It has long been expected, and is now proved in many important cases, 
that quantum algebras are more rigid than their classical limits. That is, they 
have much smaller automorphism groups. This begs the question of whether this 
broken symmetry can be recovered.

I will outline an approach to this question using the ideas of noncommutative 
projective geometry, from which we see that the correct object to study is a 
groupoid, rather than a group, and maps in this groupoid are the replacement 
for automorphisms. I will illustrate this with the example of quantum 
projective space.

This is joint work with Nicholas Cooney (Clermont-Ferrand).

The introduction of Edwards' curves in 2007 relaunched a
deeper study of the arithmetic of elliptic curves with a
view to cryptographic applications.  In particular, this
research focused on the role of the model of the curve ---
a triple consisting of a curve, base point, and projective
(or affine) embedding. From the computational perspective,
a projective (as opposed to affine) model allows one to
avoid inversions in the base field, while from the
mathematical perspective, it permits one to reduce various
arithmetical operations to linear algebra (passing through
the language of sheaves). We describe the role of the model,
particularly its classification up to linear isomorphism
and its role in the linearization of the operations of addition,
doubling, and scalar multiplication.

Permutations of finitely many elements are often drawn as permutation diagrams. We take this point of view as a motivation to construct and describe more complicated algebras arising for instance from differential operators, from operators acting on (co)homologies, from invariant theory, or from Hecke algebras. The surprising fact is that these diagrams are elementary and simple to describe, but at the same time describe relations between cobordisms as well as categories of represenetations of p-adic groups. The goal of the talk is to give some glimpses of these phenomena and indicate which role categorification plays here.
 

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