The current and upcoming topology and geometry courses at MSU:
Fundamental group, covering spaces, and homology.
Cohomology, products, duality, basic homotopy theory, bundles, obstruction theory, spectral sequences, characteristic classes, and other related topics.
The course will be an introduction to minimal surfaces and their applications in geometry. Along the way, students will learn some important methods and techniques in geometric analysis. We will start with the basic theory of submanifolds in a Riemannian manifold and then present the first and second variation formulas for the area functional. We will study many interesting examples of minimal surfaces. In the 2nd half of the course, we will study more advanced topics: the Simons identity, stability and Berstein type theorems, existence results (the Plateau problem and the Saks-Uhlenbeck theory), and Schoen-Yau's work on positive scalar curvature metrics using minimal surfaces.
Prerequisites: Riemannian geometry and basic elliptic PDE
This course will introduce students to a collection of interrelated topics in low-dimensional topology, organized around codimension one structures on surfaces and 3-manifolds. The course will discuss the basics of the theory of foliations, and with special emphasis on the setting of taut foliations on 3- manifolds. We will discuss the Thurston norm, its calculation using taut foliations, and the theory of sutured manifolds which allows for production of the latter. Fibered knots, links, and 3-manifolds will be particularly nice examples, and we will discuss dynamical properties of this class of objects. The NielsenThurston classification of surface automorphisms will be presented, and its relationship to 3-manifolds via the monodromy of fibrations. Floer homological invariants associated to area preserving surface diffeomorphisms will be discussed and calculated in terms of the Nielsen-Thurston classification. We will talk about existence and non-existence results for Anosov flows on 3-manifolds, and the more abundant examples of pseudo-Anosov flows stemming from the suspensions of corresponding surface automorphisms. We will discuss the relationship between these topics and contact and symplectic geometry in dimensions 3 and 4.
Since covering all of these topics is probably too much for one semester, those which we settle upon will be influenced by the interests of the participants and their background. Knowledge of topics covered in the first-year geometry and topology qualifying sequence (868-869) and math 960 will be assumed.
Differentiable manifolds, vector bundles, transversality, calculus on manifolds. Differential forms, tensor bundles, deRham theorem, Frobenius theorem.
A continuation of Algebraic Topology I with various topics.