Titles and Abstracts:

• James Farre

Title: Long curves and random hyperbolic surfaces

Abstract: We will fix some topological data, a pants decomposition, of a closed surface of genus g and build hyperbolic structures by gluing hyperbolic pairs of pants along their boundary. The set of all hyperbolic metrics with a pants decomposition having a given set of lengths defines a (3g-3)-dimensional immersed torus in the (6g-6)-dimensional moduli space of hyperbolic metrics, a twist torus. Mirzakhani conjectured that as the lengths of the pants curves tend to infinity, that the corresponding twist torus equidistributes in the moduli space. In joint work-in-progress with Aaron Calderon, we confirm Mirzakhani’s conjecture. In the talk, we explain how to import tools in Teichmüller dynamics on the moduli space of flat surfaces with cone points to dynamics on the moduli space of hyperbolic surfaces with geodesic laminations.

• Bill Goldman

Title: Cubic surfaces and non-Euclidean geometry

Abstract:  The classification of geometric structures on manifolds naturally leads to actions of automorphism groups, (such as mapping class groups of surfaces) on "character varieties" (spaces of equivalence classes of representations of surface groups). Just as surfaces of Euler characteristic -1 are the building blocks of surfaces, their character varieties are the building blocks of character varieties in general. They enjoy natural mapping class group-invariant Posson structures. The simplest examples are affine cubic surfaces, such as the Markoff surface

x^2 + y^2 + z^2 - x y z = 0

which parametrizes complete hyperbolic structures on the punctured torus. In general these actions are dynamically complicated. Another notable example is the

x^2 + y^2 + z^2 - x y z = 20

which relates to Clebsch's diagonal cubic surface in projective 3-space. Here the dynamics bifurcates from ergodic (with respect to the Poisson measure) when the constant term lies between 2 and 20, and when to wandering when the contant term is more thn 20 . This bifurcation can be understood in terms of lines on Clebsch's surface.

• Sara Maloni

Title: Pleated surfaces in PSL(d,C)

Abstract: Pleated surfaces are an important tool introduced by Thurston to study hyperbolic 3-manifolds, and can be described as piece-wise totally geodesic surfaces, bent along a geodesic lamination lambda. Bonahon generalized this notion to representations of surface groups in PSL(2,C),  and described a holomorphic parametrization of the resulting open charts of the character variety in term of shear-bend cocycles. 

In this talk I will discuss joint work with Martone, Mazzoli and Zhang, where we generalize this theory to representations in PSL(2,C). In particular, I will discuss the notion of d-pleated surfaces, and their holomorphic parametrization.

• Nathaniel Sagman

Title: Non-uniqueness of minimal surfaces in products of hyperbolic surfaces.

Abstract: We explain that from the data of an unstable equivariant minimal surface in R^n, one can obtain a maximal representation of a surface group into \prod_{i=1}^n PSL(2,R) such that the associated energy functional on Teichmueller space admits multiple critical points. In geometric terms, there is more than one minimal surface in the relevant homotopy class in the corresponding product of closed hyperbolic surfaces. An example of an unstable equivariant minimal surface in R^n is the lift of an unstable minimal surface in an n-torus. This is related to an important question in Higher Teichmueller theory. This is joint work with Vladimir Markovic and Peter Smillie.

Burton Jones Lecture by Bill Goldman: MAY 11, 2022 @ 4pm in Skye 284

(for Zoom information contact brian.collier@ucr.edu)