Mathematical Sciences: Topics in Topology

本课题主要研究自由环空间的拓扑，辛形态空间的同伦类型和von Neumann拓扑。更确切地说，研究：(1)自由环空间的代数拓扑与代数几何之间的类比；(2)弦上同调的应用；(3)检测辛同构空间非平凡同伦群的新方法；(4)具有无限基群的紧流形的Novikov-Shubin不变量、l2 -扭转和L2-eta不变量等数值不变量。这项研究将为代数几何、微分几何、椭圆算子和冯·诺伊曼代数等领域提供新的发展。它将拓宽现有的知识，并（希望）将发现诸如代数几何和自由循环空间，或代数几何和辛拓扑等不同领域之间的新关系。它也将回答具体的问题（解决问题）在拓扑，几何和数学物理的直接关注。＊＊＊

9501701 Burghelea This project pursues research on the topology of the free loop spaces, on the homotopy type of the space of symplectomorphisms and on von Neumann topology. More precisely, one investigates: (1) the analogy between algebraic topology of the free loop spaces and algebraic geometry; (2) applications of string cohomology; (3) new methods to detect nontrivial homotopy groups of the space of symplectomorphisms; and (4) numerical invariants such as the Novikov-Shubin invariant, L2-torsion, and the L2-eta invariant for compact manifolds with infinite fundamental group. This research will provide new developments in areas like algebraic geometry, differential geometry, elliptic operators, and von Neumann algebras. It will broaden the present knowledge in, and (hopefully) will discover new relationships between, such different areas as algebraic geometry and free loop spaces, or algebraic geometry and symplectotopology. It will also answer specific questions (solve ppoblems) of immediate concern in topology, geometry and mathematical physics. ***