Geometric Study of the Category of Matrix Factorizations

We study the geometry of matrix factorizations in this dissertation.It contains two parts. The first one is a Chern-Weil styleconstruction for the Chern character of matrix factorizations; thisallows us to reproduce the Chern character in an explicit,understandable way. Some basic properties of the Chern character arealso proved (via this construction) such as functoriality and thatit determines a ring homomorphism from the Grothendieck group ofmatrix factorizations to its Hochschild homology. The second part isa reconstruction theorem of hypersurface singularities. This isgiven by applying a slightly modified version of Balmer\u27s tensortriangular geometry to the homotopy category of matrixfactorizations. Adviser: Mark E. Walke