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This volume presents the construction of canonical modular compactifications of moduli spaces for polarized Abelian varieties (possibly with level structure), building on the earlier work of Alexeev, Nakamura, and Namikawa. This provides a different approach to compactifying these spaces than the more classical approach using toroical embeddings, which are not canonical. There are two main new contributions in this monograph: (1) The introduction of logarithmic geometry as understood by Fontaine, Illusie, and Kato to the study of degenerating Abelian varieties; and (2) the construction of canonical compactifications for moduli spaces with higher degree polarizations based on stack-theoretic techniques and a study of the theta group.

Geometry --- landmeetkunde

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In the last decade, convolution operators of matrix functions have received unusual attention due to their diverse applications. This monograph presents some new developments in the spectral theory of these operators. The setting is the Lp spaces of matrix-valued functions on locally compact groups. The focus is on the spectra and eigenspaces of convolution operators on these spaces, defined by matrix-valued measures. Among various spectral results, the L2-spectrum of such an operator is completely determined and as an application, the spectrum of a discrete Laplacian on a homogeneous graph is computed using this result. The contractivity properties of matrix convolution semigroups are studied and applications to harmonic functions on Lie groups and Riemannian symmetric spaces are discussed. An interesting feature is the presence of Jordan algebraic structures in matrix-harmonic functions.

Ordered algebraic structures --- Differential geometry. Global analysis --- Operator theory --- Differential equations --- differentiaalvergelijkingen --- algebra --- analyse (wiskunde) --- Laplacetransformatie --- differentiaal geometrie

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Ordered algebraic structures --- Geometry --- algebra --- landmeetkunde

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Industrial economics --- Numerical methods of optimisation --- Operational research. Game theory --- Mathematics --- toegepaste wiskunde --- speltheorie --- operationeel onderzoek --- industriële marketing --- ingenieurswetenschappen --- kansrekening --- optimalisatie

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Category theory. Homological algebra --- Geometry --- algebra --- landmeetkunde