Instructor: Anton Izosimov

Classes: Tuesday and Thursday, 9:30-10:45AM in MATH 501

Office hours: Tuesday 11-11:50AM, Thursday 12-12:50AM, and by appointment; S414, ENR2 building

TA: Lindsay Swift

Problem sessions (held by Lindsay): Wednesday, 1-1:50PM in MATH 203

Textbook: John M. Lee, Introduction to Smooth Manifolds, Second Edition

Course syllabus in PDF (updated on Sep 19)

Tests:

• Test 1, Solutions


• Test 2, Solutions


• Final Exam

Homework assignments (all due at 11:59pm):

• Homework 1 due Aug 30, LaTeX source


• Homework 2 due Sep 8, , LaTeX source


• Homework 3 due Sep 18, LaTeX source


• Homework 4 due Sep 27, LaTeX source


• Homework 5 due Oct 9, LaTeX source, Solutions (by Lindsay Swift)


• Homework 6 due Oct 16, LaTeX source, Solutions


• Homework 7 due Oct 30, LaTeX source, Solutions


• Homework 8 due Nov 13, LaTeX source

Covered textbook sections:

• Tue, Aug 21 -- Topological Manifolds


• Thu, Aug 23 -- Smooth Structures (started), Examples of Smooth Manifolds


• Tue, Aug 28 -- Smooth Structures (finished), Manifolds with Boundary, Smooth Functions and Smooth Maps (started)


• Thu, Aug 30 -- Smooth Functions and Smooth Maps (continued)


• Tue, Sep 4 -- Smooth Functions and Smooth Maps (finished), Tangent Vectors (started)


• Thu, Sep 6 -- Tangent Vectors (finished), Velocity Vectors of Curves, Computation in Coordinates (started)


• Tue, Sep 11 -- Alternative Definitions of the Tangent Space, The Differential of a Smooth Map (started)


• Thu, Sep 13 -- The Differential of a Smooth Map (finished), Computation in Coordinates (finished)


• Tue, Sep 18 -- Local Diffeomorphisms (subsection of Maps of Constant Rank), Submanifolds (different from textbook approach, see lecture notes)


• Thu, Sep 20 -- Submanifolds (continued, see lecture notes)


• Tue, Sep 25 -- Submanifolds (continued, see lecture notes)


• Thu, Sep 27 -- Submanifolds (continued, see lecture notes)


• Thu, Oct 4 -- Submanifolds (finished, see lecture notes)


• Tue, Oct 9 -- The Rank Theorem (subsection of Maps of Constant Rank), Submersions


• Thu, Oct 11 -- Embeddings


• Tue, Oct 16 -- Partitions of Unity (for finite covers on compact manifolds), first part of Theorem 6.15 (first four paragraphs)


• Thu, Oct 18 -- Sets of Measure Zero (started)


• Tue, Oct 23 -- Sets of Measure Zero (finished), The Tangent Bundle (up to and including Proposition 3.18), Whitney Immersion Theorem (Theorem 6.18, compact case)


• Thu, Oct 25 -- The Whitney Embedding Theorem (Theorem 6.13 + Theorem 6.15, compact case), Vector Fields on Manifolds (up to and including Proposition 8.1)


• Tue, Oct 30 -- Covectors, The Differential of a Function, Pullbacks of Covector Fields


• Thu, Nov 1 -- Restricting Covector Fields to Submanifolds (subsection of Pullbacks of Covector Fields), Line Integrals


• Thu, Nov 8 -- Multilinear Algebra (started)


• Tue, Nov 13 -- Multilinear Algebra (finished), Tensors and Tensor Fields on Manifolds (partially covered)


• Thu, Nov 15 -- Symmetric and Alternating Tensors, The Algebra of Alternating Tensors (started)


• Tue, Nov 20 -- The Algebra of Alternating Tensors (finished), Differential Forms on Manifolds


• Tue, Nov 27 -- Orientations of Vector Spaces, Orientations of Manifolds (started)


• Thu, Nov 29 -- Orientations of Manifolds (finished)


• Tue, Dec 4 -- Integration of Differential Forms (started)

Additional materials:

• Lecture notes on submanifolds, LaTeX source


• Introductory lecture notes on projective geometry by Thomas Baird