Course Syllabus for

# Mathematical Structures Matematiska strukturer

## FMAN65F, 6 credits

Valid from: Spring 2023
Decided by: Maria Sandsten
Date of establishment: 2023-01-14

## General Information

Division: Mathematics
Course type: Course given jointly for second and third cycle
The course is also given at second-cycle level with course code: FMAN65
Teaching language: Swedish

## Aim

Besides mere knowledge imparting, the course aims to give training in theorem proving, and to bring out the possibilities of a more abstract representation of mathematical concepts and the connections between them. The intention is to give an overall view elucidating the foundations of the mathematical theories in the basic courses.

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• be familiar with and in his or her own words be able to explain the concepts within analysis, algebra and geometry touched upon in the course.
• be able to give examples of how these concepts are abstractions of concepts in the basic courses, and show understanding for how the abstractions serve to simplify and clarify the theory.
• in his/her own word be able to describe the logical connections between the concepts (theorems and proofs).

Competences and Skills

For a passing grade the doctoral student must

• be able to demonstrate ability to identify problems which can be modelled with the concepts introduced.
• in the context of problem solving be able to demonstrate ability to, in simple situations, develop the theory further.
• with proper terminology, in a well-structured manner, and with clear logic be able to explain the connections between various concepts in the course.
• with proper terminology, suitable notation, in a well-structured manner and with clear logic be able to explain the solution to a problem or the proof of a theorem.
• have developed his or her ability to independently read and judge mathematical text at a high level.

## Course Contents

Sets. Real numbers. Metric spaces. Algebra (groups and linear spaces). Banach spaces and Hilbert spaces with applications.

## Course Literature

Kaplansky, I.: Set Theory and Metric Spaces. American Mathematical Society. 2001. ISBN 9780821826942.

## Instruction Details

Types of instruction: Lectures, seminars

## Examination Details

Examination formats: Written exam, oral exam. The examiner, in consultation with Disability Support Services, may deviate from the regular form of examination in order to provide a permanently disabled student with a form of examination equivalent to that of a student without a disability.