PMath 955: \"Topics in Geometry: The AtiyahSinger Index Theorem
MATH 955

Winter 2014
PMATH 955: The AtiyahSinger Index Theorem
Assignment 3; due Thursday, 27 February 2014
[1] Let (M n , g) be a compact oriented Riemannian manifold, and let : k (M ) nk (M ) be the associated
Hodge star operator.
[a] Let 1 (M ) and k (M ). Show that
= (1)
PMath 955: \"Topics in Geometry: The AtiyahSinger Index Theorem
MATH 955

Winter 2014
PMATH 955: The AtiyahSinger Index Theorem
Assignment 4; due Thursday, 13 March 2014
[1] Let (M, g) be a compact oriented Riemannian manifold, and let S be a Cliord bundle over M with Dirac
operator D. Let (D) be the spectrum of D. Given > 0, dene a bounde
PMath 955: \"Topics in Geometry: The AtiyahSinger Index Theorem
MATH 955

Winter 2014
PMATH 955: The AtiyahSinger Index Theorem
Assignment 6; due Monday, 14 April 2014
[1] Let A be a ltered algebra. This means that for all k Z, we have subspaces Ak such that Ak Ak+1 ,
with A = kZ Ak and Ak Al Ak+l . Let G be a graded algebra. This means th
PMath 955: \"Topics in Geometry: The AtiyahSinger Index Theorem
MATH 955

Winter 2014
PMATH 955: The AtiyahSinger Index Theorem
Assignment 5; due Tuesday, 1 April 2014
[1] Recall the denitions of the Sobolev spaces of Cvalued functions W k (T n ) on the torus: for a smooth
function f : T n C, we dene the Sobolev kform f k of f by
f (p
PMath 955: \"Topics in Geometry: The AtiyahSinger Index Theorem
MATH 955

Winter 2014
PMATH 955: The AtiyahSinger Index Theorem
Assignment 1; due Thursday, 23 January 2014
[1] Let K = R or C and let r 0 be an integer. Let M be a manifold with an open cover U = cfw_U , A.
Suppose that for any , A such that U U = , we are given smooth functi
PMath 955: \"Topics in Geometry: The AtiyahSinger Index Theorem
MATH 955

Winter 2014
PMATH 955: The AtiyahSinger Index Theorem
Assignment 2; due Thursday, 06 February 2014
[1] Let E be a Kr vector bundle over M . Let
be a connection on E, and denote again by
the
induced connection on End(E) and by d the induced exterior covariant derivat