# MTH 868 Geometry and Topology I, Fall 2017

**Lectures:**MWF 9.10am-10am A334 Wells Hall

**Instructor:**Thomas Walpuski

**Text book:**John M. Lee

*Introduction to Smooth Manifolds*

Here is a list of topics I plan to cover throughout the semester.

All homework sets are due at the beginning of the class on the date indicated below. You are encouraged to work in groups to find the solutions, but you need to write them up yourself.

Homework # | Due |
---|---|

Homework 10 | 2017-12-06 |

Homework 9 | 2017-11-20 |

Homework 8 | 2017-11-10 |

Homework 7 | 2017-11-03 |

Homework 6 | 2017-10-27 |

Homework 5 | 2017-10-13 |

Homework 4 | 2017-10-06 |

Homework 3 | 2017-09-29 |

Homework 2 | 2017-09-22 |

Homework 1 (PDF,LaTeX source,Hint) | 2017-09-11 |

Date | Topics | Reading |
---|---|---|

2017-12-4 | Variation of curvature, gauge invariance of curvature, Yang--Mills functional | |

2017-12-1 | Gauge transformations, covariant exterior derivatives, and curvature | |

2017-11-29 | Covariant Derivatives on associated bundles | |

2017-11-27 | Covariant Derivatives | |

2017-11-22 | Hodge Theory | |

2017-11-20 | De Rham cohomology of connected sums | pp. 440-466 |

2017-11-17 | Computation of the de Rham cohomology of CP^{n} |
pp. 440-466 |

2017-11-15 | PoincarĂ© duality | pp. 440-466 |

2017-11-13 | Computation of the de Rham cohomology of S^{n} |
pp. 440-466 |

2017-11-10 | The Mayer–Vietoris Theorem | pp. 440-466 |

2017-11-8 | Homotopy invariance of de Rham cohomology | pp. 440-466 |

2017-11-6 | De Rham cohomology | pp. 440-466 |

2017-11-3 | Stokes' Theorem | pp. 411-439 |

2017-11-1 | Stokes' Theorem | pp. 411-439 |

2017-10-30 | Integration on Manifolds | pp. 400-411 |

2017-10-27 | Integration on Manifolds | pp. 400-411 |

2017-10-25 | Lie derivatives of differential forms, Cartan's magic fomula | pp. 372-373, notes |

2017-10-23 | Exterior Derivative | pp. 362-373 |

2017-10-20 | Differential Forms (cont.) | pp. 349-361 |

2017-10-16 | Differential Forms | pp. 349-361 |

2017-10-13 | Vector bundle constructions | pp. 249-268 |

2017-10-11 | Multilinear algebra (cont.) | pp. 304-316 |

2017-10-09 | Multilinear algebra | pp. 304-316 |

2017-10-06 | Frobenius's Theorem | pp. 496-501 |

2017-10-04 | Flows and the Lie derivative | pp. 209-236 |

2017-10-02 | Vector fields and integral curves | pp. 205-209 |

2017-09-29 | Partitions of unity and the Whitney embedding theorem | pp. 40-47 and 131-136 |

2017-09-27 | Submanifolds (cont.) | pp. 98-123 and 657-662 |

2017-09-25 | Submanifolds (cont.) | pp. 98-123 and 657-662 |

2017-09-22 | Submanifolds | pp. 98-123 |

2017-09-20 | Vector fields | pp. 174-189 |

2017-09-18 | The tangent bundle | pp. 50-75 |

2017-09-15 | Review session with the TA | |

2017-09-13 | first student seminar | |

2017-09-11 | Tangent vectors and tangent spaces: the geometer's and the algebraist's approach |
pp. 50-75, my notes |

2017-09-08 | Tangent vectors and tangent spaces: the physicist's approach |
pp. 50-73 |

2017-09-04 | Smooth maps between smooth manifolds | pp. 32-40 |

2017-09-01 | What is a smooth manifold? | pp. 10-29 |

2017-08-30 | What is a topological manifold? | pp. 1-10 and Appendix A |

Here is a list of possible topics for the student seminar. If you have your own ideas for topics, please, don't hesitate to propose them.

The student seminar takes place on Thursday 2-3pm or Friday 3-4pm in C117. For details, please, see the schedule below.

You can sign up for the student seminar by simply emailing me at, talking to me after class, or coming to my office.

Date | Topics | Speakers | Room |
---|---|---|---|

2017-12-8 3-4pm | Linking numbers | Aaron Roach | C117 WH |

2017-12-7 2-3pm | Hopf–Rinow Theorem | Dimitris Vardakis | C117 WH |

2017-12-1 3-4pm | Geodesics in Riemannian Geometry | Zack Bezemek | C117 WH |

2017-11-30 2-3pm | Lie groups actions and de Rham cohomology | Jon Miles | C117 WH |

2017-11-17 3-4pm | Symplectic geometry and Hamiltonian flows | Mike Annunziata and Keshav Sutrave | C117 WH |

2017-11-16 2-3pm | Complex geometry | Christopher Grow | C117 WH |

2017-11-3 3-4pm | The connected sum of manifolds | David Storey | C117 WH |

2017-11-2 2-3pm | Morse theory | Alex Ginsberg | C117 WH |

2017-10-26 2-3pm | Brouwer's Fixed-Point Theorem | Mark Roach | C117 WH |

2017-10-5 2-3pm | Lie groups, Lie algebras | Franciska Nekaien, Rachel Domagalski | C117 WH |

2017-09-29 3-4pm | classification of 1–manifolds | Joshua Schroeder | C117 WH |

2017-09-22 3-3.30pm | The Grassmann manifold | Stan Tan | C117 WH |

2017-09-21 2.10-3.40pm | point-less topology | Brian Pinsky | C117 WH |

2017-09-13 9-10am | Manifolds with boundary, Quotient Manifolds | Alex Wilson, Joe Melby | A334 WH |

If you cannot make it to the student seminar or simply don't want to give a presentation in the student seminar, you can alternatively do a little project which will count for 10% of the grade.

Final Grade
=
20% Midterm + 25% Final Exam + 45% Homework + 10% Student Seminar.