Differentiable Manifolds
Level 4 and postgraduate (15 credits)

Lecturer: Dr Theodore Voronov
Room 2.109 (Alan Turing Bldg). Email: theodore.voronov@manchester.ac.uk
Classes in autumn semester 2017-2018: (weeks 1-5 and 7-12)
    Tuesdays 3:00-3:50 pm (lecture): Williamson G.33
    Thursdays 1:00-1:50 pm (lecture) Alan Turing G.108; 2:00-2:50 pm (example class, starts week 2): Alan Turing G.107

Course webpage: http://www.maths.manchester.ac.uk/~tv/manifolds.html
(Page last modified: Wednesday 13 (25) October) 2017. Each time refresh the browser to get updated pages.)

Prerequisites: MATH20132 Calculus of Several Variables; MATH20222 Introduction to Geometry (optional). Taking in parallel MATH31051 Introduction to Topology may be beneficial but is optional.
See more details concerning course description, prerequisites and textbooks below.

Assessment: Final mark = 20% coursework + 80% exam.
Coursework mode: take home work distributed in week 5; deadline beginning of week 8.
Exam mode: 3 hour exam at the end of semester.

Coursework (download it as a PDF file) is due on Tuesday afternoon, 14 November 2017. (Please bring coursework to the lecture on Tuesday.)

Syllabus and online materials:


Description:

Differentiable manifolds are among the most fundamental notions of modern mathematics. Roughly, they are geometrical objects that can be endowed with coordinates; using these coordinates one can apply differential and integral calculus, but the results are coordinate-independent. Examples of manifolds start with open domains in Euclidean space Rn, and include "multi-dimensional surfaces" such as the n-sphere Sn and n-torus Tn, the projective spaces RPn and CPn, matrix groups such as the rotation group SO(n), etc. Differentiable manifolds naturally arise in various applications, e.g., as configuration spaces of physical systems. They are arguably the most general objects on which calculus can be developed. On the other hand, differentiable manifolds provide for calculus a powerful invariant geometric language, which is used in almost all areas of mathematics and its applications.

Details of prerequisites: Standard calculus and linear algebra; familiarity with the statement of the implicit function theorem (students may consult any good textbook in multivariate calculus); some familiarity with algebraic concepts such as groups and rings is desirable, but no knowledge of group theory or ring theory is assumed; some familiarity with basic topological notions (topological spaces, continuous maps, Hausdorff property, connectedness and compactness) may be beneficial, but is not assumed.

Textbooks: No particular textbook is followed. Students are advised to keep their own lecture notes and study my notes posted on the web. Solving problems is essential for understanding. There are many good sources available treating various aspects of differentiable manifolds on various levels and from different viewpoints. Below is a list of texts that may be useful. More can be found by searching library shelves.


Information about the exam: (Information about exam will be here.)


http://www.maths.manchester.ac.uk/~tv/

Last modified: Wednesday 13 (25) October) 2017. Theodore Voronov.