The cover of the December 2016 AMS Notices shows an eye-like region picked out by blue and red dots and surrounded by green rays. The picture, drawn by Yasushi Yamashita, illustrates Gaven Martin’s search for the smallest volume 3-dimensional hyperbolic orbifold. It represents a family of two generator groups of isometries of hyperbolic 3-space which was recently studied, for quite different reasons, by myself, Yamashita and Ser Peow Tan.

After explaining the coloured dots and their role in Martin’s search, we concentrate on the green rays. These are Keen-Series pleating rays which are used to locate spaces of discrete groups. The theory also suggests why groups represented by the red dots on the rays in the inner part of the eye display some interesting geometry.
 

The law of quadratic reciprocity and the celebrated connection between modular forms and elliptic curves over Q are both examples of reciprocity laws. Constructing new reciprocity laws is one of the goals of the Langlands program, which is meant to connect number theory with harmonic analysis and representation theory.

In this talk, I will survey some recent progress in establishing new reciprocity laws, relying on the Galois representations attached to torsion classes which occur in the cohomology of arithmetic hyperbolic 3-manifolds. I will outline joint work in progress on better understanding these Galois representations, proving modularity lifting theorems in new settings, and applying this to elliptic curves over imaginary quadratic fields.

Nested commutators of differential operators appear frequently in the numerical solution of equations of quantum mechanics. These are expensive to compute with and a significant effort is typically made to avoid such commutators. In the case of Magnus-Lanczos methods, which remain the standard approach for solving Schrödinger equations featuring time-varying potentials, however, it is not possible to avoid the nested commutators appearing in the Magnus expansion.

We show that, when working directly with the undiscretised differential operators, these commutators can be simplified and are fairly benign, cost-wise. The caveat is that this direct approach compromises structure -- we end up with differential operators that are no longer skew-Hermitian under discretisation. This leads to loss of unitarity as well as resulting in numerical instability when moderate to large time steps are involved. Instead, we resort to working with symmetrised differential operators whose discretisation naturally results in preservation of structure, conservation of unitarity and stability
 

Multi-index methods are a generalization of multilevel methods in high dimensional problem and are based on taking mixed first-order differences along all dimensions. With these methods, we can accurately and efficiently compute a quadrature or construct an interpolation where the integrand requires some form of high dimensional discretization. Multi-index methods are related to Sparse Grid methods and the Combination Technique and have been applied to multiple sampling methods, i.e., Monte Carlo, Stochastic Collocation and, more recently, Quasi Monte Carlo.

In this talk, we describe and analyse the Multi-Index Monte Carlo (MIMC) and Multi-Index Stochastic Collocation (MISC) methods for computing statistics of the solution of a PDE with random data. Provided sufficient mixed regularity, MIMC and MISC achieve better complexity than their corresponding multilevel methods. We propose optimization procedures to select the most effective mixed differences to include in these multi-index methods. We also observe that in the optimal case, the convergence rate of MIMC and MISC is only dictated by the convergence of the deterministic solver applied to a one-dimensional spatial problem. We finally show the effectiveness of MIMC and MISC in some computational tests, including PDEs with random coefficients and Stochastic Particle Systems.