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:  9/24	I.1-I.4 and III.1	Definition of manifolds and differentiable manifolds. Examples. Products of manifolds, open submanifolds, the sphere, cut and paste techniques.
Day 2:  9/26	III.2 and III.3	Further examples of manifolds: projective plane, Grassman manifolds, homeomorphism between G(k, n) and G(n-k, n). Differentiable functions and mappings.
Day 3:  9/29	III.3 and III.4	Examples of differentiable mappings, diffeomorphism, rank of mappings, immersion, submersions, imbeddings.
Day 4:  10/1	III.5 and III.6	Submanifolds, preimage of a point as a submanifold, constant rank mappings. Definition of a Lie group and some simple examples.
Day 5:  10/3	III.6 and start III.7	Sl(n,R) and O(n), subgroups of Lie groups and closed subgroups of Lie groups. The action of a Lie group on  a manifold.
Day 6:  10/6	III.7 and III.8	Examples of Lie group actions. Discrete group actions. Quotient spaces of Lie groups by discrete subgroups.
10/7 x-hour	Oral presentation of homework exercises.	.
Day 7:  10/8	III.9	Covering Manifolds Examples. Loose ends for Chapter III
10/10 No class
Day 8:  10/13	IV.1	Tangent space to a manifold, velocity vector of a curve, rank of mapping in terms of tangent space mappings.
Day 9 10/14 x-hour instead of Friday 10/10r	IV.2 and start IV.3	Vector fields. Lie groups are parallelizable. Parallelizability of spheres. Euler class. One parameter group actions. Integral curves. Integral curves with vanishing vector fields.
Day 10:  10/15	IV.3 and IV.4	Local one parameter actions. Examples. The existence theorem for solutions of ODE. Uniqueness of integral curves.
Day 11:  10/17	IV.5 and start IV.6	Examples of one parameter group actions on manifolds. Complete vector fields and left invariant vector fields on Lie groups. Correspondence between one parameter subgroups of a Lie group G and the elements of T_eG
Day 12:  10/20	IV.6 and IV.7	Exponent map and the Lie algebra of vector fields on  a manifold.
Day 13:  10/21 x-hour instead of Friday 10/24	IV.8	Frobenius’s Theorem
Day 14:  10/22	IV.9	Homogeneous spaces.
10/24 Homecoming weekend no class
Day 15:  10/27 Takehome Midterm Exam and the lecture	V.1 and start V.2	Covector fields on manifolds. Bilinear form fields.
Day 16:  10/29	V.2 and V.3	Riemannian metric and Riemannian manifolds as metric spaces (sketch of prof).
Day 17:  10/31	V.4	Partitions of Unity, existence of Riemannian metric. imbeddings of compact manifolds into Euclidean spaces.
Day 18:  11/3	V.5	Tensor fields on manifolds, symmetrizing and alternating transformation.
Day 19:  11/5	V.6	Multiplication of tensor fields. Exterior multiplication. Exterior algebra of alternating tensors on manifolds. Pull-back of exterior algebras.
Day 20:  11/7	V.7	Orientation of manifolds, orientation covering, volume element.
Day 21:  11/10	V.8	Exterior differentiation. Loose ends. Differential forms interpretation of the Frobenius’s Theorem if we have time.
Day 22:  11/12	VI.1 and most of VI.2	Riemann Integral in R^n and integrals of forms on manifolds
Day 23:  11/14	VI.2 and VI.3	Integrals of functions on Riemannin manifolds. Some words about integration on Lie groups. Orientability of Lie groups. Some statements about invariant metrics on Lie groups (possibly without proof.).
Day 24:  11/17	VI.4	Manifolds with boundary. Induced orientation on the boundary of an oriented manifold.
Day 25:  11/19	VI.5	Stokes Theorem. Examples: Greene, Gausss etc. calculus Theorems as particular cases of the Stokes Theorem.
Day 26:  11/21	VI.6 and start VI.9	Homotopy of Mappings, fundamental group, contractible manifolds (shortly). Integration of forms along homotopic paths. Fundamental groups of a covering. Normal coverings. (Very few proofs.)
Day 27:  11/24	VI.7	Exact and closed forms.De Rhap cohomology groups. Behavior of cohomology groups of manifolds under mappings. The top cohomology group of a closed oriented manifolds. Homotopy operator.
Thanksgiving recess 11/25-11/30
Day 28:  12/1	VI.8	Applications of de Rham groups. Mappings from disks to their boundary, orientability of projective spaces. Cohomology groups of Lie groups (if we have time).
Day 29:  12/3		Bordism groups and a few tricks about integration. Loose ends.