In this talk we will study the space of projective structures on a Riemann surface $X$. We will see that a projective structure $Z$ can be repackaged as a development-holonomy pair $(f,\rho)$ where $f: \tilde{Z} \to \mathbb{CP}^1$ is a locally Mobius immersion and $\rho: \pi_1(X) \to \mathrm{PSL}(2,\mathbb{C})$ is a holonomy map. If time permits we will see that the space of projective structures forms an affine space over the space of quadratic differentials $H^0(X,K^2)$.
In this talk we will prove that Riemann surfaces which are annuli are analytically isomorphic to certain cylinders. We will take a closer look at the hyperbolic annuli, and study the geodesics on them. (Chapter 3.2-3.3 Hubbard)
We continue to learn about Coxeter groups following Anne Thomas' "Geometric and topological aspects of Coxeter groups and buildings". There are nice combinatorial properties associated to a Coxeter system. For example, the Cayley graph of a Coxeter system must have nice symmetries and is thus a reflection system. We will visualize everything with D6.
In this talk I will give an elementary proof of the Morse inequalities, which establish a relationship between the homology of a space and the critical points of certain smooth functions on the space plus some local information around these critical points. Some general form of these inequalities can be used to compute the Poincare polynomial (Betti numbers) for moduli spaces of Higgs bundles (of fixed low ranks).
How does a change of coordinates affect the Riemannian metric? When are two Riemannian manifolds locally isometric? Riemann has a famous counting argument that suggests the metric is determined by $n(n-1)/2$ functions. In this talk we will see why this is true. In particular, we will see that for any 2-dimensional subspace of the tangent space, we can find an invariant quadratic function on the space in any normal coordinate system. In an $n$-dimensional tangent space there are $\binom{n}{2}$ 2-dimensional subspaces and hence we have Riemann's claim. In preparation for this goal, we will introduce the exponential map and the Riemann normal coordinates.
Given a very concrete surface in $\mathbb{R}^3$ , we can use the curvature of a surface curve to define the curvature of the surface at points of the curve in the directions the curve travels in. Although the motivation for the definition relies on curves, the curvature of a surface does not actually depend on specific surface curves. I will formally introduce the notions of normal curvature, principal curvatures, Gaussian curvature, and mean curvature of a surface and show you how to compute them. In the process, I will talk about things like the Gauss map, the shape operator, and the second fundamental form, which lead to the list of curvatures in the previous sentence. I will also discuss some geometric interpretations.
The goal of this presentation is to understand an important tool used to study intrinsic properties of a surface - the covariant derivative. First we define the covariant derivative and look at how to compute it. Next we see why the tool is appropriate for studying intrinsic geometry. And finally we shall see some applications and generalizations of this tool.