Topics in Topology and Geometry

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9803254 Burghelea D. Burghelea plans work in topology, geometric analysis and dynamics, based on methods of linear algebra ``a la von Neumann'' and on regularized determinants of elliptic operators. He plans: 1) to attack a number of open problems on L2-invariants, torsion invariants, relation between torsion and dynamics, 2) to develop new mathematical tools, such as the Hodge theory of geometric complexes associated to Morse-Bott functions, Witten-Hellfer-Sjostrand theory with symmetry and with parameters, and 3) to test these new tools against a number of open problems in geometric analysis and spectral geometry. The research will shed more light on the nature and the power of the L2 invariants and will considerably increase the generality of some successful techniques in geometric analysis and hopefully solve some open problems in topology and spectral geometry. The above work explores the relationship between the shape of a geometric object (a Riemannian manifold), as embodied in its topology and geometry, and sound, as embodied in the spectra of various Laplace operators associated to it -- think in terms of natural frequencies of vibration. It also investigates the constraints imposed by the shape and sound on basic qualitative elements of dynamics such as closed trajectories, attractors, repulsors, and saddle points. The methods used involve unusual quantities introduced by von Neumann, like dimensions that are not integers and volumes of infinitely large objects. The research will investigate the type of additional information about the shape and sound, and about the dynamics on geometric objects, that can be obtained by using these unusual quantities. ***