Fundamental group, covering spaces, and homology.

This is an introductory course on ∞-categories and higher algebra. Higher algebraic structures in

homotopy theory encode rich geometrical information and higher categories provide convenient

framework to study the algebraic structures. In recent years, besides in algebraic topology and homotopy

theory, ∞- categories have numerous applications to low-dimensional topology, algebraic geometry,

number theory, representation theory and algebraic K-theory. In this framework, abelian groups are

replaced by the so-called ∞-loop spaces, and we shall see that all homology and cohomology theories are

entirely determined by ∞-loop spaces. We will also cover more classical tools from homotopy theory such

as simplicial sets, model categories, homotopy colimits and operads.

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.

Prerequisites (suggested): MTH 868, MTH 869, Geometry and Topology I and II (additional Riemann

geometry and topology courses may also be helpful)

Course Description: Contact topology is the study of manifolds equipped with a certain geometric

structure on their tangent bundles (a family of hyperplanes in their tangent spaces, satisfying a technical

condition). These contact manifolds arise naturally when studying Hamiltonian dynamics.

This course will introduce contact manifolds, and we will then prove some of the fundamental theorems

in this area, including Darboux’s theorem that shows that all contact manifolds are locally standard (this

result shows that contact topology and Riemannian geometry are quite, quite different from each other).

We will then introduce some of the other fundamental objects in contact topology: Legendrian and

transverse knots and their relatives.

We will then shift gears to study contact topology in low dimensions. It turns out that already in low

dimensions (meaning dimension 3 for the most part), contact manifolds already have an extremely rich

structure, and their study is very hands-on and visual. We will cover Eliashberg’s fundamental theorem

that there is a unique tight contact structure on S3 . This will be proved using convex surface theory.

If time allows, we will mention some of the many connections of modern Floer theories with contact

topology (in low and high dimension). This is a rapidly expanding field. Some important references will be:

Resources:

1) An Introduction to Contact Topology by Hansjörg Geiges

2) Topological Methods in 3-dimensional Contact Geometry by Patrick Massot.

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.

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.

Cohomology, products, duality, basic homotopy theory, bundles, obstruction theory, spectral sequences, characteristic classes, and other related topics.

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.

This is an one semester course in basic 3-manifold topology. Topics we will discuss will be chosen

from:

1) Dehn’s Lemma and the Loop and Sphere theorems,

2) Prime decompositions of 3-manifolds,

3) Incompressible surfaces,

4) Triangulations and normal surfaces

5) Seifert fibered spaces and their classification

6) Toroidal decompositions and JSJ decompositions of 3-manifolds.

7) geometric structures of 3-manifolds and in particular hyperbolic geometry

Note: In examples and specific constructions often particular emphasis will be paid to 3-manifolds

that are complements of knots in the 3-sphere.

Continuation of MTH 960.

We will cover topics in Diophantine geometry, including some subset of the following: theory of heights, Diophantine approximation (Roth's theorem, Schmidt subspace theorem, Vojta's conjectures), rational and integral points (Mordell-Weil theorem, Siegel's theorem, Falting's theorem). Basic references for the course topics include the book by Hindry and Silverman and the book by Bombieri and Gubler.

The field of topological data analysis involves creating tools for analyzing and encoding the shape and structure of data. These tools include algebraic representations, such as persistent homology and the Euler characteristic transform, as well as graphical representations, such as the Reeb graph, mapper graph, and merge tree. When analyzing data with these tools, it is important to be able to make metrics available, in particular to be able to understand the behavior of the representations in the face of noise. One such important idea arising from this view is the interleaving distance. What started as a metric for comparing persistence modules as a generalization of the bottleneck distance for persistence diagrams has become a much more broad concept. Category theoretic formulations allow for translation of these ideas across different representations of data, including both the algebraic and graph-based signatures of data. In this course, we will both introduce the relevant mathematical constructions from TDA and explore the variations of the definition of interleavings on these frameworks.

This course will cover a variety of techniques, and their applications to questions in low-dimensional topology and geometry. Topics will touch on many aspects of knot theory and 3- and 4-manifolds, and may include: Dehn surgery, unknotting numbers, contact and symplectic geometry, taut foliations and sutured manifold theory, minimal genus problems (e.g. Thurston norm, Milnor and Thom conjectures), Reidemeister-Turaev torsion, the Casson invariant, Floer homology, concordance and homology cobordism groups, mapping class and braid groups. Participants interests will be taken into account when decisions about the emphasis placed on particular subjects are made.

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.

Riemannian metrics, connections, curvature, geodesics. First and second variation, Jacobi fields, conjugate points. Rauch comparison theorems, Hodge theorem, Bochner technique, spinors. Further topics on curvature or submanifold theory.

Cohomology, products, duality, basic homotopy theory, bundles, obstruction theory, spectral sequences, characteristic classes, and other related topics.

This course intends to introduce students to aspects of Symplectic Geometry (and probably Contact Geometry, its odd dimensional cousin). Symplectic Geometry emerged from Classical Mechanics, and many peculiar phenomena were discovered since the 1980's which sets it apart from Riemannian Geometry. In particular, Rigidity and Flexibility coexist side by side. Versions of Gromov's h-principle which will be introduced in the course reduce geometric classifications to topological results. They yield constructive results and are a manifestation of Flexibility while Rigidity yields obstructive results. The aim of the course is to introduce students to some of these phenomena while keeping prerequisites rather modest (MTH868 should be sufficient).