belated update.

seven (7) new lectures available in PDF at the usual place. sorry for the delay.

to clarify matters, every monday or so is a double session of student talks for grade assessment. i've chosen not to take notes during these talks, but happily every speaker thus far has made LaTeX PDF handouts for the audience. if you would like some, then go bug these people:

a review of basic riemannian geometry (jason miller)

in which we examine the riemann curvature tensor through connections, and compute the sectional curvature of hyperbolic n-space to be constant (-1).

on liouville's theorem (aaron magid)

this is a proof of a rigidity theorem: given a domain in euclidean n-space (n ≥ 3) a thrice-differentiable conformal transformation is the restriction of a möbius transformation. the proof rests on differentiating to obtain an overdetermined system of pde, where our computations are motivated by appropriate curvature quantities from riemannian geometry.

the hyperboloid model for hyperbolic space (karl weintraub)

this is a construction of an equivalent model for hyperbolic n-space, using the lorentz metric on a light cone, defining an appropriate projection, and demonstrating that it is isometric to the disc model.

on the measurable riemann mapping theorem (marie snipes)

the outline: a quick introduction to sobolev spaces and quasiconformal mappings, then formulating the beltrami equation, and through means of clever singular integral operators (the cauchy and hilbert transforms), one proves that given two simply connected domains in the plane and a beltrami differential, there is a quasiconformal homeomorphism of the prescribed domains and with the prescribed dilatation.