3 research outputs found
Model Order Reduction
An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering, computational physics and computer science
Global logarithmic deformation theory
A classical problem in algebraic geometry is to construct smooth algebraic
varieties with prescribed properties. In the approach via smoothings, one first
constructs a degenerate scheme with the prescribed properties, and then shows
the existence of a smooth variety degenerating to this scheme. Logarithmic
geometry has given important new impulses to the second step of this approach,
which we explore in this book. Degenerations, in particular in the context of
mirror symmetry, often enjoy similar formal properties as smooth morphisms once
considered from the logarithmic perspective. Logarithmic deformation theory has
therefore become an effective tool for the construction of smoothings and the
transfer of properties between smooth nearby fibers and the singular special
fiber.
The strongest existence result for deformations in classical algebraic
geometry is the Bogomolov--Tian--Todorov theorem for Calabi--Yau varieties. A
logarithmic variant, once established, constructs log smooth deformations.
However, the logarithmic Bogomolov--Tian--Todorov theorem has resisted efforts
to its proof for a while. Finally, a method to prove it was discovered in 2019
by Chan, Leung, and Ma.
In this book, we explore this new approach to the logarithmic
Bogomolov--Tian--Todorov theorem. We prove several variants of the abstract
unobstructedness theorem, some of which are new and stronger than previously
known results. We investigate its application to the global deformation theory
of log smooth and mildly log singular spaces, obtaining unobstructedness
results for log Calabi--Yau spaces, some log Fano spaces, and line bundles.
Special care is taken to allow sufficiently mild log singularities, including
all log singularities that appear in the Gross--Siebert construction of toric
log Calabi--Yau mirror pairs.Comment: 345 pages, 16 figures; terminology changed to the more standard
curved Lie algebra (instead of pdg Lie algebra); figures improved; more
examples added; references updated and new references added; list of figures
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