Schedule:

9:00-11:00 and 14:00-16:00 on Monday 30/05, Tuesday 31/05, Wednesday 01/06 and Friday 03/06.

 

Abstract:

The course is an introduction to functions of a quaternionic variable, focusing on the theory of slice regularity developed over the last decade. It provides all the prerequisites for the subsequent courses within the INdAM intensive research period, covering selected topics among the following ones.

 

The Cauchy-Fueter operator and Fueter regularity.

Slice regular functions: definitions and basic results.

Regular power series.

Zeros of regular functions, with their algebraic and topological properties; factorization of quaternionic polynomials.

Infinite products of quaternions: convergence of an infinite product; the Weierstrass factorization theorem.

Singularities: regular reciprocal and quotients; Laurent series and expansions; the Casorati-Weierstrass theorem.

Integral Representations: Cauchy theorem and Morera theorem; Cauchy integral formula; the argument principle.

Maximum modulus theorem and applications: open mapping theorem; Phragmén-Lindelöf principles.

Spherical series and differential; symmetric analyticity.

Fractional transformations and the unit ball: transformations of the quaternionic space and of the Riemann sphere; Schwarz lemma and transformations of the unit ball; rigidity and a boundary Schwarz lemma.