Lecturer: Dr Theodore Voronov
Room 2.109 (Alan Turing Bldg). Email: theodore.voronov@manchester.ac.uk
Classes in spring semester 2012-2013:
Tuesdays 2:00-2:50 pm (lecture): Alan Turing G.209,
Fridays 9:00-10:50 am (lecture and example class): Alan Turing G.209
Course webpage: http://www.maths.manchester.ac.uk/~tv/manifolds.html
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.
Future topics requiring this course unit: there are no particular courses directly following this one; however, differentiable manifolds are used in almost all areas of mathematics and its applications, including physics and engineering.
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, and their generalizations, matrix groups such as the rotation group SO(n), etc. Differentiable manifolds naturally appear in various applications, e.g., as configuration spaces in mechanics. 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.
In this course we give an introduction to the theory of manifolds, including their definition and examples; vector fields and differential forms; integration on manifolds and de Rham cohomology.
Textbooks: No particular textbook is followed. Students are advised to keep their own lecture notes and use my notes posted on the web. 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.
Assessment:
Final mark = 20% coursework + 80% exam.
Coursework mode: take home work distributed in week 6; deadline end of week 8.
Exam mode: 2 hour exam (10 credits) or 3 hour exam (15 credits) at the end of semester.
Note: Enhanced version (15 credits) includes extra theoretical material and extra problems.
Exam and coursework contain extra questions; exam is longer (3 hours).