Mathematical Sciences: Cyclic Homology, Algebra, Topology and Geometry

Burghelea将研究代数、拓扑学和几何学中循环同调的一些方面和含义。在代数方面，他计划研究代数簇和与群在代数簇上的作用有关的交叉积代数的循环同调；他还将研究群环的陈特征标。在拓扑学中，他将研究B diff(M)在稳定范围内的同调类型，在素数2上。在几何学中，他将研究Selberg迹公式、循环同调和闭测地线之间的关系。探索循环同源的用途将使一项强大的技术变得更加通用。归根结底，理论物理和其他高等数学消费者的应用应该会受益。**8 8701643 Bergman将研究泛代数的三个重要性质：合并性、演绎和同余模簇。对融合性的研究主要围绕两个问题展开。1.确定合并性质加上剩余小是否蕴含同余扩张性质。2.品种融合碱基的特征。关于演绎的计划没有那么具体。一般的问题是试图对那些演绎的品种进行分类。第三条途径是同余模簇中交换子的表示问题。所有这些话题都与代数的基础有关，只有在它们制约了关于代数问题的思维模式的情况下，才有可能应用于代数之外。

Burghelea will study some aspects and implications of cyclic homology in algebra, topology and geometry. In algebra, he plans to study the cyclic homology of algebraic varieties and crossed product algebras associated to actions of groups on algebraic varieties; he will also study the Chern character for group rings. In topology he will study the homology type of B Diff(M) in the stability range, and at the prime 2. In geometry he will study the relationship between the Selberg trace formula, cyclic homology and closed geodesics. Exploring the uses of cyclic homology will render a powerful technique more versatile. Ultimately applications to theoretical physics and other consumers of higher mathematics should profit. **8 8701643 Bergman Three important properties of universal algebras will be studied: the amalgamation property, deductiveness, and congruence modular varieties. The research on the amalgamation property centers on two problems. 1. Determination if the amalgamation property plus residual smallness implies the congruence extension property. 2. Characterization of the amalgamation bases of the variety. The plans for deductiveness are less specific. The general issue is to attempt to classify those varieties that are deductive. A third avenue to be pursued is the representation problem for the commutator in congruence modular varieties. All of these topics concern the foundations of algebra and are likely to be applied outside of algebra only to the extent that they condition a mode of thought about algebraic questions.