Fundamental group and covering spaces, van Kampen's theorem. Homology theory, Differentiable manifolds, vector bundles, transversality, calculus on manifolds. Differential forms, tensor bundles, deRham theorem, Frobenius theorem.

Course Goal: The goal of this course is to add breadth and formal training in communication to the

current cohort of geometry and topology students at Michigan State University. This course ran

successfully in Spring 2015 and in online format in Spring 2021.

Breadth: There are numerous articles, results, or methods (that will be referred to as classics) of

which students in geometry and topology should be aware. For instance, Milnor’s results on exotic

smooth structures on the 7-sphere, Thurston’s classification of surface diffeomorphisms, the Atiyah-

Singer Index theorem, Gromov’s work on pseudo-holomorphic curves, Quillen’s work on

homotopical algebra or rational homotopy theory, the Jones polynomial of knots and its

interpretation by Witten, the Gordon-Luecke knot complement theorem, etc. (see attached

bibliography from the previous course offering for a sample of possible topics). Many of these topics

do not fit into our regularly offered courses and could not constitute a stand-alone course. Others,

while deep enough for a course (or two), can’t be taught in such a format due to the sheer number

of such results and the limitations on scheduling. Despite this, the results and techniques are clearly

fundamental, and students in geometry and topology should be exposed to them.

Formal training in communication and dissemination: We have all been disappointed with poorly

delivered seminar and colloquia lectures and frequently allow this to inform our professional

opinion of the speaker. Having the ability to deliver excellent seminar-style lectures is an extremely

important skill, particularly for freshly minted Ph.D.s. Despite this, little formal effort is paid to the

training of students in this regard.

Course Details: Students enrolling in this course will have completed the 868/869 sequence (or

demonstrated this knowledge by passing the geometry/topology qualifying exam) and at least one

additional geometry or topology course. The course will meet as an entire group twice a week for

an hour and twenty minutes. However, much of the instruction for this course will occur outside of

these two official meetings as described below.

Each student will present at least two (three depending on the class size) lectures during the

semester. The class as a whole will analyze each of these talks in order to provide feedback for

improvement. The selection of topics, preparation of the lectures, and evaluation will proceed

roughly as follows:

1) Topics will be assigned based on an individual student's level of preparation and interests.

2) Each student will have meetings with Hedden while learning the material for their lectures in

order to discuss the mathematical content, fill in possible gaps in understanding, and to discuss

strategies for giving a successful lecture on that particular topic.

3) After learning the requisite material, each student will give a practice talk to some of the other

students in the class. The observing students will provide oral and/or written (in the form of notes

taken during the lecture) feedback to the lecturing student at this point. The lecturing student will

then make changes to their talk as they see fit based on the feedback from their peers.

4) The student will deliver their lecture to the entire class during the official course meeting time.

They will meet with Hedden afterwards to discuss the talk and possible strategies for improvement.

5) For each lecture, every student in the class will be required to write a short (approximately one

page) response to the mathematics. This is not intended to be a summary; rather, an opinion. The

purpose of the response is to encourage students to think about how a piece of mathematics fits

into their current understanding, internalize some of its ideas, and to stimulate them to think

creatively about how these ideas could be applied or further developed. Encouraged as a part, or

all, of the response, are questions which the mathematics raises for the student.

Fundamental group, covering spaces, and homology.

We will cover characteristic classes, following Characteristic Classes by Milnor and Stasheff. Toward the end, we may discuss bordism theories and/or Brown Representability.

This course will introduce the rich subject of equivariant topology, which studies topological spaces with

a group action. In recent years equivariant homotopy theory has seen broad and important applications

in mathematical areas such as algebraic K-theory, classification of high-dimensional manifolds, and

invariants of knots, links, and 3-manifolds. In this course we will discuss both equivariant cohomology and

equivariant homotopy theory. For a group G, we will introduce G-spectra and the equivariant stable

category. The course will also introduce "equivariant algebra," the study of algebraic objects such as

Mackey functors that arise from invariants of equivariant objects in topology.