Explicit spectral gaps for random covers of Riemann surfaces

Spectral Geometry of Riemann Surfaces

Entropy on Riemann surfaces and the Jacobians of finite covers

This paper characterizes those pseudo-Anosov mappings whose entropy can be detected homologically by taking a limit over finite covers. The proof is via complex-analytic methods. The same methods show the natural map Mg → Ah, which sends a Riemann surface to the Jacobians of all of its finite covers, is a contraction in most directions.

Branched Covers of the Riemann Sphere

A (real) manifold of dimension n is a (Hausdorff, second countable) space which is locally homeomorphic to an open subset of Rn. If we wish to make a definition of a complex manifold, we could replace Rn by Cn, but then we see that we have simply given the definition of an even-dimensional real manifold. As with real differentiable manifolds, the key is to add some additional structure via a we...

Computing monodromy via continuation methods on random Riemann surfaces

We consider a Riemann surface X defined by a polynomial f(x, y) of degree d, whose coefficients are chosen randomly. Hence, we can suppose that X is smooth, that the discriminant δ(x) of f has d(d− 1) simple roots, ∆, and that δ(0) 6= 0 i.e. the corresponding fiber has d distinct points {y1, . . . , yd}. When we lift a loop 0 ∈ γ ⊂ C−∆ by a continuation method, we get d paths in X connecting {y...

Riemann Surfaces

Riemann introduced his surfaces in the middle of the 19th century in order to “geometrize” complex analysis. In doing so, he paved the way for a great deal of modern mathematics such as algebraic geometry, manifold theory, and topology. So this would certainly be of interest to students in these areas, as well as in complex analysis or number theory. In simple terms, a Riemann surface is a surf...