## Topic outline

• ### Lecture of 20/01/2020

Real alternative algebras: definition; complex numbers, quaternions and octonions as examples of real alternative division algebras; classification of finite dimensional division algebra, mentioned; examples of algebras admitting zero divisors; infinite-dimensional examples. A reference:

Equivalent notions in the complex case: differentiability; holomorphy; analyticity. Conformality. Elementary properties of holomorphic functions. A reference:

• ### Lecture of 23/01/2020

Integration theory, recalled: cycles; winding number; Cauchy's theorem; Cauchy's integral formula.
Complex analyticity, recalled: integral formulas for derivatives; Liouville's theorem; Taylor expansion; multiplicity of zeros; identity principle.
Classification of singularities, recalled: removable singularities; poles and their order; essential singularities and Casorati-Weierstrass theorem; argument principle.
Topological and differential properties of holomorphic functions, recalled: local degree; open mapping theorem; maximum modulus principle; minimum modulus principle; biholomorphisms and automorphisms.
A reference:

• ### Lecture of 03/02/2020

Definition of Riemann surface and holomorphic maps, recalled.

Riemann sphere: complex structure; holomorphic functions; rational functions and their degrees; automorphisms, their 3-transitivity and their fixed points; Fubini-Study distance.

Unit disc: Poincaré metric, recalled; Schwarz's lemma, recalled.

A reference:

• ### Lecture of 05/02/2020

Unit disc: automorphisms, their transitivity and their fixed points; the Schwarz-Pick lemma and its metric interpretation.

Complex plane: automorphisms, their 2-transitivity and their fixed points.

Upper half-plane: Cayley's transformations; automorphisms, their transitivity and their fixed points; standard models for elliptic, hyperbolic and parabolic automorphisms.

A reference:

Recommended exercises: Exercises200205.pdf

• ### Lecture of 10/02/2020

Classification of simply connected domains in the Riemann sphere: Hurwitz's and Montel's theorems, recalled; Riemann's mapping theorem, sketched; general classification.

Classification of simply connected Riemann surfaces: triangulation and Van der Waerden's lemma, mentioned; first part of the proof of the classification.

In the picture: Bernhard Riemann (1826 - 1866).

Some references:

• ### Lecture of 12/02/2020

Classification of simply connected Riemann surfaces: mapping of polygons into discs, mentioned; generalized Schwarz reflection principle, mentioned; second part of the proof of the classification.

Uniformization: free and properly discontinuous group actions; covering theory, recalled; uniformization theorem.

Some references:

• ### Lecture of 17/02/2020

Classification of Riemann surfaces: uniformization theorem; uniqueness of the elliptic surface; classification of parabolic surfaces; classification of hyperbolic surfaces.

A reference:

• M. Abate, Iteration theory of holomorphic maps on taut manifolds. Mediterranean Press, Cosenza, 1989, Chapter 1.1

Recommended exercises: Exercises200217.pdf

• ### Lecture of 24/02/2020

Theory of hyperbolic Riemann surfaces: examples; properties of holomorphic maps; Poincaré distance; normal families; Montel's theorem.

Holomorphic dynamics on hyperbolic Riemann surfaces: study of the general case; compact case; Ritt's theorem.

A reference:

• M. Abate, Iteration theory of holomorphic maps on taut manifolds. Mediterranean Press, Cosenza, 1989, Chapter 1.1 and Chapter 1.3

• ### Lecture of 02/03/2020

Holomorphic dynamics on hyperbolic domains: general case; unit disc case, horocycles, Wolff's lemma and Wolff-Denjoy theorem; case of hyperbolic domains of regular type, horocycles (mentioned) and Heins' theorem.

A reference:

• M. Abate, Iteration theory of holomorphic maps on taut manifolds. Mediterranean Press, Cosenza, 1989, Chapter 1.1 and Chapter 1.3

Recommended exercises: Exercises200302.pdf
• ### Lecture of 06/04/2020

Hypercomplex differentiability: definition; characterization over quaternions and other hypercomplex algebras; Liouville's theorem about conformal maps, mentioned.

Analogs of holomorphy: definition of Fueter-Regular function; properties, mentioned; definition of slice regular function; splitting lemma; examples.

Some references:

• ### Lecture of 08/04/2020

Analogs of holomorphy: quaternionic Abel theorem; comparison between different notions of regularity over quaternions.

Basic properties of slice regular functions: slice domains; identity principle; symmetric sets; representation formula; extension lemma; spherical value and derivative; basic properties of the zero set.

A reference:

• G. Gentili et al., Regular Functions of a Quaternionic Variable, Springer Berlin Heidelberg, 2013, Chapter 1.

Recommended exercises: Exercises200408.pdf

• ### Lecture of 15/04/2020

Basic properties of slice regular functions: algebraic structure of the class of regular power series centered at 0.

Regular power series centered at p: definition; study of the domains of convergence.

References:

• G. Gentili et al., Regular Functions of a Quaternionic Variable, Springer Berlin Heidelberg, 2013, Chapter 1 and Chapter 2
• ### Lecture of 17/04/2020

Regular power series center at p: regularity is equivalent to existence of regular power series expansions at all points; special case when p is real.

Operations on regular functions on a symmetric slice domain: regular product; regular conjugate; symmetrization; *-algebra structure; relation between the values of f*g and those of f and g.

A reference:

• G. Gentili et al., Regular Functions of a Quaternionic Variable, Springer Berlin Heidelberg, 2013, Chapter 1 and Chapter 2

Recommended exercises: Exercises200417.pdf
• ### Lecture of 20/04/2020

Zeros of regular functions: zero sets of slice preserving functions; relation between the zero set of f and those of fand fs; topological properties of the zeros sets; factorization of an isolated or spherical zero; fundamental theorem of algebra; factorization of a polynomial and its relation to the zero set.

A reference:

• G. Gentili et al., Regular Functions of a Quaternionic Variable, Springer Berlin Heidelberg, 2013, Chapter 3
• ### Lecture of 27/04/2020

More geometric function theory of regular functions: maximum modulus principle; regular reciprocal; minimum modulus principle; open mapping theorem.

A reference:

• G. Gentili et al., Regular Functions of a Quaternionic Variable, Springer Berlin Heidelberg, 2013, Chapter 5 and Chapter 7

Recommended exercises: Exercises200427.pdf.