Abstract: This talk mainly focus on geometric analysis of symbol curves of analytic Toeplitz operators. Under a mild condition, we establish the connection between totally Abelian property and the Riemann surface defined by the symbol of analytic Toeplitz operator. It is found that winding numbers, indices and points of self-intersection of symbol curves of Toeplitz operator altogether play an important role in this topic. Techniques of algebraic topology, complex analysis, geometry, index theory and operator theory are intrinsic in our studies. As a byproduct, under a mild condition an affirmative answer is presented to a question raised by Baker, Deddens and Ullman in Duke Math. J. (1974). We also construct some examples to show that the answer is negative if the associated conditions are weakened. This is a joint work with Hui Dan and Hansong Huang.
