Lecture Notes

Munkres: Fundamental groups, covering spaces, Seifert-Van Kampen, classification of covering spaces. Vick: Singular homology, Mayer-Vietoris, maps from spheres to spheres, hairy ball theorem, ham sandwich theorem, etc.

Differentiable manifolds, tangent vectors, vector fields, bundles, differential forms and integration, exterior derivative and Stokes theorem, de Rham cohomology, foliations, Lie groups and Lie algebras

Banach, Hilbert spaces, bounded and compact compact operators, Hahn-Banach extension theorem, Baire category, open mapping, closed graph, uniform boundedness, weak topologies, Banach-Alaoglu, Hahn-Banach separation theorem, amenability of groups, Krein-Milman

Polar decomposition, Volterra operator, Borel functional calculus

Properties of bounded operators, weak topology, ... Extreme points

Symplectic manifolds, Hamiltonian mechanics, Moser theorem, Darboux theorem, Weinstein lagrangian neighbourhood theorem, Lie groups, Lie group actions, Moment maps, Symplectic reduction, convexity properties of moment maps

Riemann surfaces, Riemann Hurwitz formula, Meromorphic functions on the complex torus, analytical continuation, holomorphic, meromorphic, harmonic differential forms, Hodge decomposition theorem, Abels' theorem, Jacobi inversion theorem

Representation theory, Character theory, Nilpotent groups, Polycylic groups, Group (co)homology, Group extensions

Connected sums and the Mazur swindle

Classification of vector bundles on spheres

A classification of topologically stable Poisson structures on a compact oriented surface

Morse Theory: A visual guide from handlebodies to the generalized Poincaré conjecture

Real numbers, limits, topology of R, (uniform) continuity, sequences of functions, derivatives of functions, series in C, metric spaces, completeness

Solutions to biweekly exercise sessions. Feedback added in green

Overview of all theorems and propositions

Contains some of the most beautiful plots I've ever made. Also covers Bairstow

Code accompanying the report

Numerical analysis of a non-linear ODE via linearization, Euler and Runge-Kutta method using Python

Dual spaces, bilinear forms, groups, rings, factor rings, polynomial rings, divisbility, field extensions, Wedderburn theorem, Cayley-Hamilton, Jordan decomposition

General topology, following Munkres

On the Zariski topology

Product topology, Cantor set

Compact versus Lindelöf

Complex numbers, power series and holomorphic functions, integration, Cauchy theorem, Morera, Schwarz reflection, meromorphic functions, residue theorem, argument and maximum modules principle, homotopies, logarithm, conformal mappings, Riemann mapping theorem

Elliptic curves and cryptography

Basics of RSA, elliptic curves and elliptic curve cryptography, algorithm of Lenstra