Mathematical Sciences: Free Loop Space, Automorphisms of Manifolds and Cyclic Homology

Burghelea教授将研究被称为紧致光滑流形的简单几何对象的自然变换的拓扑特征。它们出现在现代物理理论以及许多纯粹的数学背景中。其他相关事项在他的项目中也将扮演较小的角色。具体地说，Burghelea将研究(1)同伦型的微分同态(Rep.紧光滑的辛同构)。(2)紧致黎曼流形的自由圈空间的微分几何和拓扑。(3)椭圆算子的行列式。(4)完备度量流形的有界调和。(5)圈同调中的几何和代数问题。

Professor Burghelea will study the topological character of natural transformations of simple geometric objects known as compact smooth manifolds. These arise in modern physical theories as well as in many purely mathematical contexts. Other related matters will also play a lesser role in his project. Specifically, Burghelea will study (1) the homotopy type of diffeomorphisms (resp. symplectomorphisms) of a compact smooth (resp. symplectic) manifold M and the relation to the homology of some associated spaces defined in terms of principal bundles over M. (2) the differential geometry and topology of the free loop space of compact Riemannian manifolds. (3) the determinants of elliptic operators. (4) the bounded concordances of complete metric manifolds. (5) geometric and algebraic problems in cyclic homology.