Introduction To Differential Topology

MATH X 451.48

This course, the first of a two-quarter sequence, is a rigorous introduction to manifolds, orientability, submanifolds, embeddings, tangent spaces, critical points, and Morse functions.  

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What you can learn.

  • Learn to assign Euclidean-type coordinate systems to geometrical objects
  • Gain skills needed to include surgery on manifolds and tangent spaces
  • Obtain a solid foundation for Whitney Embedding Theorem
  • Differentiate between local and global structures in geometry

About this course:

Differential topology emerged in the 1950s as a complement to differential geometry in the study of differentiable manifolds—geometrical objects to which can be assigned Euclidean-type coordinate systems that lend themselves to methods of differential and integral calculus.  Whereas differential geometry focuses on a manifold’s local structure (e.g., curvature and geodesics), differential topology is concerned more with global structures such as orientation, immersion and submersion, and smooth mappings between manifolds.  This course, the first of a two-quarter sequence, is a rigorous introduction to the topic, but done at a level that should be accessible to those with a solid familiarity with advanced calculus and basic topology.  Topics to be discussed include manifolds, orientability, submanifolds, embeddings, tangent spaces, critical points, and Morse functions.  The second quarter will extend these studies to include surgery on manifolds and the classic Whitney Embedding Theorem. 
Prerequisites
Calculus and some exposure to advanced mathematical methods.

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