Summer Minicourses 2024

Abstract. This Course’s main objective is to introduce a classical invariant of quadratic forms on Z/2-vector space, and its many applications. We will assume familiarity with 4-manifolds, and knot theory, but these lectures should be largely self-contained. We will start with a short review of Seifert Forms, Seifert Matrices, and S-equivalence. Then we will introduce quadratic forms over Z/2 and provide a complete invariant of these quadratic forms (The Arf-invariant). Next, we will see how we can use quadratic refinements of Seifert Matrices to give us an Arf-Invariant for knots. Furthermore, we will connect the Arf-Invariant to 4-dimensional topology first by proving the Arf-invariant obstructs knots being slice , then using the Arf-Invariant to prove a famous restriction of the signature of Spin 4-manifolds (Rokhlin’s Theorem). Lastly, we will provide an invariant for Integer Homology 3-spheres, and highlight recent research using the Arf-invariant.

Abstract. Cartan geometry is an extremely general framework, based on differential geometry, that encompasses Riemannian, conformal and projective geometries and many others. We will start by quickly introducing Lie groups, Lie algebras, the Lie group-Lie algebra correspondence, principal bundles, and differential forms. Next I’ll talk about Klein pairs, which formalize the notion of a geometry, and give several examples. After that we’ll cover the Maurer-Cartan form of a Lie group, its generalization by the principal bundle definition of Cartan geometry, and the concept of curvature in this context. The example of Riemannian geometry will provide motivation for this definition. Finally, we’ll introduce mutation, which allows us to translate between closely related geometries, like the geometries based on flat, elliptic, and hyperbolic space.

Abstract. Course will use Weibel - An Introduction to Homological Algebra. We will develop the basic theory of homological algebra, culminating in the definition of Tor and Ext functors for modules and abelian groups, along with some of their properties. We will develop the theory in the generality of abelian categories (I'll introduce these!), but one should always be able to pretend that we are in the category of modules over a ring.

Time permitting (which it probably won't) I'll introduce the notion of a spectral sequence as a gadget that measures the failure of derived functors to compose, and perhaps even mention derived categories as objects that completely fix this failure. The previous sentence should make sense to you once you've taken the course!

Prerequisites:
- general algebra: rings, modules: freeness, finite generation, tensor product.
- basic category theory: Categories, functors, natural transformations.

Things that will be nice to have but not necessary:
- categorical notions: universal properties, limits, colimits & adjoints.
- more algebra: localisation, exactness.
- familairity with LES in (co)homology for topological spaces.
- sheaves may make an appearance as a source of examples and motivation.

Abstract. This course will be an overview of some topics in the theory of characteristic classes: axiomatic treatments of Euler and Chern classes, a few applications, Chern-Weil theory for complex vector bundles, and its generalization to principal G-bundles.

The tentative outline for the course is:
Lecture 1: Vector bundles and their philosophy.
Problem Session 1: Proving basic facts about vector bundles.
Lecture 2: Oriented Vector bundles, Euclidean Vector bundles, and the Euler class.
Lecture 3: Complex Vector bundles and Chern Classes.
Problem Session 2: The Spitting principle.
Lecture 4: Connections and Principal bundles.
Lecture 5: Chern Weil Theory.
Problem Session 3: A few computations and applications.

As a prerequisite, the first half of my course will require a decent familiarity with, cohomology, Poincare duality, and differential topology (specifically, concepts like transversality, the tangent bundle, and the tubular neighborhood theorem). The differential topology aspects aren't essential to understand the actual content, but most of my examples come from differential geometry/topology.
The second half of my course will need comfort with the differential topology of Lie groups and their actions on smooth manifolds, as well as some understanding of differential forms, de Rham cohomology, and integration of forms. Lee's book on smooth manifolds is a good reference for the Lie groups and differential topology/geometry content, while Bredon's Topology and Geometry, Hatcher's Algebraic topology, or tom Dieck's Algebraic Topology should be more than enough for the required algebraic topology.

Abstract. We will introduce the notion of Gromov hyperbolic metric spaces along with their Gromov boundary at infinity and investigate the finitely generated groups which admit such structures on their Cayley graphs, which are called word-hyperbolic groups. The hyperbolic metric structure for a word-hyperbolic group has strong implications on the group theoretic properties of the group that heavily reflect the properties of isometry groups of hyperbolic space and their action on its visual boundary. We will survey the quasiconformal and quasi-Möbius structures on the boundary, ways to detect hyperbolicity such as Bowditch’s convergence group condition, the relationship between local connectivity of the boundary and splittings of the group, cohomological properties of the boundary, and results and questions regarding boundaries of small dimension..