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WHEN: MWF 11 – 11:50 am
WHERE: WEBER 223 // in the oval with good weather
WHAT: A graduate level introduction to the theory of
differentiable manifolds.
TEXTBOOK: Introduction
to Smooth Manifolds (John
M. Lee)
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DATE DUE: |
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Jan 28th |
1) Exercises 1.1, 1.3 page 21. |
Feb 4th |
Let
f(x,y) and g(x,y) be two polynomials in two variables. By setting z
= f(x,y) w
= g(x,y) we get a smooth function from the
plane (with coordinates x,y) to the plane (with
coordinates z,w). What
are the conditions on f and g for this function to descend to a smooth
function from the projective line to the projective line? Can
you generalize this construction to more than two variables and give an
example of a smooth function from P^3 to P^5? |
Feb 11th |
Exercises
3-1, page 64. |
Feb 18th |
Exercises
3-4, 3-5 page 64. |
Feb 25th |
Prove that the following conditions are
equivalent: 1) V is a smooth vector field on X 2) Every component of every local
representation of V is a smooth function. 3) For every smooth function f: X->IR , the function Vf : X-> IR
is smooth. |
Mar 4th |
Exercise 4.1, page 90 |
Mar 11th |
Exercises 5-1, 5-5,
5-18 pages 126, 128 |
Apr 8th |
Exercises 6-4,6-5 page 144 |
May 6th |
Compute the de Rham cohomology vecor spaces of: ·
The
n-dimensional sphere. ·
Real
projective plane. ·
The
figure eight. ·
The
union at a point of two manifolds. For the last two spaces, what we intend is “some manifold
which is homotopy equivalent to…” but
you can use the singular space in your constructions. |
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