(February 9, 2011)
Modular forms and number theory exercises 12
Paul Garrett garrett@math.umn.edu
http:/
/www.math.umn.edu/ garrett/
[mfms 12.1] Show that the space of homogeneous degree d harmonic polynomials on R3 invariant under
rotations of the y, z-p
(February 28, 2011)
Harmonic analysis on spheres, I
Paul Garrett garrett@math.umn.edu
1.
2.
3.
4.
5.
6.
7.
8.
http:/
/www.math.umn.edu/ garrett/
Calculus on spheres
Existence of the spherical Laplacian
Polynomial eigenvectors for the spherical Laplacian
C
(February 13, 2011)
Modular forms and number theory exercises 13
Paul Garrett garrett@math.umn.edu
http:/
/www.math.umn.edu/ garrett/
[mfms 13.1] Show that the matrix exponential series
eM =
=0
M
!
(for n-by-n complex matrix M )
converges absolutely. To m
(February 28, 2011)
S. Bernsteins proof of Weierstra approximation theorem
Paul Garrett garrett@math.umn.edu
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R
Weierstra proved that polynomials are dense in C o ( n ). Decades later, Stone greatly abstracted this. Prior
(February 19, 2011)
Modular forms and number theory exercises 14
Paul Garrett garrett@math.umn.edu
http:/
/www.math.umn.edu/ garrett/
[mfms 14.1] Show that
exp
exp
0 t
t 0
0
0
=
t
0
1
0
=
cos t
sin t
0
exp 0
0
t
1
exp
sin t
cos t
a
0
0
s
0
0
t
exp
0
1
b
(January 26, 2011)
Modular forms and number theory exercises 11
Paul Garrett garrett@math.umn.edu
http:/
/www.math.umn.edu/ garrett/
[mfms 11.1] Fill in the sketches from class:
Identify the irreducible (complex) representations of the dihedral group G wi
(March 18, 2011)
[0.0.1] Claim: On a simple complex Lie algebra g, up to scalars there is at most one g-invariant nondegenerate bilinear form , .
Proof: A non-degenerate g-equivariant bilinear form B gives an g-isomorphism B of g (under ad g) to its
dual
(March 12, 2011)
Modular forms and number theory exercises 15
Paul Garrett garrett@math.umn.edu
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/www.math.umn.edu/ garrett/
[mfms 15.1] Let g be a one-dimensional Lie algebra, with a basis cfw_x, and [x, x] = 0. Show that both
the universal associa
(December 21, 2010)
Abelian topological groups and (A/k) k
Paul Garrett garrett@math.umn.edu
1.
2.
3.
4.
http:/
/www.math.umn.edu/garrett/
Compact-discrete duality
(A/k) k
Appendix: compact-open topology
Appendix: no small subgroups
The specic goal here i
(May 29, 2014)
Harmonic analysis of dihedral groups
Paul Garrett garrett@math.umn.edu
1.
2.
3.
4.
http:/
/www.math.umn.edu/ garrett/
Dihedral groups
Products on dihedral groups
Appendix: nite abelian groups on nite-dimensional spaces
Appendix: tensor prod