Random Lectures in Geometry

Schedule and Information

Thursday June 18th:
16:00 - 16:45: Ara Basmajian
17:00 - 17:45: Juan Souto

Friday June 19th:
11:30 - 12:15 Robert Young
14:00 - 14:45 Greg McShane
15:30 - 16:15 Jeff Brock

All lectures will take place in lecture room 2.52 (in the Physics building).

Everyone is welcome but please let one of the organizers know if you plan on participating (Hugo Parlier or Stefan Wenger).

There will be a dinner on Thursday night.

Titles and Abstracts:

Ara Basmajian: Arcs, orthogeodesics, and deformation spaces

Let X be a compact hyperbolic surface with either geodesic or horocyclic boundary. The homotopy class (rel the boundary) of a non-trivial arc from the boundary to itself can be realized by an orthogeodesic - a geodesic segment perpendicular to the boundary at its initial and terminal points. This talk is about a special subclass of orthogeodesics called primitive orthogeodesics. In work with Hugo Parlier and Ser Peow Tan we show that the primitive orthogeodesics arise naturally in the study of maximal immersed pairs of pants in X and are intimately connected to regions of X in the complement of the natural collars. These considerations lead to continuous families of equations (so called identities) that remain constant on the deformation space of hyperbolic structures.

Juan Souto: Counting geodesics in the punctured torus

Let C be a, possibly non-simple, closed geodesic in a hyperbolic punctured torus X and let n_C(L) be the number of geodesics in X in the mapping class group orbit of C and of length at most L. I will prove that the limit when L tends to infinity of n_C(L)/L^2 exists and that in fact, the ratio between any two such limits for different choices of C is independent of X. This is joint work with Viveka Erlandsson.

Robert Young: Filling multiples of embedded curves

Filling a curve with an oriented surface can sometimes be "cheaper by the dozen". For example, L. C. Young constructed a smooth curve drawn on a projective plane in R^n which is only about 1.5 times as hard to fill twice as it is to fill once and asked whether this ratio can be bounded below. We will use methods from geometric measure theory to answer this question and pose some open questions about systolic inequalities for surfaces embedded in R^n.

Greg McShane: Renormalized volume and the double limit theorem

Schlenker and Krasnov have established a remarkable Schlaffli-type formula for the (renormalized) volume of a quasi-Fuchsian manifold. Using this, classical results in complex analysis and convergence in Thurston's Double Limit Theorem one obtains explicit upper bounds for the volume of a mapping torus in terms of the translation distance of the monodromy on Teichmueller space. We will explain Brock-Bromberg's approach to the double limit theorem and in particular the "coarse geometry" of the convex core of a quasi fuchsian manifold.

Jeff Brock: On the geometry of random Heegaard splittings

Using random walks on the mapping class group, Dunfield and Thurston defined a notion of 'random Heegaard splittings' in an effort to understand the likelihood of Thurston's 'virtual fiber conjecture'. In this talk I'll discuss a proof of their conjectured almost sure hyperbolicity and linear volume growth with respect to word-length of a random Heegaard splitting. This is joint work with Igor Rivin and Juan Souto.

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