**MATH 439-001 40631
ST: Intro to Diff Manifold **

**Class time
MWF 11:00 am-11:50am Location SMLC 356.**

**NOTE: class and office hours will be through Microsoft Office Teams at the usual times.**

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** Instructor: Dimiter Vassilev Office : ** SMLC, Office 326 **Email: vassilev@unm.edu
Phone
Number**:
505 277 2136

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** Textbook**: Introduction to Geometry and Topology, ISBN 9783034809825/ 3034809824, Ballmann, Werner

(free on the UNM network)

**Supplemental books:**

- Baez J., Muniain J., Gauge Fields, Knots and Gravity,
- do Carmo M.P. Differential geometry of curves and surfaces. (free on the UNM network)
- Lafontaine J., An Introduction to Differential Manifolds (free on the UNM network)
- Hamilton M., Mathematical Gauge Theory, (free on the UNM network).

** Description**: The main goal of the course is to familiarize you with geometric structures through a gentle introduction to some topics motivated by physics and geometry, such as, Maxwell's Equations, Manifolds, Vector Fields, Differential Forms and DeRham Theory, Lie groups and Lie Algebras, Bundles and Connections, Curvature and the Yang-Mills Equation. **Prerequisites**: 264, 321. 431 will be helpful.

** Please
note the following guidelines for the course:**

**Grades: **The final grade will be determined by attendance (100
pts), homework (100 pts), one midterm exam (100 pts) and
a final exam (100 points). The Final Exam score will
replace the midterm score if the latter is lower than the Final Exam score. The exams will mainly cover material from Part I. All grades will
be posted on UNM Learn.

** Homework: **You
can work together on the homework, but you do need to write up your own
solutions in your own words. To help the grader, please __write__ __your
solutions up neatly and clearly__ and __staple the sheets__.
It is best if you work and
discuss the problems together and with me as questions arise. __The highest 10
homeworks will be counted.__

** Missed
Exams: ** Make-up exams can be
arranged for exams missed with a VALID excuse (illness, family emergency,
active participation in scholarly or athletic activities), and ONLY if prior
notice is given.

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** Disability
Statement:** We will accommodate students with documented disabilities. During the first two
weeks of the semester, those students should inform the instructor of their
particular needs and they should also contact Accessibility Services in Mesa
Vista Hall, Room 2021, phone 277-3506. In addition, they should see CATS-
Counseling and Therapy Services;

**Syllabus
and Homework – Spring 2020 **

**(please check after class as advanced postings of homework
could change)
**

Class Week |
Topics and Section |
Homework (due the 2nd class of the following week) in MS Teams |

1. Jan 20 | HW | |

2. Jan 27 | Manifold construction lemma. Smooth bump functions, smooth partition of unity. Lie groups. Tangent vectors. | HW |

3. Feb 3 | The tangent space. | HW |

4. Feb 10 | The tangent bundle. The rank theorem. Discussion of HW Problem 3.4. Injective immersions and embeddings. | HW |

5. Feb 17 | Embeddings - slice theorem; level sets of maps; Lie subgroups | HW |

6. Feb 24 | Closed Lie subgroups. Examples. Quaternions, the group Sp(1). | HW |

7. March 2 | SU(2)=Sp(1) =Spin (3) and SO(3). Spin(4) and SO(4). The Lie bracket. | HW |

8. March 9 | F-related vector fields. The Lie algebra of a Lie group. Free and proper Lie group actions and quotient spaces. | HW Homework 8 due March 25 |

9. March 16 | Spring Break | |

10. March 23 | Principle fiber bundles. Transition functions. Vector bundles. | HW9 |

11. March 30 | Vector bundles - examples, constriction, cotangent bundle, differential forms. | HW10 |

12. Apr 6 | Differential forms and tensors. Midterm Exam (take home) |
HW11 |

13. Apr 13 | Derivations and contractions of tensor fields. Derivations and skew-derivations of differential forms - interior product, exterior derivative, Lie derivative. de Rham cohomology. | HW12 |

14. Apr 20 | Cohomology and homotopy. Complete vector fields and their flows, the Lie derivative. Cohomology of R^n. Orientation and volume form. Stokes' theorem on a manifold without boundary | HW13 |

15. Apr 27 | Stokes' theorem. Computing cohomology, cohomology with compact support. Semi-Riemannian metrics. | HW14 |

16. May 4 | Semi-Riemannian metrics, Hodge-*. Geometric form of Maxwell's system. | |

Wednesday May 13 | Final Exam 10:00am.‐12:00pm, available in Teams at 9:30am. |

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