Lax matrices for Yang-Baxter maps

It is shown that for a certain class of Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) the Lax representation can be derived straight from the map itself. A similar phenomenon for 3D consistent equations on quad-graphs has been recently discovered by A. Bobenko and one of the authors, and by F. Nijhoff

The number of ramified coverings of the sphere by the double torus, and a general form for higher genera

An explicit expression is obtained for the generating series for the number of ramified coverings of the sphere by the double torus, with elementary branch points and prescribed ramification type over infinity. Thus we are able to prove a conjecture of Graber and Pandharipande, giving a linear recurrence equation for the number of these coverings with no ramification over infinity. The general form of the series is conjectured for the number of these coverings by a surface of arbitrary genus that is at least two.Comment: 14pp.; revised version has two additional results in Section

A proof of a conjecture for the number of ramified coverings of the sphere by the torus

An explicit expression is obtained for the generating series for the number of ramified coverings of the sphere by the torus, with elementary branch points and prescribed ramification type over infinity. This proves a conjecture of Goulden, Jackson and Vainshtein for the explicit number of such coverings.Comment: 10 page

Transitive factorizations of permutations and geometry

We give an account of our work on transitive factorizations of permutations. The work has had impact upon other areas of mathematics such as the enumeration of graph embeddings, random matrices, branched covers, and the moduli spaces of curves. Aspects of these seemingly unrelated areas are seen to be related in a unifying view from the perspective of algebraic combinatorics. At several points this work has intertwined with Richard Stanley's in significant ways.Comment: 12 pages, dedicated to Richard Stanley on the occasion of his 70th birthda

Mode-locked dysprosium fiber laser: picosecond pulse generation from 2.97 to 3.30 {\mu}m

Mode-locked fiber laser technology to date has been limited to sub-3 {\mu}m wavelengths, despite significant application-driven demand for compact picosecond and femtosecond pulse sources at longer wavelengths. Erbium- and holmium-doped fluoride fiber lasers incorporating a saturable absorber are emerging as promising pulse sources for 2.7--2.9 {\mu}m, yet it remains a major challenge to extend this coverage. Here, we propose a new approach using dysprosium-doped fiber with frequency shifted feedback (FSF). Using a simple linear cavity with an acousto-optic tunable filter, we generate 33 ps pulses with up to 2.7 nJ energy and 330 nm tunability from 2.97 to 3.30 {\mu}m (3000--3400 cm^-1)---the first mode-locked fiber laser to cover this spectral region and the most broadly tunable pulsed fiber laser to date. Numerical simulations show excellent agreement with experiments and also offer new insights into the underlying dynamics of FSF pulse generation. This highlights the remarkable potential of both dysprosium as a gain material and FSF for versatile pulse generation, opening new opportunities for mid-IR laser development and practical applications outside the laboratory.Comment: Accepted for APL Photonics, 22nd August 201