Introduction to Formal Analysis

Timetable

Module aim (aims)

The main goal of Formal Analysis is to investigate formal power series and formal Laurent
series as well. Various applications of that series ranging from algebra to classical
analysis. In particular, it is worth to mention here that it has some applications to
the investigation of the boundary convergence behavior of power series.
The main goal of this course is to introduce basic notions and tools using in
Formal Analysis as well as some of its applications.
A particular emphasis will be put on some new achievements of that theory.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

One has to have basic knowledge in mathematical analysis. Moreover, some knowledge of basic definitions and facts of functional analysis and metric topology would be also useful.

Syllabus

Week 1: Definitions and algebraic properties of formal power series.
Week 2: The composition of a formal power series with a nonunit.
Week 3: Right distributive law for composition.
Week 4: General composition of formal power series.
Week 5: General composition of formal power series- continuation.
Week 6: General right distributive law for composition.
Week 7: Calculus of formal power series.
Week 8: Calculus of formal power series- continuation
Week 9: The generalized chain rule.
Week 10: Boundary convergence of regular power series.
Week 11: Boundary convergence of regular power series- continuation.
Week 12: Topology on the space of formal power series
Week 13: Basic algebra of formal Laurent series.
Week 14: Composition of formal Laurent series and formal power series.
Week 15: Topology on the space of formal Laurent series.

Reading list

X. -X. Gan, Selected Topics of Formal Analysis, Lecture Notes in Nonlinear Analysis, to appear
D. Bugajewski and X. -X. Gan, A note on formal power series, Commentationes Mathematicae Universitatis Carolinae 51(4)(2010), 595-604
D. Bugajewski and X. -X. Gan, On formal Laurent series, Bulletin of the Brazilian Mathematical Society, New Series 42(3)(2011), 415-437