2019-01-012024-05-18https://scholars.lib.ntu.edu.tw/handle/123456789/700316Abstract: We have two themes: (A) To develop the foundations of irregular Hodge theory from the bottom up and explore applications to arithmetic. (B) To investigate quantum invariants and Calabi-Yau varieties under birational transformations and through degenerations. The two directions are intertwining and each provides indispensable aspects to further deepen the other and the scopes of applications. Specific joint researches include (1) expansions of the theory of Frobenius manifolds from aspects of algebraic analysis and A+B models in string theory and (2) studies of generalized motives equipped with irregular or quantum structures.