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Mathematical Truths: Experiment, Proof, and Understanding Nathan M. - - PowerPoint PPT Presentation
Mathematical Truths: Experiment, Proof, and Understanding Nathan M. - - PowerPoint PPT Presentation
Mathematical Truths: Experiment, Proof, and Understanding Nathan M. Dunfield University of Illinois What is topology? The study of objects up to rubbery stretching. What is a topologist? Someone who cant tell a coffee cup from a doughnut.
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1-manifold: Locally like a line. 2-manifold (surface): Locally like a plane. 3-manifold: Locally like 3-dimensional space.
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3-manifolds with boundary a torus:
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A 3-manifold is fibered if it is swept out by surfaces. For instance, the doughnut is a circle’s worth of disks.
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Not every 3-manifold fibers. For instance, this one does not: Main Question: How common is it for a 3-manifold to fiber?
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A special kind of 3-manifold:
+ = + =
Revised question: How common is it for such a 3-manifold to fiber?
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Coordinates for curves on a surface: Weights: 1 2 2 Twists: 1
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Procedure which computes whether the associated manifold fibers: Stallings (1962) + K. Brown (1987).
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Experimental Results: 20 40 60 80 100
1018 1015 1012 109 106 103 1
% of manifolds which fiber
Size of coordinates
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Q1: Does a typical 3-manifold fiber? A: No, at least for the type of 3-manifold we’ve looked at. In particular, the more complicated the manifold, the closer the odds of it fibering is to 0%.
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Q1: Does a typical 3-manifold fiber? A: No, at least for the type of 3-manifold we’ve looked at. In particular, the more complicated the manifold, the closer the odds of it fibering is to 0%. Q2: Why?
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Q1: Does a typical 3-manifold fiber? A: No, at least for the type of 3-manifold we’ve looked at. In particular, the more complicated the manifold, the closer the odds of it fibering is to 0%. Q2: Why? Q3: How can we prove this?
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Alternate sampling method 20 40 60 80 100
103000 102500 102000 101500 101000 10500 1
% of manifolds which fiber Size of coordinates
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Two random walks in the plane
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Good books about topology:
- Colin Adams, Why Knot? An Introduction to the Mathematical The-
- ry of Knots, 2004. ISBN 1-931914-22-2
- Jeffrey Weeks, The Shape of Space, 2001. ISBN 0-8247-0709-5
Original Sources:
- N. M. Dunfield and D. P
. Thurston, A random tunnel number one 3-manifold does not fiber over the circle, Geometry and Topology (2006) 2431–2499. http://arxiv.org/abs/math/0510129
- This presentation: http://dunfield.info/preprints/