EAN: 9783031277030

SPRINGERBRIEFS IN MATHEMATICS TWISTED ISOSPEC...

The question of reconstructing a geometric shape from spectra of operators (such as the Laplace
operator) is decades old and an active area of research in mathematics and mathematical
physics. This book focusses on the case of compact Riemannian manifolds  and  in particular 
the question whether one can find finitely many natural operators that determine whether two
such manifolds are isometric (coverings). The methods outlined in the book fit into the
tradition of the famous work of Sunada on the construction of isospectral  non-isometric
manifolds  and thus do not focus on analytic techniques  but rather on algebraic methods: in
particular  the analogy with constructions in number theory  methods from representation theory
and from algebraic topology. The main goal of the book is to present the construction of
finitely many twisted Laplace operators whose spectrum determines covering equivalence of two
Riemannian manifolds. The book has a leisure pace and presents details and examples that are
hard to find in the literature  concerning: fiber products of manifolds and orbifolds  the
distinction between the spectrum and the spectral zeta function for general operators  strong
isospectrality  twisted Laplacians  the action of isometry groups on homology groups  monomial
structures on group representations  geometric and group-theoretical realisation of coverings
with wreath products as covering groups  and class field theory for manifolds. The book
contains a wealth of worked examples and open problems. After perusing the book  the reader
will have a comfortable working knowledge of the algebraic approach to isospectrality. This is
an open access book.

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