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"Geometric analysis."
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Gromov’s Theory of Multicomplexes with Applications to Bounded Cohomology and Simplicial Volume
The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper
The first aim of this paper is to lay the foundation of the theory of
multicomplexes. After setting the main definitions, we construct the singular multicomplex
In the second part of this work we apply the theory of multicomplexes to the study of the bounded
cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov’s Mapping Theorem (which
implies in particular that the bounded cohomology of a space only depends on its fundamental group) and of Gromov’s Vanishing Theorem,
which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity.
The
third and last part of the paper is devoted to the study of locally finite chains on non-compact spaces, hence to the simplicial volume
of open manifolds. We expand some ideas of Gromov to provide detailed proofs of a criterion for the vanishing and a criterion for the
finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the
eBook
The shape of a life : one mathematician's search for the universe's hidden geometry
\"Harvard geometer and Fields medalist Shing-Tung Yau has provided a mathematical foundation for string theory, offered new insights into black holes, and mathematically demonstrated the stability of our universe. In this autobiography, Yau reflects on his improbable journey to becoming one of the world's most distinguished mathematicians. Beginning with an impoverished childhood in China and Hong Kong, Yau takes readers through his doctoral studies at Berkeley during the height of the Vietnam War protests, his Fields Medal-winning proof of the Calabi conjecture, his return to China, and his pioneering work in geometric analysis. This new branch of geometry, which Yau built up with his friends and colleagues, has paved the way for solutions to several important and previously intransigent problems. With complicated ideas explained for a broad audience, this book offers readers not only insights into the life of an eminent mathematician, but also an accessible way to understand advanced and highly abstract concepts in mathematics and theoretical physics\"--Publisher's website.
Book
Modeling and analysis of compositional data
Modeling and Analysis of Compositional Data presents a practical and comprehensive introduction to the analysis of compositional data along with numerous examples to illustrate both theory and application of each method. Based upon short courses delivered by the authors, it provides a complete and current compendium of fundamental to advanced methodologies along with exercises at the end of each chapter to improve understanding, as well as data and a solutions manual which is available on an accompanying website.
Complementing Pawlowsky-Glahn's earlier collective text that provides an overview of the state-of-the-art in this field, Modeling and Analysis of Compositional Data fills a gap in the literature for a much-needed manual for teaching, self learning or consulting.
eBook
Probabilistic Methods in Geometry, Topology and Spectral Theory
This volume contains the proceedings of the CRM Workshops on Probabilistic Methods in Spectral Geometry and PDE, held from August 22-26, 2016 and Probabilistic Methods in Topology, held from November 14-18, 2016 at the Centre de Recherches Mathématiques, Université de Montréal, Montréal, Quebec, Canada. Probabilistic methods have played an increasingly important role in many areas of mathematics, from the study of random groups and random simplicial complexes in topology, to the theory of random Schrödinger operators in mathematical physics. The workshop on Probabilistic Methods in Spectral Geometry and PDE brought together some of the leading researchers in quantum chaos, semi-classical theory, ergodic theory and dynamical systems, partial differential equations, probability, random matrix theory, mathematical physics, conformal field theory, and random graph theory. Its emphasis was on the use of ideas and methods from probability in different areas, such as quantum chaos (study of spectra and eigenstates of chaotic systems at high energy); geometry of random metrics and related problems in quantum gravity; solutions of partial differential equations with random initial conditions. The workshop Probabilistic Methods in Topology brought together researchers working on random simplicial complexes and geometry of spaces of triangulations (with connections to manifold learning); topological statistics, and geometric probability; theory of random groups and their properties; random knots; and other problems. This volume covers recent developments in several active research areas at the interface of Probability, Semiclassical Analysis, Mathematical Physics, Theory of Automorphic Forms and Graph Theory.
eBook
Degree Theory of Immersed Hypersurfaces
The authors develop a degree theory for compact immersed hypersurfaces of prescribed K-curvature immersed in a compact, orientable Riemannian manifold, where K is any elliptic curvature function. They apply this theory to count the (algebraic) number of immersed hyperspheres in various cases: where K is mean curvature, extrinsic curvature and special Lagrangian curvature and show that in all these cases, this number is equal to -\\chi(M), where \\chi(M) is the Euler characteristic of the ambient manifold M.
eBook
Lipschitz graphs and currents in Heisenberg groups
The main result of the present article is a Rademacher-type theorem for intrinsic Lipschitz graphs of codimension
$k\\leq n$
in sub-Riemannian Heisenberg groups
${\\mathbb H}^{n}$
. For the purpose of proving such a result, we settle several related questions pertaining both to the theory of intrinsic Lipschitz graphs and to the one of currents. First, we prove an extension result for intrinsic Lipschitz graphs as well as a uniform approximation theorem by means of smooth graphs: both of these results stem from a new definition (equivalent to the one introduced by B. Franchi, R. Serapioni and F. Serra Cassano) of intrinsic Lipschitz graphs and are valid for a more general class of intrinsic Lipschitz graphs in Carnot groups. Second, our proof of Rademacher’s theorem heavily uses the language of currents in Heisenberg groups: one key result is, for us, a version of the celebrated constancy theorem. Inasmuch as Heisenberg currents are defined in terms of Rumin’s complex of differential forms, we also provide a convenient basis of Rumin’s spaces. Eventually, we provide some applications of Rademacher’s theorem including a Lusin-type result for intrinsic Lipschitz graphs, the equivalence between
${\\mathbb H}$
-rectifiability and ‘Lipschitz’
${\\mathbb H}$
-rectifiability and an area formula for intrinsic Lipschitz graphs in Heisenberg groups.
Journal Article
The conjugacy problem for $\\operatorname {Out}(F_3)
We present a solution to the conjugacy problem in the group of outer automorphisms of
$F_3$
, a free group of rank 3. We distinguish according to several computable invariants, such as irreducibility, subgroups of polynomial growth and subgroups carrying the attracting lamination. We establish, by considerations on train tracks, that the conjugacy problem is decidable for the outer automorphisms of
$F_3$
that preserve a given rank 2 free factor. Then we establish, by consideration on mapping tori, that it is decidable for outer automorphisms of
$F_3$
whose maximal polynomial growth subgroups are cyclic. This covers all the cases left by the state of the art.
Journal Article