Introduction to Linear and Nonlinear Integrable Theories in Discrete Complex Analysis

Chapter
Part of the CRM Series in Mathematical Physics book series (CRM)

Abstract

The field of discrete differential geometry lies on the border of classical differential geometry and discrete geometry. Its aim is to develop discrete geometric theories which respect fundamental aspects of the corresponding smooth ones. Also, these discretizations often clarify structures of the smooth theory.

In our presentation, we focus on the area of discrete complex analysis. In particular, we introduce several concepts of discrete holomorphic functions based on a linear approach and on nonlinear theories concerning cross-ratio systems, circle patterns and discrete conformal equivalence. These examples are used to illustrate some characteristic features in discrete differential geometry like integrability as consistency and Bäcklund–Darboux transformations.

Notes

Acknowledgements

The author is supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.”

Furthermore, I am grateful to Nikolay Dimitrov for discussions on integrability in the context of discrete differential geometry, in particular concerning cross-ratio systems and conformally equivalent triangulations. Many thanks also to the referee for his careful reading and advice.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

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