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This overview is based on the talk [101] given at the mini-workshop
0648c "Dirac operators in di?erential and non-commutative
geometry", Mat- matisches Forschungsinstitut Oberwolfach. Intended
for non-specialists, it explores the spectrum of the fundamental
Dirac operator on Riemannian spin manifolds, including recent
research and open problems. No background in spin geometry is
required; nevertheless the reader is assumed to be fam- iar with
basic notions of di?erential geometry (manifolds, Lie groups,
vector and principal bundles, coverings, connections, and
di?erential forms). The surveys [41, 132], which themselves provide
a very good insight into closed manifolds, served as the starting
point. We hope the content of this book re?ects the wide range of
?ndings on and sometimes amazing applications of the spin side of
spectral theory and will attract a new audience to the topic. vii
Acknowledgements
TheauthorwouldliketothanktheMathematischesForschungsinstitutOb-
wolfach for its friendly hospitality and stimulating atmosphere, as
well as the organizers and all those who participated in the
mini-workshop. This s- vey would not have been possible without the
encouragement and advice of Christian B.. ar and Oussama Hijazi as
well as the support of the German Research Foundation's
Sonderforschungsbereich 647 "Raum, Zeit, Materie. Analytische und
geometrische Strukturen" (Collaborative Research Center 647/Space,
Time and Matter. Analytical and Geometric Structures). We would
also like to thank Bernd Ammann and Nadine Grosse for their enlig-
eningdiscussions and usefulreferences.
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