Talks

A hyperbolic quasifuchsian (or more generally convex co-compact) manifold $M$ contains a smallest non-empty geodesically convex subset, its convex core. The boundary of this convex core has a hyperbolic induced metric, and is pleated along a measured geodesic lamination. Thurston asked whether the induced metric, or the the measured pleating lamination, uniquely determine $M$. In the first part, we will explain why the answer is positive for the measured pleating lamination (joint w/ Bruno Dular). In the second part, we will put this problem in a more general frramework concerning the boundary data of convex subsets in hyperbolic manifolds or in hyperbolic space.

In 1914, Ramanujan wrote down 17 intriguing formulas for 1/pi, which motivated the modern-day machinery of computing trillions of digits of pi. Most of these formulas lay unproven until the 1980s. The Canadian Borwein brothers wrote a comprehensive treatise proving these formulas in the 1980s. We can now ask, “What is the physics behind Ramanujan’s pi”? I will argue that the physics connection can be found via logarithmic conformal field theories, for instance, those studied in the fractional quantum hall effect, polymers, and percolation. The CFT connection gives rise to an infinite number of new formulas for pi. In contrast, the myriad Ramanujan formulas provoke us into looking into faster converging basis for conformal correlators, which is, in turn, provided by the stringy dispersion relation mentioned above.

The cosmic neutrino background is like the cosmic microwave background, but less photon-y and more neutrino-ey. The CNB is also less talked about than the CMB, mostly because it's nearly impossible to detect directly. But if it could be detected, it would be interesting in several ways that are discussed.

We depict harmonic map ray structures on Teichmüller space as a geometric transition between Teichmüller ray structures and Thurston geodesic ray structures

As an application, while there may be many Thurston metric geodesics between a pair of points in Teichmüller space, we find that by imposing an additional energy minimization constraint on the geodesics, thought of as limits of harmonic map rays, we select a unique Thurston geodesic through those points. There are applications to the envelopes of Thurston geodesics between a pair of points.

Anosov representations can be considered a generalization of convex-cocompact representations for groups of higher-rank. In this talk we are considering connected components of Anosov representations from the fundamental group of a closed hyperbolic manifold N, and which contains Fuchsian representations, and their associated domains of discontinuity. We will prove that the quotient of these domains of discontinuity are always smooth fiber bundles over N. Determining the topology of the fiber is hard in general, but we are able to describe it for representations in Sp(4,C), and for the domain of discontinuity in the space of complex Lagrangians in C^4 by using the classification of smooth 4-manifolds. This is joint work with Daniele Alessandrini, Nicolas Tholozan and Anna Wienhard.

We study the growth of log-spectra in subgroups of SL(n,R). Specifically, we study their extremal (rather than statistically predominant) behavior, with connections to the topic of ergodic optimization in dynamics. The main theme is that extremal elements seem to be quite special, similar to how laminations play a special role in the study of Thurston's Lipschitz metric on Teichmüller space.
Joint work with J.Danciger and F.Kassel.