The coarse Baum–Connes conjecture via coarse geometry

The C₀ coarse structure on a metric space is a refinement of the bounded structure and is closely related to the topology of the space. In this paper we will prove the C₀ version of the coarse Baum–Connes conjecture and show that K_*(C^*X₀) is a topological invariant for a broad class of metric spaces. Using this result we construct a ‘geometric’ obstruction group to the coarse Baum–Connes conjecture for the bounded coarse structure. We then show under the assumption of finite asymptotic dimension that the obstructions vanish, and hence we obtain a new proof of the coarse Baum–Connes conjecture in this context.     

Property (RD) for Cocompact Lattices in a Finite Product of Rank One Lie Groups with Some Rank Two Lie Groups

We apply V. Lafforgues techniques to establish property (RD) for cocompact lattices in a finite product of rank one Lie groups with Lie groups whose restricted root system is of type A ₂.     

The maximal coarse Baum–Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space

We introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromov?s notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum–Connes conjecture holds for metric spaces with bounded geometry which admit a fibred coarse embedding into Hilbert space.     

群与代数的表示理论（迎接ICM2002特约文章）

有限群模表示论,代数群与量子群,代数表示论,同调代数与代数K-理论是当前国际数学研究的前沿重点领域,在20世纪得到巨大发展,它们之间的相互交叉融合与综合统一在数学发展中具有鲜明特色。本文简要概述上述研究领域的发展历史与最新研究进展。     

Twisted Equivariant KK-Theory and the Baum–Connes Conjecture for Group Extensions

In this paper we study the stability of the Baum–Connes conjecture with coefficients undertaking group extensions. For this, it is necessary to extend Kasparov's equivariant KK-theory to an equivariant theory for twisted actions of groups on C ^*-algebras. As a consequence of our stability results, we are able to reduce the problem of whether closed subgroups of connected groups satisfy the Baum–Connes conjecture, with coefficients to the special case of center-free semi-simple Lie groups.     

Group Pairs with Property (T), from Arithmetic Lattices

Let Γ be an arithmetic lattice in an absolutely simple Lie group G with trivial centre. We prove that there exists an integer N ≥ 2, a subgroup Λ of finite index in Γ, and an action of Λ on such that the pair ( ) has property (T). If G has property (T), then so does . If G is the adjoint group of Sp(n, 1), then is a property (T) group satisfying the Baum–Connes conjecture. If Γ is an arithmetic lattice in SO(n, 1), then the associated von Neumann algebra is a II₁-factor in Popa’s class . Elaborating on this result of Popa, we construct a countable family of pairwise nonstably isomorphic group II₁-factors in the class , all with trivial fundamental groups and with all L²-Betti numbers being zero.Mathematics Subject Classiffications (2000). 22E40, 22E47, 46L80, 37A20     

A Complete Formulation of the Baum–Connes Conjecture for the Action of Discrete Quantum Groups

We formulate a version of the Baum–Connes conjecture for a discrete quantum group, building on our earlier work. Given such a quantum group , we construct a directed family -algebras (F varying over some suitable index set), borrowing the ideas of Cuntz such that there is a natural action of satisfying the assumptions of Goswami and Kuku which makes it possible to define the analytical assembly map, say , i= 0, 1, as in our previous work, from the -equivariant K-homolgy groups of to the K-theory groups of the reduced dual (c.f. [9] and the references therein for more details). As a result, we can define the Baum–Connes maps , and in the classical case, i.e. when for a discrete group, the isomorphism of the above maps for i= 0, 1 is equivalent to the Baum–Connes conjecture. Furthermore, we verify its truth for an arbitrary finite-dimensional quantum group and obtain partial results for the dual of (2).     

Baum–Connes Conjecture and Group Actions on Trees

We check, that the Baum–Connes conjecture with coefficients, for groups acting on oriented trees, is true if and only if it is true for the stabilizer groups of the vertices.     

Analogues of the Kolosov–Muskhelishvili General Representation Formulas and Cauchy–Riemann Conditions in the Theory of Elastic Mixtures

Analogues of the well-known Kolosov–Muskhelishvili formulas of general representations are obtained for nonhomogeneous equations of statics in the case of the theory of elastic mixtures. It is shown that in this theory the displacement and stress vector components, as well as the stress tensor components, are represented through four arbitrary analytic functions.The usual Cauchy–Riemann conditions are generalized for homogeneous equations of statics in the theory of elastic mixtures.     

The Baum–Connes Conjecture and Discrete Group Actions on Trees

We introduce a property (BC) for discrete groups, which we prove to imply the Baum–Connes conjecture with coefficients and the K-amenability of the group. Then, we show that if is a discrete group which acts on a tree X such that X/ is compact, and the stabilizers of the vertices and the stabilizers of the edges satisfy (BC), then itself satisfies (BC) Finally, we indicate a couple of applications.     

Local-Global Principle for the Baum–Connes Conjecture with Coefficients

We establish the Hasse principle (local-global principle) in the context of the Baum–Connes conjecture with coefficients. We illustrate this principle with the discrete group GL(2,F) where F is any global field.     

A Descent Theorem in Topological K-Theory

We prove the Lichtenbaum–Quillen conjecture in the topological context: in other words, real K-theory can be deduced from complex K-theory via the usual descent spectral sequence. More precise results are proved, however, and new applications are stated. The main ingredients in the proof are Atiyah's KR-theory and the definition of higher K-groups via Clifford algebras.     

