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Recent Developments in Geometry
This volume is the outgrowth of a Special Session on Geometry, held at the November 1987 meeting of the AMS at the University of California at Los Angeles. The unusually well-attended session attracted more than sixty participants and featured over forty addresses by some of the day's outstanding geometers. By common consent, it was decided that the papers to be collected in the present volume should be surveys of relatively broad areas of geometry, rather than detailed presentations of new research results. A comprehensive survey of the field is beyond the scope of a volume such as this. Nonetheless, the editors have sought to provide all geometers, whatever their specialties, with some insight into recent developments in a variety of topics in this active area of research.
A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials
In particular, for b = 1 and λ = 0, we find a sharp condition on h such that the origin is a removable singularity for all non-negative solutions of [[eqref]]one, thus addressing an open question of Vázquez and Véron.
Entropy and Multivariable Interpolation
We define a new notion of entropy for operators on Fock spaces and positive multi-Toeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (e.g., multi-Toeplitz, multi-analytic, etc.) operators on Fock spaces. These results lead to entropy inequalities and entropy formulas for positive multi-Toeplitz kernels on free semigroups (resp. multi-analytic operators) and consequences concerning the extreme points of the unit ball of the noncommutative analytic Toeplitz algebra $F ninfty$. We obtain several geometric characterizations of the central intertwining lifting, a maximal principle, and a permanence principle for the noncommutative commutant lifting ...
Differential Geometry and Global Analysis
This volume contains the proceedings of the AMS Special Session on Differential Geometry and Global Analysis, Honoring the Memory of Tadashi Nagano (1930–2017), held January 16, 2020, in Denver, Colorado. Tadashi Nagano was one of the great Japanese differential geometers, whose fundamental and seminal work still attracts much interest today. This volume is inspired by his work and his legacy and, while recalling historical results, presents recent developments in the geometry of symmetric spaces as well as generalizations of symmetric spaces; minimal surfaces and minimal submanifolds; totally geodesic submanifolds and their classification; Riemannian, affine, projective, and conformal connections; the $(M_{+}, M_{-})$ method and its applications; and maximal antipodal subsets. Additionally, the volume features recent achievements related to biharmonic and biconservative hypersurfaces in space forms, the geometry of Laplace operator on Riemannian manifolds, and Chen-Ricci inequalities for Riemannian maps, among other topics that could attract the interest of any scholar working in differential geometry and global analysis on manifolds.
Semisolvability of Semisimple Hopf Algebras of Low Dimension
The author proves that every semisimple Hopf algebra of dimension less than $60$ over an algebraically closed field $k$ of characteristic zero is either upper or lower semisolvable up to a cocycle twist.
The Role of True Finiteness in the Admissible Recursively Enumerable Degrees
When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of $\alpha$-finiteness. As examples we discuss bothcodings of models of arithmetic into the recursively enumerable degrees, and non-distributive lattice embeddings into these degrees. We show that if an admissible ordinal $\alpha$ is effectively close to $\omega$ (where this closeness can be measured by size or by cofinality) then such constructions maybe performed in the $\alpha$-r.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the first-order language of partially ordered sets, and so these results also show that there are natu
The Ricci Flow in Riemannian Geometry
This book focuses on Hamilton's Ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for Riemannian manifolds, and Perelman's noncollapsing results, and culminating in a detailed analysis of the evolution of curvature, where recent breakthroughs of Böhm and Wilking and Brendle and Schoen have led to a proof of the differentiable 1/4-pinching sphere theorem.
On Maps from Loop Suspensions to Loop Spaces and the Shuffle Relations on the Cohen Groups
The maps from loop suspensions to loop spaces are investigated using group representations in this article. The shuffle relations on the Cohen groups are given. By using these relations, a universal ring for functorial self maps of double loop spaces of double suspensions is given. Moreover the obstructions to the classical exponent problem in homotopy theory are displayed in the extension groups of the dual of the important symmetric group modules Lie$(n)$, as well as in the top cohomology of the Artin braid groups with coefficients in the top homology of the Artin pure braid groups.
The Hilbert Function of a Level Algebra
Let $R$ be a polynomial ring over an algebraically closed field and let $A$ be a standard graded Cohen-Macaulay quotient of $R$. The authors state that $A$ is a level algebra if the last module in the minimal free resolution of $A$ (as $R$-module) is of the form $R(-s)a$, where $s$ and $a$ are positive integers. When $a=1$ these are also known as Gorenstein algebras. The basic question addressed in this paper is: What can be the Hilbert Function of a level algebra? The authors consider the question in several particular cases, e.g., when $A$ is an Artinian algebra, or when $A$ is the homogeneous coordinate ring of a reduced set of points, or when $A$ satisfies the Weak Lefschetz Property. The authors give new methods for showing that certain functions are NOT possible as the Hilbert function of a level algebra and also give new methods to construct level algebras. In a (rather long) appendix, the authors apply their results to give complete lists of all possible Hilbert functions in the case that the codimension of $A = 3$, $s$ is small and $a$ takes on certain fixed values.
Michele Sce's Works in Hypercomplex Analysis
This book presents English translations of Michele Sce’s most important works, originally written in Italian during the period 1955-1973, on hypercomplex analysis and algebras of hypercomplex numbers. Despite their importance, these works are not very well known in the mathematics community because of the language they were published in. Possibly the most remarkable instance is the so-called Fueter-Sce mapping theorem, which is a cornerstone of modern hypercomplex analysis, and is not yet understood in its full generality. This volume is dedicated to revealing and describing the framework Sce worked in, at an exciting time when the various generalizations of complex analysis in one variabl...
