Welcome to Advanced Mathematics for PhD Candidates in the Engineering Sciences: Analysis and Topology**Description:** The aim of this course is to give the participants an idea of how the mathematical vocabulary is used and the strength of using it properly. This is needed by engineering PhD's when reading papers, which will often use the common language of mathematics, and in particular when writing papers. Moreover, it is crucial, that engineers understand when a given mathematical toolbox is applicable and when it is not. As an example: In favourable cases, differential equations have unique solutions, but this is not always true, and trying to approximate a solution in such cases may lead to results which are simply wrong. The topics covered are: metric spaces, convergence, continuity, compactness,

completeness. Vector Spaces and linearity. Korovkins theorem on polynomial approximations via Bernstein polynomials. The Banach Fixed Point Theorem. Existence and uniqueness results for ordinary differential equations. The approach in the course is to stress the necessity of precise mathematical formulation, and, in particular, to give examples where the intuitive answer is not correct.**Organizer:** Professor Morten Nielsen, email: mnielsen@math.aau.dk**Lecturers:** Professor Morten Nielsen and Associate Professor Lisbeth Fajstrup**ECTS:** 3**Time:** 1, 3, 5, 12, 15 and 17 December, 2014, 9:00-16:00**Place:** Aalborg University, Niels Jernes Vej 12A/6 room 104**Zip code: **9220**City:** Aalborg**Number of seats:** 30**Deadline:** 10 November, 2014

- Teacher: Lisbeth Fajstrup
- Teacher: Morten Nielsen