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Register a new userPure metric geometry: introductory lectures
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Anton Petrunin
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We discuss only domestic affairs of metric spaces, leaving all external applications aside. Topics include universal spaces, injective spaces, Gromov--Hausdorff convergence, and ultralimits.
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These lecture notes introduce the multiple membrane theories known as BLG and ABJM. We assume the reader is familiar with string theory, but not with M-theory, 11-dimensional supergravity or membranes. We therefore start with a background on M-theory and its extended objects before discussing BLG and ABJM. The link to string theory via dimensional reduction will be maintained throughout.
We give an elementary introduction to classical and quantum bosonic string theory.
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This is the text of a series of five lectures given by the author at the Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras held at Vanderbilt University in May 2004. It is meant as an overview of recent results illustrating the interplay between noncommutative geometry and arithmetic geometry/number theory.
Let $mathfrak{M}$ be a class of metric spaces. A metric space $Y$ is minimal $mathfrak{M}$-universal if every $Xinmathfrak{M}$ can be isometrically embedded in $Y$ but there are no proper subsets of $Y$ satisfying this property. We find conditions under which, for given metric space $X$, there is a class $mathfrak{M}$ of metric spaces such that $X$ is minimal $mathfrak{M}$-universal. We generalize the notion of minimal $mathfrak{M}$-universal metric space to notion of minimal $mathfrak{M}$-universal class of metric spaces and prove the uniqueness, up to an isomorphism, for these classes. The necessary and sufficient conditions under which the disjoint union of the metric spaces belonging to a class $mathfrak{M}$ is minimal $mathfrak{M}$-universal are found. Examples of minimal universal metric spaces are constructed for the classes of the three-point metric spaces and $n$-dimensional normed spaces. Moreover minimal universal metric spaces are found for some subclasses of the class of metric spaces $X$ which possesses the following property. Among every three distinct points of $X$ there is one point lying between the other two points.
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