This article is an interdisciplinary review and an on-going progress report
over the last few years made by myself and collaborators in certain fundamental
subjects on two major theoretic branches in mathematics and theoretical
physics: algebraic geometry and quantum physics. I shall take a practical
approach, concentrating more on explicit examples rather than formal
developments. Topics covered are divided in three sections: (I) Algebraic
geometry on two-dimensional exactly solvable statistical lattice models and its
related Hamiltonians: I will report results on the algebraic geometry of
rapidity curves appeared in the chiral Potts model, and the algebraic Bethe
Ansatz equation in connection with quantum inverse scattering method for the
related one-dimensional Hamiltonion chain, e.g., XXZ, Hofstadter type
Hamiltonian. (II) Infinite symmetry algebras arising from quantum spin chain
and conformal field theory: I will explain certain progress made on Onsager
algebra, the relation with the superintegrable chiral Potts quantum chain and
problems on its spectrum. In conformal field theory, mathematical aspects of
characters of N=2 superconformal algebra are discussed, especially on the
modular invariant property connected to the theory. (III). Algebraic geometry
problems on orbifolds stemming from string theory: I will report recent
progress on crepant resolutions of quotient singularity of dimension greater
than or equal to three. The direction of present-day research of engaging
finite group representations in the geometry of orbifolds is briefly reviewed,
and the mathematical aspect of various formulas on the topology of string
vacuum will be discussed.