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Multiscale and Multiresolution Methods
Many computionally challenging problems omnipresent in science and engineering exhibit multiscale phenomena so that the task of computing or even representing all scales of action is computationally very expensive unless the multiscale nature of these problems is exploited in a fundamental way. Some diverse examples of practical interest include the computation of fluid turbulence, structural analysis of composite materials, terabyte data mining, image processing, and a multitude of others. This book consists of both invited and contributed articles which address many facets of efficient multiscale representation and scientific computation from varied viewpoints such as hierarchical data representations, multilevel algorithms, algebraic homogeni- zation, and others. This book should be of particular interest to readers interested in recent and emerging trends in multiscale and multiresolution computation with application to a wide range of practical problems.
Cyclic Cohomology at 40: Achievements and Future Prospects
This volume contains the proceedings of the virtual conference on Cyclic Cohomology at 40: Achievements and Future Prospects, held from September 27–October 1, 2021 and hosted by the Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada. Cyclic cohomology, since its discovery forty years ago in noncommutative differential geometry, has become a fundamental mathematical tool with applications in domains as diverse as analysis, algebraic K-theory, algebraic geometry, arithmetic geometry, solid state physics and quantum field theory. The reader will find survey articles providing a user-friendly introduction to applications of cyclic cohomology in such areas as higher ca...
The Dirichlet Problem for Parabolic Operators with Singular Drift Terms
This memoir considers the Dirichlet problem for parabolic operators in a half space with singular drift terms. Chapter I begins the study of a parabolic PDE modelled on the pullback of the heat equation in certain time varying domains considered by Lewis-Murray and Hofmann-Lewis. Chapter II obtains mutual absolute continuity of parabolic measure and Lebesgue measure on the boundary of this halfspace and also that the $L DEGREESq(R DEGREESn)$ Dirichlet problem for these PDEs has a solution when $q$ is large enough. Chapter III proves an analogue of a theorem of Fefferman, Kenig, and Pipher for certain parabolic PDEs with singular drift terms. Each of the chapters that comprise this memoir has its own numbering system and list
Rings, Modules, and Algebras in Stable Homotopy Theory
This book introduces a new point-set level approach to stable homotopy theory that has already had many applications and promises to have a lasting impact on the subject. Given the sphere spectrum $S$, the authors construct an associative, commutative, and unital smash product in a complete and cocomplete category of ``$S$-modules'' whose derived category is equivalent to the classical stable homotopy category. This construction allows for a simple and algebraically manageable definition of ``$S$-algebras'' and ``commutative $S$-algebras'' in terms of associative, or associative and commutative, products $R\wedge SR \longrightarrow R$. These notions are essentially equivalent to the earlier notions of $A {\infty $ and $E {\infty $ ring spectra, and the older notions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of $R$-modules in terms of maps $R\wedge SM\longrightarrow M$. When $R$ is commutative, the category of $R$-modules also has a
Computers, Rigidity, and Moduli
This book is the first to present a new area of mathematical research that combines topology, geometry, and logic. Shmuel Weinberger seeks to explain and illustrate the implications of the general principle, first emphasized by Alex Nabutovsky, that logical complexity engenders geometric complexity. He provides applications to the problem of closed geodesics, the theory of submanifolds, and the structure of the moduli space of isometry classes of Riemannian metrics with curvature bounds on a given manifold. Ultimately, geometric complexity of a moduli space forces functions defined on that space to have many critical points, and new results about the existence of extrema or equilibria follow...
Special Groups
This monograph presents a systematic study of Special Groups, a first-order universal-existential axiomatization of the theory of quadratic forms, which comprises the usual theory over fields of characteristic different from 2, and is dual to the theory of abstract order spaces. The heart of our theory begins in Chapter 4 with the result that Boolean algebras have a natural structure of reduced special group. More deeply, every such group is canonically and functorially embedded in a certain Boolean algebra, its Boolean hull. This hull contains a wealth of information about the structure of the given special group, and much of the later work consists in unveiling it. Thus, in Chapter 7 we in...
Ruelle Operators: Functions which Are Harmonic with Respect to a Transfer Operator
Let $N\in\mathbb{N}$, $N\geq2$, be given. Motivated by wavelet analysis, this title considers a class of normal representations of the $C DEGREES{\ast}$-algebra $\mathfrak{A}_{N}$ on two unitary generators $U$, $V$ subject to the relation $UVU DEGREES{-1}=V DEGREES{N}$. The representations are in one-to-one correspondence with solutions $h\in L DEGREES{1}\left(\mathbb{T}\right)$, $h\geq0$, to $R\left(h\right)=h$ where $R$ is a certain transfer operator (positivity-preserving) which was studied previously by D. Ruelle. The representations of $\mathfrak{A}_{N}$ may also be viewed as representations of a certain (discrete) $N$-adic $ax+b$ group which was considered recently
Non-Uniform Lattices on Uniform Trees
This title provides a comprehensive examination of non-uniform lattices on uniform trees. Topics include graphs of groups, tree actions and edge-indexed graphs; $Aut(x)$ and its discrete subgroups; existence of tree lattices; non-uniform coverings of indexed graphs with an arithmetic bridge; non-uniform coverings of indexed graphs with a separating edge; non-uniform coverings of indexed graphs with a ramified loop; eliminating multiple edges; existence of arithmetic bridges. This book is intended for graduate students and research mathematicians interested in group theory and generalizations.
Frames, Bases and Group Representations
This work develops an operator-theoretic approach to discrete frame theory on a separable Hilbert space. It is then applied to an investigation of the structural properties of systems of unitary operators on Hilbert space which are related to orthonormal wavelet theory. Also obtained are applications of frame theory to group representations, and of the theory of abstract unitary systems to frames generated by Gabor type systems.
A Geometric Setting for Hamiltonian Perturbation Theory
In this text, the perturbation theory of non-commutatively integrable systems is revisited from the point of view of non-Abelian symmetry groups. Using a co-ordinate system intrinsic to the geometry of the symmetry, the book generalizes and geometrizes well-known estimates of Nekhoroshev (1977), in a class of systems having almost $G$-invariant Hamiltonians. These estimates are shown to have a natural interpretation in terms of momentum maps and co-adjoint orbits. The geometric framework adopted is described explicitly in examples, including the Euler-Poinsot rigid body.
