Topic #1 (Wang): Holography of the Loewner Energy; Topic #2 (Hutchcroft): Critical Behaviour in Long-Range and Hierarchical Percolation
Topic #1 (Wang): The Loewner energy is a Möbius invariant quantity that measures the roundness of Jordan curves on the Riemann sphere. It arises from large deviation deviations of SLE0+ and is also a Kähler potential on the Weil-Petersson Teichmüller space. Motivated by AdS/CFT correspondence and the fact that Möbius transformations extend to isometries of the hyperbolic 3-space $H^3$, we look for quantities defined geometrically in $H^3$ which equal the Loewner energy of a curve in the conformal boundary. We show that the Loewner energy equals the renormalized volume of a submanifold of $H^3$ constructed using the Epstein surfaces associated to the hyperbolic metric on both sides of the curve. This is a work in progress with Martin Bridgeman (Boston College), Ken Bromberg (Utah), and Franco Vargas-Pallete (Yale).
Topic #2 (Hutchcroft): Statistical mechanics models undergoing a phase transition often exhibit rich, fractal-like behaviour at their critical points, which are described in part by critical exponents, the indices governing the power-law growth or decay of various quantities of interest. These exponents are expected to depend on the dimension but not on the microscopic details of the model such as the choice of lattice. After much progress over the last 30 years, we now understand two-dimensional and high-dimensional models rather well, but intermediate dimensions such as three remain mysterious. I will discuss these issues in the context of long-range and hierarchical percolation, and in particular how we can now compute some critical exponents for the hierarchical model in all dimensions.