luke
Referee:Prof. Dr. Max Wardetzky
Referee:Prof. Dr. Stephan Huckemann
Referee:Prof. Dr. Thomas Schick
Persistent Address: http://dx.doi.org/10.53846/goediss-8802

 

 

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Abstract

English

This thesis investigates the interplay between geometry and convex analysis in Hadamard spaces. Motivated by numerous applications of CAT(0) geometry, our work builds upon the results in convex analysis and Alexandrov geometry of many previous authors. Our investigations answer several questions in the theory of CAT(0) spaces some of which were posed as open problems in recent literature. In a nutshell our thesis develops along the following lines: 1. Weak topologies in Hadamard spaces, 2. Convex hulls of compact sets, 3. Mean tree problem in phylogenetic tree spaces, 4. Mosco convergence in Hadamard spaces, 5. Firmly nonexpansive operators and their applications in Hadamard spaces.
Keywords: Hadamard space; geodesics; nonpositive curvature; weak topology; weak convergence; geodesically monotone; convex hulls; phylogenetic trees; Fréchet mean; threading; threading degree; Mosco convergence; proximal mappings; firmly nonexpansive operators; convex combinations of operators; compositions of operators; metric subregularity

German

Diese Doktorarbeit untersucht das Zusammenspiel zwischen Geometrie und konvexer Analyse in Hadamardräumen. Motiviert durch zahlreiche Anwendungen der CAT(0)-Geometrie baut unsere Arbeit auf den Ergebnissen vieler früherer Autoren in der konvexen Analysis und der Alexandrov-Geometrie auf. Unsere Untersuchungen beantworten mehrere Fragen in der Theorie von CAT(0)-Räumen, von denen einige in der neueren Literatur als offene Probleme gestellt wurden. Zusammengefasst entwickelt sich unsere Dissertation in folgende Richtungen: 1. Schwache Topologien in Hadamard-Räumen, 2. Konvexe Hüllen kompakter Mengen, 3. Mittleres Baumproblem in phylogenetischen Baumräumen, 4. Mosco-Konvergenz in Hadamard-Räumen, 5. Fest nichtexpansive Operatoren und ihre Anwendungen in Hadamard-Räumen.

Other Languages

Kjo tezë e doktoratës hulumton ndërveprimin midis gjeometrisë dhe analizës konvekse në hapësirat Hadamard. E motivuar nga aplikime të shumta të gjeometrisë CAT(0), puna jonë bazohet në rezultatet e shumë autorëve të mëparshëmnë mbi analizën konvekse dhe gjeometrinë në sensin e Alexandrovit. Hetimet tona u përgjigjen disa pyetjeve në teorinë e hapësirave CAT(0) prej të cilave disa janë parashtruar si probleme të hapura në literaturën e fundit. Teza jonë e doktoratës zhvillohet sipas linjave të mëposhtme: 1. Topologjitë e dobëta në hapësirat Hadamard, 2. Konveksifikimi i bashkësive kompakte, 3. Problemi i pemës mesatare në hapësirat e pemëve filogjenetike, 4. Konvergjenca Mosko në hapësirat Hadamard, 5. Operatorët (plotësisht) jo-ekspansivë dhe aplikimet e tyre në hapësirat Hadamard.
Schlagwörter: Hadamard space; geodesics; nonpositive curvature; weak topology; weak convergence; geodesically monotone; convex hulls; phylogenetic trees; Fréchet mean; threading; threading degree; Mosco convergence; proximal mappings; firmly nonexpansive operators; convex combinations of operators; compositions of operators; metric subregularity
 
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