Vector Fields on Spheres and Clifford Algebras

PhD Student Seminar
Open to the Public
Zrinyi u. 14
Wednesday, November 19, 2014 - 2:30pm
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Wednesday, November 19, 2014 - 2:30pm

The next department seminar will be at the usual place, but it starts
half an hour later than usual, at 2.30pm.

Abstract: By the well-known "hedgehog theorem", every (tangential)
vector field on the sphere S^2 must vanish somewhere. More generally,
Poincaré proved that one can find a nowhere zero vector field on the
n-dimensional sphere S^n if and only if n is odd. This result raised
the question how many pointwise linearly independent vector fields can
be found on S^n for a given n. The solution of this problem consists
of two parts. First, we have to construct as many linearly independent
vector fields on S^n as we can, then it has to be proved that the
maximal number is reached. In this lecture we focus on the first part
and show how representation theory of Clifford algebras tells us the
maximal number of pointwise linearly independent linear vector fields
on S^n. This number cannot be exceeded even if we drop the linearity
condition on the vector fields, by a theorem of Adams.

You can watch the lecture here: