Student Analysis

We will discuss the density of closed geodesics on hyperbolic manifolds and the trace of the resolvents as applications of Selberg's trace formula. Time permitting, we may also define Selberg's zeta function and/or prove the prime geodesic theorem.

We will study properties general hyperbolic surfaces. We will discuss the spectrum of a compact surface, Selberg's pre-trace and trace formulas. Time permitting, we may also study the heat kernel and Weyl's law.

We will discuss the Green's function on the hyperbolic plane and derive a formula which (in following talks) will ultimately descend to the Selberg trace formula on hyperbolic surfaces. We will show how to get a version of the trace formula on a hyperbolic cylinder, and hint at the connection to…

I'll introduce trace formulas, with the Poisson summation formula as a model case and a trace formula for S^2 as a slightly more involved example. Time permitting, I'll also start developing some of the theory we'll need for the hyperbolic plane.

Last time we have seen that the conformal area of a compact surface introduced by Li—Yau gives an upper bound of the first eigenvalue. In this talk, I will talk about more results on the first eigenvalue of compact surfaces, including Hersch’ result on S², Li—Yau’s result on RP²…

We will look at the notion of conformal volume introduced by Li-Yau for a compact Riemannian manifold. One use of this is that conformal area of a compact surface gives an upper bound of the first eigenvalue, and rigidity results for the equality case. This also allows us to estimate the area of…

From the uniformization of simply connected Riemann surfaces it is only a small step to uniformizing closed Riemann surfaces. We will also discuss how branched covering maps between Riemann surfaces can be realized up to homeomorphisms as holomorphic maps. An application is the classification of…

This week, we will describe another proof of the Koebe uniformisation theorem, which states that given a simply connected Riemann surface W, there exists a one-to-one analytic map of W onto the open disc, complex plane, or Riemann sphere. How do we know which of the three W is? This is…

How does the spectral theory of the Laplacian on a surface relate to its geometry? In this expository talk, I will present a remarkable answer to this question, as found by Osgood, Phillips and Sarnak in 1988 (while at Stanford!). In particular, by carefully studying the eigenvalues of the…

I will introduce a few motivating questions in projection theory (Marstrand projection thm, Falconer’s…