Syllabus for Math 124

This syllabus is tentative and will be updated irregularly. The homework page will be updated on the regular basis.

Lectures	Sections in Text	Brief Description
Day 1, September 22	Chapter 1	Manifolds and Submanifolds, examples and properties, Manifolds with boundary
Day 2, September 27	Chapter 2	Differentiable structures, atlases, smooth manifolds, critical point, regular value, expression of functions in different coordinate systems,
Day 3, September 29	Chapter 2	Measure zero sets, Sard Theorem, presentations of functions of fixed rank, immersion, immersed and imbedded submanifolds,
Day 4, October 4	Chapter 2	Boundary of manifold as an imbedded submanifold. Preimage of a point under a constant rank mapping is a submanifold, GL(n), SL(n) as submanifolds
Day 5, October 6	Chapter 2	Covers, Refinements of covers. Partitions of unity and their applications, imbeddings of manifolds into Euclidian spaces
Day 6, October 11	Chapter 3	Tangent vectors, fibers and projection maps, n-dimensional vector bundles, equivalence of vector bundles
Day 8, October 13	Chapter 3	Trivial vector bundles, bundle maps, examples of trivial and nontrivial bundles. sections of vector bundles, vector fields, Euler class Tangent bundle. Euler characteristic of the manifold is the Euler class of the tangent bundle TM, tangent vectors as derivations.
October 18, No class
Day 8, October 19 x-hour instead of a class on 10/18		Tangent bundle. Euler characteristic of the manifold is the Euler class of the tangent bundle TM, tangent vectors as derivations.
Day 9, October 20	Chapter 3	Orientation of a bundle, orientability, summation of bundles and induced bundles (briefly if time permits)
Day 10, October 25	Chapter 4	Dual bundle, Cotangent bundle, coordinate description for the differential, covariant tensor fields
Day 11, October 27 TENTATIVELY the Take home Midterm exam is given out. It will be due Thursday November 3	Chapter 4	Tensors in local coordinates, contraction of tensors, covariant and contravariant functors
Day 12, November 1	Chapter 5	Integral curves, existence of solution theorems for differential equations without proofs, flows, compact support case, straightening of a vector field
Day 13, November 3	Chapter 5	Lie derivative and its properties, Lie bracket of vector fields and Lie algebra
Day 14, November 8	Chapter 5 and 6	Existence of coordinate system for tangent to vector fields with vanishing brackets, Distributions, integrable distributions
Day 15, November 10	Chapter 6	Frobenius Theorem and maximal integral submanifolds
Day 16, November 15	Chapter 7	Differential forms, differential of a form, closed and exact form, cohomology groups
x-hour November 16 Oral homework presentations
Day 17, November 17	Chapter 8	Bordism groups, integration Stokes Theorem, pairing between bordism groups and cohomology
Day 18, November 22	Chapter 9	Riemann metric, its existence via partition of unity length of curves
Day 19, November 29 TENTATIVELY the take home Final Exam will be distributed on December 1 and it will be due in the evening of December 7	Chapter 10	Lie groups and Lie algebras, one parametric subgroups, examples

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