We introduce the class of n‐extremal holomorphic maps, a class that generalizes both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc into the symmetrized bidisc Γ. We show that a well‐known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater. We introduce a sequence 𝒞ν, where ν⩾0, of necessary conditions for solvability, prove that they are of strictly increasing strength and show that 𝒞n−3 is insufficient for the solvability of an n‐point problem for n⩾3. We propose the conjecture that condition 𝒞n−2 is necessary and sufficient for the solvability of an n‐point interpolation problem for Γ and we explore the implications of this conjecture. We introduce a classification of rational Γ‐inner functions, that is, analytic functions from the disc into Γ whose radial limits at almost all points on the unit circle lie in the distinguished boundary of Γ. The classes are related to n‐extremality and the conditions 𝒞ν; we prove numerous strict inclusions between the classes.

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We establish the basic complex geometry and function theory of the pentablock P, which is the bounded domainwhere B denotes the open unit ball in the space of 2 × 2 complex matrices. We prove several characterisations of the domain. We show that P arises naturally in connection with a certain robust stabilisation problem in control theory, the problem of μ-synthesis. We describe the distinguished boundary of P and exhibit a 4-parameter group of automorphisms of P. We demonstrate connections between the function theories of P and B. We show that P is polynomially convex and starlike, and we show that the real pentablock P ∩ R 3 is a convex set bounded by five faces, three of them flat and two curved.

Preface vii Chapter 1. Introduction Chapter 2. An overview Chapter 3. Extremal problems in the symmetrized bidisc G 3.1. The Carathéodory and Kobayashi extremal problems 3.2. The Carathéodory extremal problem Car(δ) for G 3.3. Five types of datum δ in G 3.4. The Kobayashi extremal problem Kob(δ) for G Chapter 4. Complex geodesics in G 4.1. Complex geodesics and datums in G 4.2. Uniqueness of complex geodesics for each datum in G 4.3. Flat C-geodesics 4.4. Rational Γ-inner functions Chapter 5. The retracts of G and the bidisc D 2 5.1. Retracts and geodesics of G 5.2. Retracts of D 2 5.3. Geodesics in G are varieties Chapter 6. Purely unbalanced and exceptional datums in G Chapter 7. A geometric classification of geodesics in G Chapter 8. Balanced geodesics in G Chapter 9. Geodesics and sets V with the norm-preserving extension property in G 9.1. V and Car(δ) 9.2. V and balanced datums 9.3. V and flat or royal datums Chapter 10. Anomalous sets R ∪ D with the norm-preserving extension property in G 10.1. Definitions and lemmas 10.2. The proof of the norm-preserving extension property for R ∪ D Chapter 11. V and a circular region R in the plane Chapter 12. Proof of the main theorem iii iv CONTENTS 12.1. Preliminary lemmas 12.2. σ : R → V is analytic on R − 12.3. Degree considerations Chapter 13. Sets in D 2 with the symmetric extension property Chapter 14. Applications to the theory of spectral sets Chapter 15. Anomalous sets with the norm-preserving extension property in some other domains Appendix A. Some useful facts about the symmetrized bidisc A.1. Basic properties of G and Γ A.2. Complex C-geodesics in G A.3. Automorphisms of G A.4. A trichotomy theorem A.5. Datums for which all Φ ω are extremal Appendix B. Types of geodesic: a crib and some cartoons B.1. Crib sheet B.2. Cartoons Bibliography Index

The sethas intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface homeomorphic to the Möbius band and it has a special subvariety which is the only complex geodesic of G that is invariant under all automorphisms. We exploit the geometry of G to develop an explicit and detailed structure theory for the rational maps from the unit disc to the closure Γ of G that map the boundary of the disc to the distinguished boundary of Γ.