Mathematics

Majorana fermions -- charge-neutral spin-1/2 particles that are their own antiparticles -- have been detected in one- and two-dimensional topological superconductors. Due to the non-Abelian exchange statistics that they obey, Majorana fermions open the door to new and powerful methods of quantum information processing. Motivated by the recent experimental discovery of superconductivity in a quasicrystal, we study the possible occurrence of non-Abelian topological superconductivity (TSC) in two-dimensional quasicrystals by the same mechanism as in crystalline counterparts. We show that the TSC phase can be realised in Penrose and Ammann-Beenker quasicrystals, where the Bott index is unity. Furthermore, we confirm the existence of Majorana zero modes along the surfaces and in a vortex at the centre of the system, consistently with the bulk-boundary correspondence.

I will discuss recent results in the theory of symmetry-enriched topological phases, with a focus on the (2+1) case. I will review the classification of symmetry-enriched topological order and present general formula to compute relative 't Hooft anomaly for bosonic topological phases. I will also discuss partial results for fermionic topological phases and open questions.

The notions of Bloch wave, crystal momentum, and energy bands are commonly regarded as unique features of crystalline materials with commutative translation symmetries. Motivated by the recent realization of hyperbolic lattices in circuit QED, I will present a hyperbolic generalization of Bloch theory, based on ideas from Riemann surface theory and algebraic geometry. The theory is formulated despite the non-Euclidean nature of the problem and concomitant absence of commutative translation symmetries. The general theory will be illustrated by examples of explicit computations of hyperbolic Bloch wavefunctions and bandstructures.

Narrow escape (NE) problems are concerned with the calculation of the mean first passage time (MFPT) for a Brownian particle to escape a domain whose boundary contains N small windows (traps). NE problems arise in escape kinetics modeling in chemistry and cell biology, including receptor trafficking in synaptic membranes and RNA transport through nuclear pores. The related Narrow capture (NC) problems are characterized by the presence of small traps within the domain volume; such traps may be fully absorbing, or have absorbing and reflecting boundary parts. The MFPT of Brownian particles traveling in domains with traps is commonly modeled using a linear Poisson problem with Dirichlet-Neumann boundary conditions. We provide an overview of recent analytical and numerical work pertaining to the understanding and solution of different variants of NE and NC problems in three dimensions. The discussion includes asymptotic MFPT expressions in in the limit of small trap sizes, the cases of spherical and non-spherical domains, same and different trap sizes, the dilute trap fraction limit and MFPT scaling laws for N 1 traps, and the global optimization of trap positions to seek globally and locally optimal MFPT-minimizing trap arrangements. We also present recent comparisons of asymptotic and numerical solutions of NE problems to results of full numerical Brownian motion simulations, in the usual case of constant diffusivity, as well as considering more realistic anisotropic diffusion near the domain boundary.