Math 868 -- Fall 2024

Geometry and Topology

Class: MWF 3--3:50 pm in A130 Wells
Recitation: Th 10:20-11:40 in A318 Wells

Professor: T. ParkerOffice: D-216 Wells Hall parker@msu.edu
Office hours: Monday 12:40-1:30 Tuesday 11-12 Thursday 12:40-1:30 or by appointment
TA: Nick Robino robinoni@msu.edu Office hour: Tuesday: 1-2 am Thursday 3-4

Goals: This course introduces the intuition and techniques used to study manifolds. Manifolds are the natural setting for calculus in its most appealing and flexible form, and are the primary objects of study in much of modern geometry and topology.The course will start with a rapid introduction (which should be a review for most students) to multi-variable calculus, metric spaces and linear algebra. Then come the main topics of the course: differentiable manifolds and tangent spaces, vector bundles, calculus on manifolds, differential forms, tensor bundles, the deRham Theorem and cohomology groups. If time permits, we will cover the beginnings of Riemannian geometry.
Prerequisites: The official prerequisites are a 400-level course on Abstract Algebra and one on Real Analysis. In reality, the main prerequisite is a solid knowledge of multi-variable calculus and linear algebra and some knowledge of basic point set topology (open sets, compactness).
Textbook: Introduction to Smooth Manifolds by John M. Lee. Supplemental reading from An Introduction to Manifolds by Loring Tu. For additional books and material, see the Textbooks and Supplemental Material pages.
Exams and grades: There will be weekly homework assignments, an in-class Midterm Exam, and an in-person Final Exam. Course grades will be determined by the weighted sum of Homework and Quizzes (64%), Midterm (12%) and Final (24%).
Recitation: The (required) recitation section will cover additional material (coordinated with the lectures), provide help with homework problems, and practice with some qualifying exam problems.

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