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This book provides a highly accessible approach to discrete surface theory, within the unifying frameworks of Moebius and Lie sphere geometries, from the perspective of transformation theory of surfaces rooted in integrable systems. It elucidates how the transformation theory for smooth surfaces can be used as a springboard for understanding the discretization process of certain types of surfaces, and it is aimed at high-level undergraduate students, graduate students and professional mathematicians alike. The reader will benefit from the detailed exploration of the transformation theory of surfaces, including Christoffel, Calapso and Darboux transformations of particular classes of surfaces, as well as becoming more familiar with integrable systems via zero curvature representation, including flat connections and conserved quantities, in both smooth and discrete settings.
List of contents
Chapter 1. Introduction.- Chapter 2.Isothermic surfaces in Möbius geometry.- Chapter 3. From smooth to discrete via permutability.- Chapter 4. Discrete Isothermic surfaces.- Chapter 5. -surfaces in Lie sphere geometry.- Chapter 6. Integrability of -surfaces via isothermicity.- Chapter 7. Discrete -surfaces.
About the author
Joseph Cho is a differential geometer with a primary interest in the integrable systems approach to smooth and discrete surface theory. His studies in mathematics led to a bachelor's degree from the University of California, Berkeley, a master's degree from Korea University, and a doctorate from Kobe University. He has also worked as a postdoctoral researcher at TU Wien and is currently enjoying life as an assistant professor at Handong Global University, Republic of Korea.
Kosuke Naokawa is an associate professor at Hiroshima Institute of Technology in Japan. He received his PhD from Tokyo Institute of Technology in 2013, and subsequently conducted postdoctoral research in Kobe and Vienna. His research focuses on the geometry and topology of surfaces with singularities and their discretizations.
Yuta Ogata is a researcher in the field of surface theory, integrable systems, and singularity theory. He obtained his PhD from Kobe University in 2017. After working in Okinawa, Japan, he is currently an associate professor at Kyoto Sangyo University, Japan.
Mason Pember is a lecturer of mathematics at the University of Bath, United Kingdom. He obtained his PhD from the University of Bath in 2015 and subsequently held postdocs in Kobe, Vienna and Turin. His research is in differential geometry, specialising in topics such as integrable systems, surface theory and Lie sphere geometry.
Wayne Rossman is a professor of mathematics at Kobe University, Japan. He earned his PhD from the University of Massachusetts, Amherst in 1992, and has been working ever since in the Japanese differential geometry community, interspersed with post-doctoral positions and sabbaticals in California, Brazil, Bath England, Berlin and Vienna. His expertise lies in the differential geometry of surfaces and its discretization.
Masashi Yasumoto is an associate professor at Tokushima University, Japan. He obtained his PhD from Kobe University in 2015, and subsequently conducted his research at Tübingen, Osaka, and Fukuoka as a postdoctoral researcher. His research focuses on differential and discrete differential geometry of surfaces.
Summary
This book provides a highly accessible approach to discrete surface theory, within the unifying frameworks of Moebius and Lie sphere geometries, from the perspective of transformation theory of surfaces rooted in integrable systems. It elucidates how the transformation theory for smooth surfaces can be used as a springboard for understanding the discretization process of certain types of surfaces, and it is aimed at high-level undergraduate students, graduate students and professional mathematicians alike. The reader will benefit from the detailed exploration of the transformation theory of surfaces, including Christoffel, Calapso and Darboux transformations of particular classes of surfaces, as well as becoming more familiar with integrable systems via zero curvature representation, including flat connections and conserved quantities, in both smooth and discrete settings.
