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Topics in Teichmuller Theory and Kleinian Groups |
| November 12, 2007 to November 16, 2007 |
| Organized By: Jeff Brock, Ken Bromberg, Richard Canary, Howard Masur, Alan Reid, Maryam Mirzakhani, and John Smillie |
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| Parent Programs: |
| Teichmuller Theory and Kleinian Groups |
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| Return to Workshop Description |
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Positivity of the universal pairing in 3 dimensions, and the
topological Cauchy-Schwarz inequality
Tuesday November 13, 2007
09:30AM - 10:30AM
Speakers:
Danny Calegari
Abstract:
Fix a topological surface S, and let V be the complex vector space spanned by all (compact, orientable) 3-manifolds which bound S. There is a Hermitian pairing on V, with values in the complex vector space spanned by all closed 3-manifolds. The main result is that this pairing is nondegenerate: if <v,v>=0 then v=0.
The proof involves the construction of a suitable complexity function c on all closed 3-manifolds so that if A and B are two 3-manifolds which bound S, there is an inequality
c(AB) ≤ max(c(AA), c(BB))
with equality if and only if A=B. We discuss some details of the construction of the function c, which involves input ranging from finite group TQFT's to Perelman's recent proof of the geometrization conjecture. This is joint work with Mike Freedman and Kevin Walker.
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