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 style construction for the Chern character of matrix factorizations; this allows us to reproduce the Chern character in an explicit, understandable way. Some basic properties of the Chern character are also proved (via this construction) such as functoriality and that it determines a ring homomorphism from the Grothendieck group of matrix factorizations to its Hochschild homology. The second part is a reconstruction theorem of hypersurface singularities. This is given by applying a slightly modified version of Balmer’s tensor triangular geometry to the homotopy category of matrix factorizations. iii DEDICATION To my parents, for their love and support. iv ACKNOWLEDGMENTS First, I would like to thank my advisor, Prof. Mark Walker, for his guidance an