This monograph presents an approachable proof of Mirzakhani’s curve counting theorem, both for simple and non-simple curves. Designed to welcome readers to the area, the presentation builds intuition with elementary examples before progressing to rigorous proofs. This approach illuminates new and established results alike, and produces versatile tools for studying the geometry of hyperbolic surfaces, Teichmüller theory, and mapping class groups.
Beginning with the preliminaries of curves and arcs on surfaces, the authors go on to present the theory of geodesic currents in...
This monograph presents an approachable proof of Mirzakhani’s curve counting theorem, both for simple and non-simple curves. Designed to welcome ...
This monograph provides a state-of-the-art, self-contained account on the effectiveness of the method of boundary layer potentials in the study of elliptic boundary value problems with boundary data in a multitude of function spaces. Many significant new results are explored in detail, with complete proofs, emphasizing and elaborating on the link between the geometric measure-theoretic features of an underlying surface and the functional analytic properties of singular integral operators defined on it. Graduate students, researchers, and professionals interested in a modern account of...
This monograph provides a state-of-the-art, self-contained account on the effectiveness of the method of boundary layer potentials in the study of ell...
This work gives a coherent introduction to isoperimetric inequalities in Riemannian manifolds, featuring many of the results obtained during the last 25 years and discussing different techniques in the area.
Written in a clear and appealing style, the book includes sufficient introductory material, making it also accessible to graduate students. It will be of interest to researchers working on geometric inequalities either from a geometric or analytic point of view, but also to those interested in applying the described techniques to their field.
This work gives a coherent introduction to isoperimetric inequalities in Riemannian manifolds, featuring many of the results obtained during ...
In this monograph, for elliptic systems with block structure in the upper half-space and t-independent coefficients, the authors settle the study of boundary value problems by proving compatible well-posedness of Dirichlet, regularity and Neumann problems in optimal ranges of exponents. Prior to this work, only the two-dimensional situation was fully understood. In higher dimensions, partial results for existence in smaller ranges of exponents and for a subclass of such systems had been established. The presented uniqueness results are completely new, and the authors...
In this monograph, for elliptic systems with block structure in the upper half-space and t-independent coefficients, the authors settle the ...
This work gives a coherent introduction to isoperimetric inequalities in Riemannian manifolds, featuring many of the results obtained during the last 25 years and discussing different techniques in the area. Written in a clear and appealing style, the book includes sufficient introductory material, making it also accessible to graduate students. It will be of interest to researchers working on geometric inequalities either from a geometric or analytic point of view, but also to those interested in applying the described techniques to their field.
This work gives a coherent introduction to isoperimetric inequalities in Riemannian manifolds, featuring many of the results obtained during the las...
This monograph addresses two significant related questions in complex geometry: the construction of a Chern character on the Grothendieck group of coherent sheaves of a compact complex manifold with values in its Bott-Chern cohomology, and the proof of a corresponding Riemann-Roch-Grothendieck theorem. One main tool used is the equivalence of categories established by Block between the derived category of bounded complexes with coherent cohomology and the homotopy category of antiholomorphic superconnections. Chern-Weil theoretic techniques are then used to construct...
This monograph addresses two significant related questions in complex geometry: the construction of a Chern character on the Grothendieck group o...
The purpose of this monograph is to provide a systematic account of the theory of noncommutative integration in semi-finite von Neumann algebras. It is designed to serve as an introductory graduate level text as well as a basic reference for more established mathematicians with interests in the continually expanding areas of noncommutative analysis and probability. Its origins lie in two apparently distinct areas of mathematical analysis: the theory of operator ideals going back to von Neumann and Schatten and the general theory of rearrangement invariant Banach lattices of measurable...
The purpose of this monograph is to provide a systematic account of the theory of noncommutative integration in semi-finite von Neumann algebras. It i...
This monograph introduces readers to locally conformally Kähler (LCK) geometry and provides an extensive overview of the most current results. A rapidly developing area in complex geometry dealing with non-Kähler manifolds, LCK geometry has strong links to many other areas of mathematics, including algebraic geometry, topology, and complex analysis. The authors emphasize these connections to create a unified and rigorous treatment of the subject suitable for both students and researchers.
Part I builds the necessary foundations for those approaching LCK geometry for...
This monograph introduces readers to locally conformally Kähler (LCK) geometry and provides an extensive overview of the most current results. &nb...
This monograph offers a self-contained introduction to the regularity theory for integro-differential elliptic equations, mostly developed in the 21st century. This class of equations finds relevance in fields such as analysis, probability theory, mathematical physics, and in several contexts in the applied sciences. The work gives a detailed presentation of all the necessary techniques, with a primary focus on the main ideas rather than on proving all the results in their greatest generality.
The basic building blocks are presented first, with the study of the square root of...
This monograph offers a self-contained introduction to the regularity theory for integro-differential elliptic equations, mostly developed in the 2...
This award-winning monograph explores advanced topics in harmonic analysis, addressing both classical and contemporary problems. Several connections to number theory, crystallography or atomic theory are also surveyed. The term “suprematism” refers to a certain geometric point of view underlying proofs and arguments.
The opening of the book is dedicated to a few results, with short statements and proofs, that could be called “mathematical haikus”. Then, in the first part of the book, singular integrals beyond the classical Calderón-Zygmund theory, such as Vitali-type...
This award-winning monograph explores advanced topics in harmonic analysis, addressing both classical and contemporary problems. Several connection...
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