Clarence Kineider studied mathematics at the École Normale Supérieure de Rennes and the University of Strasbourg between 2016 and 2019. He completed his Ph.D. at the University of Strasbourg, where his research focused on higher Teichmüller spaces, Fock–Goncharov coordinates, and spectral networks. Since 2024, he has been a postdoctoral fellow at the Max Planck Institute for Mathematics in the Sciences in Leipzig.Georgios Kydonakis studied Mathematics (B.Sc.) at the University of Athens in Greece and obtained his Ph.D. degree from the University of Illinois at Urbana-Champaign in the USA. He has held postdoctoral positions in Strasbourg, Bonn, and Heidelberg as an Alexander von Humboldt fellow. Since 2023, he has been an Assistant Professor at the University of Patras in Greece. His research interests lie on the crossroad between complex geometry and mathematical physics.Eugen Rogozinnikov studied mathematics at the University of Heidelberg, where he completed his master's degree in 2016 and received his Ph.D. in mathematics in 2020. After that, he held a postdoctoral position at the Université de Strasbourg. In 2023–2024, he was a DAAD fellow at the University of Notre Dame (USA). He later worked at the Max Planck Institute for Mathematics in the Sciences in Leipzig and is currently a research fellow at the Korea Institute for Advanced Study (KIAS) in Seoul. His research interests include differential geometry, geometric topology, and higher Teichmüller theory.Valdo Tatitscheff studied mathematics and physics at the École normale supérieure de la rue d'Ulm in Paris from 2014 to 2018, and earned a Ph.D. in mathematics from the University of Strasbourg and the Institut de Recherche Mathématique Avancée (IRMA) in 2022. Currently a Postdoctoral Fellow at the Institute for Mathematics at Heidelberg University, Tatitscheff's main research interests lie at the interface of geometry and gauge theories.Alexander Thomas studied mathematics at the École Normale Supérieure de Lyon from 2013 to 2017 and completed his Ph.D. at the University of Strasbourg in 2020. He held postdoctoral positions at the Max Planck Institute for Mathematics in Bonn (2020–2022) and at the University of Heidelberg (2022–2024). Since 2024, he has been a Maître de Conférences (Associate Professor) at Université Claude Bernard Lyon 1. His research focuses primarily on gauge theory and quantum algebra.

• Preface.-Motivation and Historical Perspective.-I. Mathematical Foundation.- 1.The Fundamental Group.-2.Covering Spaces.-3.Bundles and Connections.-4.Hyperbolic Geometry.- 5.Cluster Varieties.- II Higher-rank Teichmüller Theory.-6. The Teichmüller Space.-7.Coordinates on Teichmüller Space.-8.differentials on a Riemann Surface.-9.Higher-rank Teichmüller Spaces.- 10.Hitchin Representations.-11.Maximal Representations.-12.Positivity.- III Spectral Networks in Geometry.-13.Non-degenerate Spectral Networks.-14.Combinatorial and Analytic Construction.-15.Non-abelianization and Abelianization.-16.WKB Method and Stokes Graphs.- IV Counting BPS States.-17.Four-dimensional Supersymmetric Quantum Field Theories.-18.Four-dimensional N=2 Gauge Theories.-19.Electric-magnetic Duality.-20.Seiberg-Witten Theory.-21.Theories of Class S.-22.BPS States in Class S Theories.- V Generalizations.-23.Exponential Networks.-24.Three-dimensional Networks.-25.WKB Cameral Networks.-26.Nonabelianization for Conformal Virasoro Blocks.

Clarence Kineider studied mathematics at the École Normale Supérieure de Rennes and the University of Strasbourg between 2016 and 2019. He completed his Ph.D. at the University of Strasbourg, where his research focused on higher Teichmüller spaces, Fock–Goncharov coordinates, and spectral networks. Since 2024, he has been a postdoctoral fellow at the Max Planck Institute for Mathematics in the Sciences in Leipzig.Georgios Kydonakis studied Mathematics (B.Sc.) at the University of Athens in Greece and obtained his Ph.D. degree from the University of Illinois at Urbana-Champaign in the USA. He has held postdoctoral positions in Strasbourg, Bonn, and Heidelberg as an Alexander von Humboldt fellow. Since 2023, he has been an Assistant Professor at the University of Patras in Greece. His research interests lie on the crossroad between complex geometry and mathematical physics.Eugen Rogozinnikov studied mathematics at the University of Heidelberg, where he completed his master's degree in 2016 and received his Ph.D. in mathematics in 2020. After that, he held a postdoctoral position at the Université de Strasbourg. In 2023–2024, he was a DAAD fellow at the University of Notre Dame (USA). He later worked at the Max Planck Institute for Mathematics in the Sciences in Leipzig and is currently a research fellow at the Korea Institute for Advanced Study (KIAS) in Seoul. His research interests include differential geometry, geometric topology, and higher Teichmüller theory.Valdo Tatitscheff studied mathematics and physics at the École normale supérieure de la rue d'Ulm in Paris from 2014 to 2018, and earned a Ph.D. in mathematics from the University of Strasbourg and the Institut de Recherche Mathématique Avancée (IRMA) in 2022. Currently a Postdoctoral Fellow at the Institute for Mathematics at Heidelberg University, Tatitscheff's main research interests lie at the interface of geometry and gauge theories.Alexander Thomas studied mathematics at the École Normale Supérieure de Lyon from 2013 to 2017 and completed his Ph.D. at the University of Strasbourg in 2020. He held postdoctoral positions at the Max Planck Institute for Mathematics in Bonn (2020–2022) and at the University of Heidelberg (2022–2024). Since 2024, he has been a Maître de Conférences (Associate Professor) at Université Claude Bernard Lyon 1. His research focuses primarily on gauge theory and quantum algebra.

