## Fall 2023

### MATH GR6263 – Topics in Algebraic Geometry

**Instructor**: John Morgan

**Title**: Differential Forms, Rational Homotopy Theory, and Smooth Complex Algebraic Varieties

**Day/Time:** TR 2:40PM – 3:55PM

**Room:** 507 Math

**Abstract**: The course will begin with a brief overview of homotopy theory including Postnikov Towers and localization in homotopy theory. We will then introduce differential graded algebras (DGA’s) over the rationals and prove that these form a model for rational homotopy theory. Lastly, the lectures will develop and use the Kahler identities for differential forms on compact Kahler manifolds to study the rational homotopy theory of smooth Complex Algebraic Varieties, both projective and quasi-projective.

Course Prerequisites: The level of the course will be a second-year graduate course. We shall assume a basic understanding of (i) homotopy theory and algebraic topology, including homology and cohomology as well as the fundamental group, and (ii) differential topology including the basics of differential forms on smooth manifolds. No knowledge of more advanced topics in these domains will be assumed.

### MATH GR8507 – Topics in Topology

**Instructor**: Soren Galatius

**Title**: Topics in high-dimensional topology

**Day/Time:** MW 11:40AM – 12:55PM

**Room:** 507 Math

**Abstract**: This course will cover classical topics in high dimensional manifold theory, starting with the h- and s-cobordism theorems. We will then discuss L-theory and surgery theory, roughly following parts of Wall’s book. These classical results attempt to describe “structure sets” — sets of diffeomorphism classes of manifolds of a given homotopy type. The rest of the course will discuss efforts to upgrade from structure sets to structure spaces.

The course should be accessible to PhD students familiar with the basics of differential topology and homotopy theory, for instance including the following concepts.

Differential Topology: smooth manifolds, embeddings, regular values and transversality, vector bundles. Prior encounters with Morse functions helpful, but will not be assumed.

Homotopy Theory: homotopy groups, weak equivalence, homotopy fiber. Prior encounters with simplicial sets helpful, but will not be assumed.

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