Mathematics as a logical system, including sentential and some predicate logic and Gödel's theorems. The number system, especially Peanos axioms for the natural numbers and Dedekind's construction of the reals. Cardinality. Some aspects of Euclidean and non-Euclidean geometry. Basic set-theory and topology, metric spaces. Convergence, continuity, compactness, connectivity. Contractions and fix-point theorems. In-depth study of differential and integral calculus, including the inverse function theorem, Weierstrass' approximation theorem and Picard's theorem on existence and uniqueness of solutions to ordinary differential equations. An orientation about Lebesgue integration, complex analysis and functional analysis.
An individual project is mandatory.
After taking this course the student is supposed to be able to
- explain the structure of the number system, both intuitively and axiomatically, especially Peano's axioms for the natural numbers and Dedekind's construction of the reals
- carry out cardinality arguments showing the denumerability of the rational numbers and the non-denumerability of the real numbers
- give an account of mathematics as a logical system with axioms, rules of inference, definitions, theorems and proofs, and carry out deductions in sentential and predicate logic, and explain the content of Gödel's theorems
- understand and use set-theoretical and topological notions in mathematical reasoning, and have some knowledge of naive and axiomatic set-theory
- analyze different types of convergence in different types of spaces, for example Euclidean spaces, general metric spaces and different function spaces, and understand and analyze compactness, continuity and connectivity
- use the supremum property of the real numbers to prove some properties of continuous functions
- understand, prove and apply certain important theorems from differential and integral calculus, such as the inverse function theorem, Weierstrass' approximation theorem and Picard's theorem on existence and uniqueness of solutions to differential equations
- elucidate connections and differences between notions in analysis using examples, i.e. give examples of a nowhere differentiable continuous function, a connected set that is not pathwise connected, a sequence of functions that is pointwise but not uniformly convergent, etcetera
- explain introductory complex analysis and some of its applications
- give an account of the basic facts in euclidean and non-euclidean geometry
Furthermore, the student is supposed to carry out an individual project in some area of mathematics that can be chosen freely. Possible examples are
- explain the basic ideas of Lebesgue integration
- explain the classical problems of construction, i.e. doubling the cube, trisecting the angle and the quadrature of the circle
- give an account of the basic ideas of Galois theory
- give an account of the basic ideas of functional analysis.
