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"Functions of several complex variables"
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Weighted Bergman spaces induced by rapidly increasing weights
This monograph is devoted to the study of the weighted Bergman space $A^p_\\omega$ of the unit disc $\\mathbb{D}$ that is induced by a radial continuous weight $\\omega$ satisfying $\\lim_{r\\to 1^-}\\frac{\\int_r^1\\omega(s)\\,ds}{\\omega(r)(1-r)}=\\infty.$ Every such $A^p_\\omega$ lies between the Hardy space $H^p$ and every classical weighted Bergman space $A^p_\\alpha$. Even if it is well known that $H^p$ is the limit of $A^p_\\alpha$, as $\\alpha\\to-1$, in many respects, it is shown that $A^p_\\omega$ lies ``closer'' to $H^p$ than any $A^p_\\alpha$, and that several finer function-theoretic properties of $A^p_\\alpha$ do not carry over to $A^p_\\omega$.
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Topics in several complex variables : first USA-Uzbekistan conference analysis and mathematical physics, May 20-23, 2014, California State University, Fullerton, CA
This volume contains the proceedings of the Special Session on Several Complex Variables, which was held during the first USA-Uzbekistan Conference on Analysis and Mathematical Physics from May 20-23, 2014, at California State University, Fullerton.This volume covers a wide variety of topics in pluripotential theory, symplectic geometry and almost complex structures, integral formulas, holomorphic extension, and complex dynamics. In particular, the reader will find articles on Lagrangian submanifolds and rational convexity, multidimensional residues, S-parabolic Stein manifolds, Segre varieties, and the theory of quasianalytic functions.
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Functional Analysis and Geometry
This is the first of two volumes dedicated to the centennial of the distinguished mathematician Selim Grigorievich Krein. The companion volume is Contemporary Mathematics, Volume 734.Krein was a major contributor to functional analysis, operator theory, partial differential equations, fluid dynamics, and other areas, and the author of several influential monographs in these areas. He was a prolific teacher, graduating 83 Ph.D. students. Krein also created and ran, for many years, the annual Voronezh Winter Mathematical Schools, which significantly influenced mathematical life in the former Soviet Union.The articles contained in this volume are written by prominent mathematicians, former students and colleagues of Selim Krein, as well as lecturers and participants of Voronezh Winter Schools. They are devoted to a variety of contemporary problems in functional analysis, operator theory, several complex variables, topological dynamics, and algebraic, convex, and integral geometry.
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Local dynamics of non-invertible maps near normal surface singularities
We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs
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Analysis and geometry in several complex variables : workshop on Analysis and Geometry in Several Complex Variables, January 4-8, 2015, Texas A&M University at Qatar, Doha, Qatar
This volume contains the proceedings of the workshop on Analysis and Geometry in Several Complex Variables, held from January 4-8, 2015, at Texas AM University at Qatar, Doha, Qatar.This volume covers many topics of current interest in several complex variables, CR geometry, and the related area of overdetermined systems of complex vector fields, as well as emerging trends in these areas.Papers feature original research on diverse topics such as the rigidity of CR mappings, normal forms in CR geometry, the d-bar Neumann operator, asymptotic expansion of the Bergman kernel, and hypoellipticity of complex vector fields. Also included are are two survey articles on complex Brunn-Minkowski theory and the regularity of systems of complex vector fields and their associated Laplacians.
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Holomorphic Automorphic Forms and Cohomology
We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic
cocycles. For integral weights at least
For real weights that are not an integer at least
A tool in establishing these results is the relation to
cohomology groups with values in modules of “analytic boundary germs”, which are represented by harmonic functions on subsets of the
upper half-plane. It turns out that for integral weights at least
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Cohomology of the Moduli Space of Cubic Threefolds and Its Smooth Models
We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic
threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily–Borel and toroidal compactifications of the ball
quotient model, due to Allcock–Carlson–Toledo. Our starting point is Kirwan’s method. We then follow by investigating the behavior of
the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a
detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli space
of cubic surfaces is discussed in an appendix.
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Nil Bohr-sets and almost automorphy of higher order
Two closely related topics: higher order Bohr sets and higher order almost automorphy are investigated in this paper. Both of them
are related to nilsystems.
In the first part, the problem which can be viewed as the higher order version of an old question
concerning Bohr sets is studied: for any
In the second part, the notion
of
eBook
Szegő kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds
Let X be an abstract not necessarily compact orientable CR manifold of dimension 2n-1, n\\geqslant 2, and let L^k be the k-th tensor power of a CR complex line bundle L over X. Given q\\in \\{0,1,\\ldots ,n-1\\}, let \\Box ^{(q)}_{b,k} be the Gaffney extension of Kohn Laplacian for (0,q) forms with values in L^k. For \\lambda \\geq 0, let \\Pi ^{(q)}_{k,\\leq \\lambda} :=E((-\\infty ,\\lambda ]), where E denotes the spectral measure of \\Box ^{(q)}_{b,k}. In this work, the author proves that \\Pi ^{(q)}_{k,\\leq k^{-N_0}}F^*_k, F_k\\Pi ^{(q)}_{k,\\leq k^{-N_0}}F^*_k, N_0\\geq 1, admit asymptotic expansions with respect to k on the non-degenerate part of the characteristic manifold of \\Box ^{(q)}_{b,k}, where F_k is some kind of microlocal cut-off function. Moreover, we show that F_k\\Pi ^{(q)}_{k,\\leq 0}F^*_k admits a full asymptotic expansion with respect to k if \\Box ^{(q)}_{b,k} has small spectral gap property with respect to F_k and \\Pi^{(q)}_{k,\\leq 0} is k-negligible away the diagonal with respect to F_k. By using these asymptotics, the authors establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR S^1 action.
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