“A nice and unique feature of the book that every chapter contains a section on ‘Complements and open problems’. All in all, I find this text very well written and it would definitely be recommended to graduate students and researchers who would like to learn about the fine properties of Sobolev functions valued in the circle.” (Alpár R. Mészáros, zbMATH 1501.46001, 2023)“This monograph offers a rigorous discussion to a fascinating topic of mathematics with multiple relevant applications to various fields. Also for this reason it is highly recommended both for mathematicians and physicists working in the various fields involved in the theory of Sobolev maps to the circle … . the book may also be useful for Ph.D students who are interested in the topic, and the content of the individual chapters could be used in advanced courses and seminars.” (Antonia Chinnì, Mathematical Reviews, February, 2023)

• Lifting in $W^{1,p}$.- The Geometry of $J(u)$ and $\Sigma(u)$ in 2D; Point Singularities and Minimal Connections.- The Geometry of $J(u)$ and $\Sigma(u)$ in 3D (and higher); Line Singularities and Minimal Surfaces.- A Digression: Sphere-Valued Maps.- Lifting in Fractional Sobolev Spaces and in $VMO$.- Uniqueness of Lifting and Beyond.- Factorization.- Applications of the Factorization.- Estimates of Phases: Positive and Negative Results.- Density.- Traces.- Degree.- Dirichlet Problems, Gaps, Infinite Energies.- Domains with Topology.- Appendices.

“A nice and unique feature of the book that every chapter contains a section on ‘Complements and open problems’. All in all, I find this text very well written and it would definitely be recommended to graduate students and researchers who would like to learn about the fine properties of Sobolev functions valued in the circle.” (Alpár R. Mészáros, zbMATH 1501.46001, 2023)“This monograph offers a rigorous discussion to a fascinating topic of mathematics with multiple relevant applications to various fields. Also for this reason it is highly recommended both for mathematicians and physicists working in the various fields involved in the theory of Sobolev maps to the circle … . the book may also be useful for Ph.D students who are interested in the topic, and the content of the individual chapters could be used in advanced courses and seminars.” (Antonia Chinnì, Mathematical Reviews, February, 2023)

• Lifting in $W^{1,p}$.- The Geometry of $J(u)$ and $\Sigma(u)$ in 2D; Point Singularities and Minimal Connections.- The Geometry of $J(u)$ and $\Sigma(u)$ in 3D (and higher); Line Singularities and Minimal Surfaces.- A Digression: Sphere-Valued Maps.- Lifting in Fractional Sobolev Spaces and in $VMO$.- Uniqueness of Lifting and Beyond.- Factorization.- Applications of the Factorization.- Estimates of Phases: Positive and Negative Results.- Density.- Traces.- Degree.- Dirichlet Problems, Gaps, Infinite Energies.- Domains with Topology.- Appendices.

“A nice and unique feature of the book that every chapter contains a section on ‘Complements and open problems’. All in all, I find this text very well written and it would definitely be recommended to graduate students and researchers who would like to learn about the fine properties of Sobolev functions valued in the circle.” (Alpár R. Mészáros, zbMATH 1501.46001, 2023)“This monograph offers a rigorous discussion to a fascinating topic of mathematics with multiple relevant applications to various fields. Also for this reason it is highly recommended both for mathematicians and physicists working in the various fields involved in the theory of Sobolev maps to the circle … . the book may also be useful for Ph.D students who are interested in the topic, and the content of the individual chapters could be used in advanced courses and seminars.” (Antonia Chinnì, Mathematical Reviews, February, 2023)

• Lifting in $W^{1,p}$.- The Geometry of $J(u)$ and $\Sigma(u)$ in 2D; Point Singularities and Minimal Connections.- The Geometry of $J(u)$ and $\Sigma(u)$ in 3D (and higher); Line Singularities and Minimal Surfaces.- A Digression: Sphere-Valued Maps.- Lifting in Fractional Sobolev Spaces and in $VMO$.- Uniqueness of Lifting and Beyond.- Factorization.- Applications of the Factorization.- Estimates of Phases: Positive and Negative Results.- Density.- Traces.- Degree.- Dirichlet Problems, Gaps, Infinite Energies.- Domains with Topology.- Appendices.