**Welcome to Advanced Mathematics for PhD Candidates****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, e-mail: mnielsen@math.aau.dk**Lecturers: **Professor Morten Nielsen and Associate Professor Lisbeth Fajstrup**ECTS:** 4**Time:** 30 november + 2, 4, 9, 11 and 14 December 2020**Place:** ~~Fibigerstræde 13. Room 51.~~ **ONLINE**

Zip code: ~~9220~~**City: **Aalborg Øst

**Number of seats:**30

**Deadline:**9 November 2020

**Important information concerning PhD courses:** We have over some time experienced problems with no-show for both project and general courses. It has now reached a point where we are forced to take action. Therefore, the Doctoral School has decided to introduce a no-show fee of DKK 3,000 for each course where the student does not show up. Cancellations are accepted no later than 2 weeks before start of the course. Registered illness is of course an acceptable reason for not showing up on those days. Furthermore, all courses open for registration approximately four months before start. This can hopefully also provide new students a chance to register for courses during the year. We look forward to your registrations.

- Teacher: Lisbeth Fajstrup
- Teacher: Morten Nielsen