Differential Geometry Of Curves And Surfaces With Singularities
Differential Geometry Of Curves And Surfaces With Singularities

Author: Masaaki Umehara

Publisher: World Scientific

Published: 2021-11-29

Total Pages: 387

ISBN-13: 9811237158

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This book provides a unique and highly accessible approach to singularity theory from the perspective of differential geometry of curves and surfaces. It is written by three leading experts on the interplay between two important fields — singularity theory and differential geometry.The book introduces singularities and their recognition theorems, and describes their applications to geometry and topology, restricting the objects of attention to singularities of plane curves and surfaces in the Euclidean 3-space. In particular, by presenting the singular curvature, which originated through research by the authors, the Gauss-Bonnet theorem for surfaces is generalized to those with singularities. The Gauss-Bonnet theorem is intrinsic in nature, that is, it is a theorem not only for surfaces but also for 2-dimensional Riemannian manifolds. The book also elucidates the notion of Riemannian manifolds with singularities.These topics, as well as elementary descriptions of proofs of the recognition theorems, cannot be found in other books. Explicit examples and models are provided in abundance, along with insightful explanations of the underlying theory as well. Numerous figures and exercise problems are given, becoming strong aids in developing an understanding of the material.Readers will gain from this text a unique introduction to the singularities of curves and surfaces from the viewpoint of differential geometry, and it will be a useful guide for students and researchers interested in this subject.

Centres of Centralizers of Unipotent Elements in Simple Algebraic Groups
Centres of Centralizers of Unipotent Elements in Simple Algebraic Groups

Author: Ross Lawther

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 201

ISBN-13: 0821847694

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Let G be a simple algebraic group defined over an algebraically closed field k whose characteristic is either 0 or a good prime for G, and let uEG be unipotent. The authors study the centralizer CG(u), especially its centre Z(CG(u)). They calculate the Lie algebra of Z(CG(u)), in particular determining its dimension; they prove a succession of theorems of increasing generality, the last of which provides a formula for dim Z(CG(u)) in terms of the labelled diagram associated to the conjugacy class containing u.

Second Order Analysis on $(\mathscr {P}_2(M),W_2)$
Second Order Analysis on $(\mathscr {P}_2(M),W_2)$

Author: Nicola Gigli

Publisher: American Mathematical Soc.

Published: 2012-02-22

Total Pages: 173

ISBN-13: 0821853090

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The author develops a rigorous second order analysis on the space of probability measures on a Riemannian manifold endowed with the quadratic optimal transport distance $W_2$. The discussion includes: definition of covariant derivative, discussion of the problem of existence of parallel transport, calculus of the Riemannian curvature tensor, differentiability of the exponential map and existence of Jacobi fields. This approach does not require any smoothness assumption on the measures considered.

Jumping Numbers of a Simple Complete Ideal in a Two-Dimensional Regular Local Ring
Jumping Numbers of a Simple Complete Ideal in a Two-Dimensional Regular Local Ring

Author: Tarmo Järvilehto

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 93

ISBN-13: 0821848119

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The multiplier ideals of an ideal in a regular local ring form a family of ideals parameterized by non-negative rational numbers. As the rational number increases the corresponding multiplier ideal remains unchanged until at some point it gets strictly smaller. A rational number where this kind of diminishing occurs is called a jumping number of the ideal. In this manuscript the author gives an explicit formula for the jumping numbers of a simple complete ideal in a two-dimensional regular local ring. In particular, he obtains a formula for the jumping numbers of an analytically irreducible plane curve. He then shows that the jumping numbers determine the equisingularity class of the curve.

On $L$-Packets for Inner Forms of $SL_n$
On $L$-Packets for Inner Forms of $SL_n$

Author: Kaoru Hiraga

Publisher: American Mathematical Soc.

Published: 2012

Total Pages: 110

ISBN-13: 0821853643

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The theory of $L$-indistinguishability for inner forms of $SL_2$ has been established in the well-known paper of Labesse and Langlands (L-indistinguishability forSL$(2)$. Canad. J. Math. 31 (1979), no. 4, 726-785). In this memoir, the authors study $L$-indistinguishability for inner forms of $SL_n$ for general $n$. Following the idea of Vogan in (The local Langlands conjecture. Representation theory of groups and algebras, 305-379, Contemp. Math. 145 (1993)), they modify the $S$-group and show that such an $S$-group fits well in the theory of endoscopy for inner forms of $SL_n$.

Hardy Spaces Associated to Non-Negative Self-Adjoint Operators Satisfying Davies-Gaffney Estimates
Hardy Spaces Associated to Non-Negative Self-Adjoint Operators Satisfying Davies-Gaffney Estimates

Author: Steve Hofmann

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 91

ISBN-13: 0821852388

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Let $X$ be a metric space with doubling measure, and $L$ be a non-negative, self-adjoint operator satisfying Davies-Gaffney bounds on $L^2(X)$. In this article the authors present a theory of Hardy and BMO spaces associated to $L$, including an atomic (or molecular) decomposition, square function characterization, and duality of Hardy and BMO spaces. Further specializing to the case that $L$ is a Schrodinger operator on $\mathbb{R}^n$ with a non-negative, locally integrable potential, the authors establish additional characterizations of such Hardy spaces in terms of maximal functions. Finally, they define Hardy spaces $H^p_L(X)$ for $p>1$, which may or may not coincide with the space $L^p(X)$, and show that they interpolate with $H^1_L(X)$ spaces by the complex method.

The Internally 4-Connected Binary Matroids with No $M(K_{3,3})$-Minor
The Internally 4-Connected Binary Matroids with No $M(K_{3,3})$-Minor

Author: Dillon Mayhew

Publisher: American Mathematical Soc.

Published: 2010

Total Pages: 110

ISBN-13: 0821848267

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The authors give a characterization of the internally $4$-connected binary matroids that have no minor isomorphic to $M(K_{3,3})$. Any such matroid is either cographic, or is isomorphic to a particular single-element extension of the bond matroid of a cubic or quartic Mobius ladder, or is isomorphic to one of eighteen sporadic matroids.