FROM THE PERIPHERY: A CASE STUDY OF ASPIRING ACADEMICS JOURNEY INTO FULL-TIME FACULTY POSITIONS THROUGH THE LENS OF COMMUNITY OF PRACTICE

FROM THE PERIPHERY: A CASE STUDY OF ASPIRING ACADEMICS JOURNEY INTO FULL-TIME FACULTY POSITIONS THROUGH THE LENS OF COMMUNITY OF PRACTIC

Characteristics of professional development research in Aotearoa New Zealand's early childhood education sector: A systematic literature review

Teachers’ professional learning and development (PLD) is an essential component in the provision of quality education. Through objective 3.6 in the Early Learning Action Plan 2019-2029 (Ministry of Education, 2019a) the Ministry of Education has signalled a need for a managed, coherent system of PLD to support the professional learning needs of early childhood teachers in Aotearoa New Zealand. Over time, research has sought to enhance understanding of PLD in ways that can contribute to more effective PLD programmes. Yet, gaps remain between PLD research, policy and practice. Synthesising extant research is important to identify existing and cumulative knowledge, and reveal research-to-practice gaps. This article reports the results of a systematic literature review, conducted to identify characteristics of PLD research within Aotearoa New Zealand’s early childhood education sector. Fifty-six research articles and reports were systematically reviewed. Findings identify that the predominantly descriptive body of research is characterised by a convergence of researchers’ and teachers’ roles, largely positive outcomes, and a broad content focus with less attention paid to PLD processes

Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights

We study a model of $n$ non-intersecting squared Bessel processes in the confluent case: all paths start at time $t = 0$ at the same positive value $x = a$, remain positive, and are conditioned to end at time $t = T$ at $x = 0$. In the limit $n \to \infty$, after appropriate rescaling, the paths fill out a region in the $tx$-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at $x = 0$, but at a certain critical time $t^*$ the smallest paths hit the hard edge and from then on are stuck to it. For $t \neq t^*$ we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time $t$ constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a $3 \times 3$ matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large $n$ limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.Comment: 59 pages, 11 figure

A pedestrian's view on interacting particle systems, KPZ universality, and random matrices

These notes are based on lectures delivered by the authors at a Langeoog seminar of SFB/TR12 "Symmetries and universality in mesoscopic systems" to a mixed audience of mathematicians and theoretical physicists. After a brief outline of the basic physical concepts of equilibrium and nonequilibrium states, the one-dimensional simple exclusion process is introduced as a paradigmatic nonequilibrium interacting particle system. The stationary measure on the ring is derived and the idea of the hydrodynamic limit is sketched. We then introduce the phenomenological Kardar-Parisi-Zhang (KPZ) equation and explain the associated universality conjecture for surface fluctuations in growth models. This is followed by a detailed exposition of a seminal paper of Johansson that relates the current fluctuations of the totally asymmetric simple exclusion process (TASEP) to the Tracy-Widom distribution of random matrix theory. The implications of this result are discussed within the framework of the KPZ conjecture.Comment: 52 pages, 4 figures; to appear in J. Phys. A: Math. Theo

Finite-gap Solutions of the Vortex Filament Equation: Isoperiodic Deformations

We study the topology of quasiperiodic solutions of the vortex filament equation in a neighborhood of multiply covered circles. We construct these solutions by means of a sequence of isoperiodic deformations, at each step of which a real double point is "unpinched" to produce a new pair of branch points and therefore a solution of higher genus. We prove that every step in this process corresponds to a cabling operation on the previous curve, and we provide a labelling scheme that matches the deformation data with the knot type of the resulting filament.Comment: 33 pages, 5 figures; submitted to Journal of Nonlinear Scienc