QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday, March 12 (DayLet X be a compact n-dimensional differentiable manifold, and Y ⊂ X a closed submanifold of dimension m. Show that the E

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• Topology
• Mathematics
• Algebra
• Differential geometry
• Manifolds
• Differential topology
• Fiber bundles
• Codimension
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• Moving frame
• Frame bundle
• Vector bundle

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• Topology
• Differential topology
• Manifolds
• Theoretical physics
• Foliations
• Differential geometry
• Symmetry
• Invariant
• Submanifold
• Lie group
• Web

Graduate Texts in Mathematics 218 Graduate Texts in Mathematics Series Editors:

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• Manifolds
• Differential topology
• Geometric topology
• Differential geometry
• Differentiable manifold
• Riemannian geometry
• Topology
• Contact geometry
• Riemannian manifold
• Lie group
• Submanifold
• Transversality

401 Documenta Math. Tight Embeddings of Simply Connected 4-Manifolds ¨hnel

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• Geometric topology
• 4-manifold
• Differentiable manifold
• Triangulation
• Topological manifold
• Manifold
• Handle decomposition
• Immersion
• 3-manifold
• Submanifold
• Connected sum
• Intersection form

QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday, March 12 (DayLet X be a compact n-dimensional differentiable manifold, and Y ⊂ X a closed submanifold of dimension m. Show that the E

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Finding the Homology of Submanifolds with High Confidence from Random Samples∗ P. Niyogi†, S. Smale‡, S. Weinberger§ March 15, 2006

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• Topology
• Mathematics
• Algebra
• Homology theory
• Algebraic topology
• Differential topology
• Homology
• Topological space
• Betti number
• Submanifold
• Cohomology
• BorelMoore homology

Neighborly embedded manifolds G. Kalai∗ and A. Wigderson† 1 Introduction

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• Topology
• Mathematics
• Mathematical analysis
• Differential topology
• General topology
• Maps of manifolds
• Manifolds
• Category theory
• Embedding
• Submanifold
• Convex function
• Neighborly polytope