270,168 research outputs found
Hypercomplex Algebraic Geometry
It is well-known that sums and products of holomorphic functions are holomorphic, and the holomorphic functions on a complex manifold form a commutative algebra over C. The study of complex manifolds using algebras of holomorphic functions upon them is called complex algebraic geometry
Phylogenetic Algebraic Geometry
Phylogenetic algebraic geometry is concerned with certain complex projective
algebraic varieties derived from finite trees. Real positive points on these
varieties represent probabilistic models of evolution. For small trees, we
recover classical geometric objects, such as toric and determinantal varieties
and their secant varieties, but larger trees lead to new and largely unexplored
territory. This paper gives a self-contained introduction to this subject and
offers numerous open problems for algebraic geometers.Comment: 15 pages, 7 figure
Derived Algebraic Geometry
This text is a survey of derived algebraic geometry. It covers a variety of
general notions and results from the subject with a view on the recent
developments at the interface with deformation quantization.Comment: Final version. To appear in EMS Surveys in Mathematical Science
Dagger Geometry As Banach Algebraic Geometry
In this article, we apply the approach of relative algebraic geometry towards
analytic geometry to the category of bornological and Ind-Banach spaces
(non-Archimedean or not). We are able to recast the theory of Grosse-Kl\"onne
dagger affinoid domains with their weak G-topology in this new language. We
prove an abstract recognition principle for the generators of their standard
topology (the morphisms appearing in the covers). We end with a sketch of an
emerging theory of dagger affinoid spaces over the integers, or any Banach
ring, where we can see the Archimedean and non-Archimedean worlds coming
together
Frobenius techniques in birational geometry
This is a survey for the 2015 AMS Summer Institute on Algebraic Geometry
about the Frobenius type techniques recently used extensively in positive
characteristic algebraic geometry. We first explain the basic ideas through
simple versions of the fundamental definitions and statements, and then we
survey most of the recent algebraic geometry results obtained using these
techniques
Algebraic geometry over algebraic structures II: Foundations
In this paper we introduce elements of algebraic geometry over an arbitrary
algebraic structure. We prove Unification Theorems which gather the description
of coordinate algebras by several ways.Comment: 55 page
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