• Preface.-Motivation and Historical Perspective.-I. Mathematical Foundation.- 1.The Fundamental Group.-2.Covering Spaces.-3.Bundles and Connections.-4.Hyperbolic Geometry.- 5.Cluster Varieties.- II Higher-rank Teichmüller Theory.-6. The Teichmüller Space.-7.Coordinates on Teichmüller Space.-8.differentials on a Riemann Surface.-9.Higher-rank Teichmüller Spaces.- 10.Hitchin Representations.-11.Maximal Representations.-12.Positivity.- III Spectral Networks in Geometry.-13.Non-degenerate Spectral Networks.-14.Combinatorial and Analytic Construction.-15.Non-abelianization and Abelianization.-16.WKB Method and Stokes Graphs.- IV Counting BPS States.-17.Four-dimensional Supersymmetric Quantum Field Theories.-18.Four-dimensional N=2 Gauge Theories.-19.Electric-magnetic Duality.-20.Seiberg-Witten Theory.-21.Theories of Class S.-22.BPS States in Class S Theories.- V Generalizations.-23.Exponential Networks.-24.Three-dimensional Networks.-25.WKB Cameral Networks.-26.Nonabelianization for Conformal Virasoro Blocks.

Clarence Kineider studied mathematics at the École Normale Supérieure de Rennes and the University of Strasbourg between 2016 and 2019. He completed his Ph.D. at the University of Strasbourg, where his research focused on higher Teichmüller spaces, Fock–Goncharov coordinates, and spectral networks. Since 2024, he has been a postdoctoral fellow at the Max Planck Institute for Mathematics in the Sciences in Leipzig.Georgios Kydonakis studied Mathematics (B.Sc.) at the University of Athens in Greece and obtained his Ph.D. degree from the University of Illinois at Urbana-Champaign in the USA. He has held postdoctoral positions in Strasbourg, Bonn, and Heidelberg as an Alexander von Humboldt fellow. Since 2023, he has been an Assistant Professor at the University of Patras in Greece. His research interests lie on the crossroad between complex geometry and mathematical physics.Eugen Rogozinnikov studied mathematics at the University of Heidelberg, where he completed his master's degree in 2016 and received his Ph.D. in mathematics in 2020. After that, he held a postdoctoral position at the Université de Strasbourg. In 2023–2024, he was a DAAD fellow at the University of Notre Dame (USA). He later worked at the Max Planck Institute for Mathematics in the Sciences in Leipzig and is currently a research fellow at the Korea Institute for Advanced Study (KIAS) in Seoul. His research interests include differential geometry, geometric topology, and higher Teichmüller theory.Valdo Tatitscheff studied mathematics and physics at the École normale supérieure de la rue d'Ulm in Paris from 2014 to 2018, and earned a Ph.D. in mathematics from the University of Strasbourg and the Institut de Recherche Mathématique Avancée (IRMA) in 2022. Currently a Postdoctoral Fellow at the Institute for Mathematics at Heidelberg University, Tatitscheff's main research interests lie at the interface of geometry and gauge theories.Alexander Thomas studied mathematics at the École Normale Supérieure de Lyon from 2013 to 2017 and completed his Ph.D. at the University of Strasbourg in 2020. He held postdoctoral positions at the Max Planck Institute for Mathematics in Bonn (2020–2022) and at the University of Heidelberg (2022–2024). Since 2024, he has been a Maître de Conférences (Associate Professor) at Université Claude Bernard Lyon 1. His research focuses primarily on gauge theory and quantum algebra.

• Preface.-Motivation and Historical Perspective.-I. Mathematical Foundation.- 1.The Fundamental Group.-2.Covering Spaces.-3.Bundles and Connections.-4.Hyperbolic Geometry.- 5.Cluster Varieties.- II Higher-rank Teichmüller Theory.-6. The Teichmüller Space.-7.Coordinates on Teichmüller Space.-8.differentials on a Riemann Surface.-9.Higher-rank Teichmüller Spaces.- 10.Hitchin Representations.-11.Maximal Representations.-12.Positivity.- III Spectral Networks in Geometry.-13.Non-degenerate Spectral Networks.-14.Combinatorial and Analytic Construction.-15.Non-abelianization and Abelianization.-16.WKB Method and Stokes Graphs.- IV Counting BPS States.-17.Four-dimensional Supersymmetric Quantum Field Theories.-18.Four-dimensional N=2 Gauge Theories.-19.Electric-magnetic Duality.-20.Seiberg-Witten Theory.-21.Theories of Class S.-22.BPS States in Class S Theories.- V Generalizations.-23.Exponential Networks.-24.Three-dimensional Networks.-25.WKB Cameral Networks.-26.Nonabelianization for Conformal Virasoro Blocks.