Applications of algebra to a problem in topology - by Michael Hopkins
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Applications of algebra to a problem in topology
Tuesday, April 21, 2009
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Joint work with
Mike Hill
and
Doug Ravenel
Tuesday, April 21, 2009
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Pontryagin (1930’s)
Tuesday, April 21, 2009
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Pontryagin (1930’s)
cobordism group of stably framed k-manifolds
Tuesday, April 21, 2009
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Pontryagin (1930’s)
k=0
k=1
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Pontryagin (1930s) k=2
Tuesday, April 21, 2009
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Pontryagin (1930s) k=2
Tuesday, April 21, 2009
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Pontryagin (1930s) k=2
Tuesday, April 21, 2009
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Pontryagin (1930s) This defines a fuction
If genus , and so
You can always lower the genus with surgery
Tuesday, April 21, 2009
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Pontryagin (?)
it’s quadratic and refines the intersection pairing
is not linear
Tuesday, April 21, 2009
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Pontryagin (?)
Tuesday, April 21, 2009
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Kervaire (1960)(framed)
defined
showed
quadratic refinement of the intersection pairing
Tuesday, April 21, 2009
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produced a piecewise linear
Kervaire (1960)
with
has no smooth structurehence
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represented bythere exists
Browder (1969)
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Barratt-Jones-Mahowald (1969, 1984)
so the first open dimension is 126
The elements exist
for
dimensions
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The Kervaire invariant problem
In which dimensions can
be non-zero?
Tuesday, April 21, 2009
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Doomsday Theorem (Hill, H., Ravenel)
Tuesday, April 21, 2009
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Adams SpectralSequence
Adams-NovikovSpectral Sequence
Something easierto compute
Tuesday, April 21, 2009
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Tuesday, April 21, 2009
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Adams SpectralSequence
Adams-NovikovSpectral Sequence
Something easierto compute
Tuesday, April 21, 2009
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Tuesday, April 21, 2009
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Tuesday, April 21, 2009
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supports afornon-zero differential
Tuesday, April 21, 2009
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Adams SpectralSequence
Adams-NovikovSpectral Sequence
Something easierto compute
Tuesday, April 21, 2009
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Tuesday, April 21, 2009
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Rochlin’s Theorem
periodicity Theorem
Tuesday, April 21, 2009
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K-theory and reality (Atiyah, 1966)
space with a action
vector bundles with compatible conjugate-
linear action
Tuesday, April 21, 2009
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Assemble K-theory from the equivariant
chains on
slice filtration(Dugger, Hu-Kriz, H.-Morel, Voevodsky)
Tuesday, April 21, 2009
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slice filtration
Tuesday, April 21, 2009
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Tuesday, April 21, 2009
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Tuesday, April 21, 2009
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periodicity
Tuesday, April 21, 2009
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periodicity
Tuesday, April 21, 2009
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level 5 topological modular forms
Like with instead of
Tuesday, April 21, 2009
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Tuesday, April 21, 2009
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32
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32
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Tuesday, April 21, 2009
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the period2 below the period
Tuesday, April 21, 2009
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the period2 below the period
Tuesday, April 21, 2009
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Assemble tmf(5) from the equivariant chains
on
slice filtration
the 4 dimensional real regular representation of
Tuesday, April 21, 2009
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Tuesday, April 21, 2009
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0-2
Tuesday, April 21, 2009
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the period2 below the period
Tuesday, April 21, 2009
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gap + periodicity
differentials on the
Tuesday, April 21, 2009
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The actual proof
Show that all the choices of are distinguished
Step 2:
Prove a gap theorem (easy)Step 3:
Use and an appropriateStep 1:cohomology theory
Prove a periodicity theorem (of period 256)Step 4:
Tuesday, April 21, 2009
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Relation to Geometry/Physics?
4 dimensional field theory?
generalization of Clifford algebras with periodicity of
(maybe twice that)
Tuesday, April 21, 2009
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Question
Tuesday, April 21, 2009