Exploring Convergence and Symmetry of Discrete Period Matrices

In recent decades, discrete complex analysis has seen remarkable advancements, despite its relatively late inception compared to its continuous counterpart. While continuous complex analysis boasts a long-established history, the discrete analog has emerged more recently and remains an active area of research and exploration.

This presentation focuses on a pivotal aspect of Riemann surfaces: their period matrices. After providing a concise introduction to the linear theory of discrete Riemann surfaces, we introduce the discrete period matrix—a direct counterpart to its continuous counterpart. We also delve into the larger complete discrete period matrix, whose calculation is based on a broader class of discrete holomorphic differentials, thereby giving more information on the underlying discrete Riemann surface.

Our discussion extends to the convergence of discrete period matrices when a sequence of increasingly fine discretizations is applied to a given compact polyhedral surface. We outline the proof strategy for the convergence of these discrete period matrices to the period matrix of the original surface, and elucidate the limits of the blocks within the complete discrete period matrix.

In closing, we explore a discretization of real Riemann surfaces and their period matrices, a collaborative effort with Johanna Düntsch. As one expects, we find that the corresponding discrete period matrices exhibit the same symmetrical properties as their continuous counterparts, with the complete discrete period matrices following a similar pattern.