Lau, Siu-CheongNan, Junzheng2022-11-042022-11-042022https://hdl.handle.net/2144/45304Noncommutative(nc) deformations have been very important to the study of quantum physics and geometric representation theory. In the thesis, we develop a local-to-global approach to study nc geometry. Given a nc crepant resolution A in the sense of Van den Bergh and its nc deformations A^h, we develop a mirror-symmetric method to construct an algebroid stack X^h, and a universal twisted complex which gives a functor between the dg categories of A^h and X^h. We apply this method to the conifold singularity and show that the functor gives a derived equivalence. We also apply the method to K_{P2} and find interesting phenomena occur when h≠0.en-USMathematicsThesis/Dissertation2022-11-030000-0002-6740-6481