Quantum Mechanics on Manifolds

In this paper, we investigate the relationships between quantum mechanics and the theory of partial differential equations. We closely follow the De Broglie and Schrödinger picture. Namely, we consider the well-known wave-particle duality as a relation between solutions of partial differential equations, describing waves, and singularities of solutions, that is particles. Our analysis of these relations shows that the necessary ingredients of any quantum mechanical picture are two connections. The first one is a connection in the tangent bundle of the configuration manifold and the second one is a connection in the trivial linear bundle.We also consider mechanical systems equipped with an inner structure and show that quantization of these systems requires a linear connection in the corresponding vector bundle.These are gravity and electromagnetic fields, or Yang–Mills fields if the configuration space is the Minkowski space. In the case of general mechanical systems, they should be considered as natural generalizations of these fields.Explicit formulas for quantizations of some mechanical systems and the corresponding star-products are given.     

Phantom Maps and Purity in Modular Representation Theory, II

In the second part of this paper, we use the theory described in the first part to construct an example of a counterintuitive phenomenon. We show how to produce examples of filtered systems in the stable module category stmod(kG) which do not lift to filtered systems in the module category mod(kG). Our main tool is the Extended Milnor Sequence, a five-term exact sequence which specializes to the classical Milnor sequence under certain countability conditions.     

Generic Representation Theory of the Heisenberg Group

In this article we extend a result for representations of the additive group G_a given in [4 Suslin , A. , Friedlander , E. M. , Bendel , C. P. ( 1997 ). Infinitesimal 1-parameter subgroups and cohomology . Journal of the AMS 10 ( 3 ): 693 – 728 .[Web of Science ®] , [Google Scholar]] to the Heisenberg group H₁. Namely, if p is greater than 2d, then all d-dimensional characteristic p representations for H₁ can be factored into commuting products of representations, with each factor arising from a representation of the Lie algebra of H₁, and conversely any commuting collection of Lie algebra representations gives rise to a representation of H₁ in this fashion. In this sense, for a fixed dimension and large enough p, all representations for H₁ look generically like representations for direct powers of it over a field of characteristic zero.

The following originally appeared as Chapter 13 of the author's dissertation [1 Crumley , M. ( 2010 ). Ultraproducts of Tannakian Categories and Generic Representation Theory of Unipotent Algebraic Groups. Ph.D. dissertation, The University of Toledo, Department of Mathematics . [Google Scholar]].     

Representation Theory of some Infinite-dimensional Algebras Arising in Continuously Controlled Algebra and Topology

In this paper we determine the representation type of some algebras of infinite matrices continuously controlled at infinity by a compact metrizable space. We explicitly classify their finitely presented modules in the finite and tame cases. The algebra of row-column-finite (or locally finite) matrices over an arbitrary field is one of the algebras considered in this paper, its representation type is shown to be finite.Received October 2003     

代数表示论简介与综述

代数表示论是本世纪七十年代初兴起的代数学的一个新的分支。它的基本内容是研究一个Ａｒｔｉｎ代数上的模范畴。由于各国代数学家的共同努力，这一理论于最近二十年代有了异常迅猛的发展并逐步趋于完善。本文介绍了代数表示论的理论基础；几乎可裂序列；箭图和赋值箭图的表示；Ｃｏｘｅｔｅｒ函子；ＡＲ－箭图的覆盖及代数的Ｇａｌｏｉｓ覆盖。并简单介绍了在代数表示论中普遍应用的工具：Ｔｉｌｔ理论，以及著名的Ｄｐｏｚｇ定理证     

New Perturbation Theory for Quantum Field Theory: Convergent Series Instead of Asymptotic Expansions

We present a new approach to perturbation theory for quantum field theory based on convergent series instead of asymptotic expansions. This approach could be considered as the next step after traditional perturbation theory calculations, which allows more comprehensive use of previously obtained information in finding numerical values with greater accuracy.     

La conjecture de Novikov pour les feuilletages hyperboliques

Nous définissons la notion de bolicité pour les feuilletages, qui est une notion plus faible que l'hyperbolicité de Gromov, et nous démontrons la conjecture de Novikov pour les feuilletages boliques à base compacte dont le groupoï de d'holonomie est séparé en établissant l'injectivité de l'application de Baum–Connes. Ce résultat généralise celui de Kasparov et Skandalis obtenu dans le cas des groupes boliques.We define the notion of bolicity for foliations, which is a weaker notion than Gromov's hyperbolicity, and we prove the Novikov conjecture for foliations with compact base and whose holonomy groupoid is Hausdorff, by showing that the Baum–Connes map is injective. This result generalizes that of Kasparov and Skandalis in the case of bolic groups.     

Generic Representation Theory of the Unipotent Upper Triangular Groups

It is generally believed (and for the most part it is probably true) that Lie theory, in contrast to the characteristic zero case, is insufficient to tackle the representation theory of algebraic groups over prime characteristic fields. However, in this article we show that, for a large and important class of unipotent algebraic groups (namely the unipotent upper triangular groups U_n), and under a certain hypothesis relating the characteristic p to both n and the dimension d of a representation (specifically, p ≥ max(n, 2d)), Lie theory is completely sufficient to determine the representation theories of these groups. To finish, we mention some important analogies (both functorial and cohomological) between the characteristic zero theories of these groups and their “generic” representation theory in characteristic p.