## Australian Mathematical Society Lift-Off Fellows

The *Australian Mathematical Society Lift-off Fellowships* are designed to help recent PhD graduates in Mathematics and Statistics jump-start their career by giving them financial support during the period between the submission of their PhD thesis and their first postdoctoral position. The Lift-off Fellowships are awarded on the basis
of academic merit.

The rules and application forms for the Lift-Off Fellowships can be found on the Lift-Off Fellowship information page.

**The Australian Mathematics Society Lift-Off Fellows to date are**:

- 2019 – Scott B. LindstromThe principal focus of Scott Lindstrom's research has been on proximal point methods and their associated operators. An especially important, and commonly employed, example of such an algorithm is the iteration of a Bregman proximity operator. Scott has approached such algorithms from two broad perspectives: experimental mathematics and variational analysis. Recently, A/Prof Regina Burachik (Uni. S.A.) and J. E. Martínez-Legaz have introduced a brand new distance, which generalizes the Bregman distance. Scott's current research with Professor Burachik and Dr Minh Dao (U. Newcastle) has revealed that many such distances retain the necessary attributes to consider the iteration of their associated proximity operators. At Hong Kong Polytechnic University, Scott will analyse the new associated proximal point algorithms which arise from iterating the proximity operators for these new distances. Possible outcomes include faster proximal point algorithms and new insights about old ones. He will also continue investigation of the celebrated Douglas–Rachford proximal point algorithm, which is being used by Dr. Jeff Hogan, David Franklin, and Neil Dizon at University of Newcastle for finding wavelets.
- 2018 – Dr Jens GrimmJens Grimm's research is based in the field of mathematical physics. In recent years, the surprising role of boundary effects in high-dimensional physical systems of critical phenomena such as the zero-field ferromagnetic Ising model have been the subject of considerable debate. The current lack of theoretical understanding and the numerical difficulty for simulating large systems strongly motivate the need for a rigorous approach. We address a number of open questions in the debate by introducing a random-length random walk model on a hypercubic lattice with various boundary conditions which we then study rigorously.
- 2018 – Dr Alex CasellaIn his PhD dissertation, Casella studied Fock and Goncharov’s coordinates parametrising decorated characters relative to the fundamental group of a punctured surface with negative Euler characteristic. He extended these coordinates to
*ℙGL(3, ℂ)*–characters of hyperbolic once-punctured torus bundles using monodromy ideal triangulations. The proposed project is to study geometric structures on low dimensional manifolds, with particular interest in projective and flag geometry. Alex Casella will work on this project with Sam Ballas, at the Florida State University (FSU) in August/September 2018. Sam is an assistant professor in the Department of Mathematics at FSU, leading expert in various types of geometric structures on manifolds as hyperbolic and projective structures. His expertise in convex projective geometry is perfectly in line with Casella's research. - 2018 – Dr Mark BugdenMark Bugden's research is on the mathematical structures of dualities in string theory. During his PhD, he published results proving that the new so-called non-isometric
*T*-duality was simply the well-known non-abelian*T*-duality in disguise. Together with collaborators, he has recently obtained a result that generalises the putative non-isometric*T*-duality in such a way as to incorporate Poisson–Lie*T*-duality. Additional projects are also underway on limits of spacetime and the cosmological constant, geometric realisations of spherical*T*-duality in*M*-theory and 11-dimensional supergravity. The work to be undertaken in the fellowship consists of three main components: preparing an publishing results from the PhD, obtaining and publishing new results, and promoting work by giving talks at conferences and as a visiting researcher. - 2018 – Dr Kyle WrightKyle Wright's research interests lie at the intersection of geometry and physics. In particular, he is interested in `stringy geometry' — investigating the appropriate geometric framework for describing string theory.
*T*-duality is a symmetry in string theory which identifies equivalent string theory solutions on geometrically and topologically distinct backgrounds. Some string backgrounds are `non-geometric’, exhibiting non-commutative and even non-associative geometric behaviour. Wright is interested in studying various aspects of higher structures in geometry, and their relationship to string theory and quantum field theory.The Lift-Off fellowship will allow him to attend conferences related to his area of research. - 2018 – Dr Branko NikolicI will use the fellowship to cover my living expenses while completing results from different chapters of my thesis and preparing them for publication. In particular, the first chapter (out of three) is a construction of ‘strictification tensor product’ generalizing the free distributive law, and will be one publication. The second chapter simplifies Cauchy completeness conditions for certain bases of enrichment, and it is already submitted to a journal. The third chapter deals with obtaining differetial graded abelian groups, and categories enriched in them, as a two step process of constructing the category of coalgebras. It is a joint work with my supervisor and will lead to two papers.
- 2018 – Dr Sudeep StephenI plan to extend the results on the research I have done during my Ph.D tenure and present the results in an international conference. My PhD thesis consists of two published research articles and have 3 submitted articles. I also have 6 other published articles that arose during my PhD tenure but not included in my thesis. In my PhD thesis, we solved a conjecture of Davila and Kenter concerning a lower bound on the zero forcing number
*Z(G)*. But this opened up a lot of open questions which I intend to tackle. The inequality we proved is*Z(G) \geq (g-3)(δ-2) + δ*where*δ*is the minimum degree and*g*is the girth (length of the smallest cycle in*G*). This inequality can be thought of as*f(gδ) ≥ (g-3)(δ-2) + δ*where*f(g,δ)*is the zero forcing number over all graphs with girth*g*and minimum degree*δ*. The bound is tight only for smaller values of*δ*and*g*.

This motivated the following questions:

1. Find better lower and upper bound for*f(g,δ)*?

2. What is the asymptotic behaviour of*f(g,δ)*?

In my PhD thesis, we also solved the zero forcing problem in butterfly networks. Benes networks are closely related to butterfly networks. Our computations with Sage confirms a strong relation between the zero forcing number of these graphs which we stated as a conjecture in my PhD thesis. We would like to settle the conjecture in the affirmative. - 2018 – Dr Christopher TaylorThe work to be undertaken will be both new research and preparing material from my thesis for publication. Specifically,

• I will be writing up a paper in conjunction with my supervisors, titled “A non-splitting theorem in algebraic logic.” The mathematical content is based largely on work from my thesis, but also includes some new results.

• I will be continuing research initiated by results in my thesis.

My own interests are in lattice-based algebras; particularly those connected to algebraic logic. In my thesis, I found some new connections between these types of algebras and graph theory. Specifically, the lattice of subgraphs of a graph forms a type of algebra known as a regular double*p*-algebra. The converse question requires something more general, namely incidence structures. I proved a characterisation in my thesis. This prompts some interesting graph-theoretic questions, and also provides novel techniques applicable to algebraic logic. For example, a well-known open problem in graph theory, known as the graph reconstruction problem, conjectures that graphs are uniquely determined by a careful selection of their subgraphs. As a result of my work, the graph reconstruction problem has a purely lattice-theoretic formulation. On the algebraic side, I proved that small finitely generated subvarieties of regular double*p*-algebras are in one-to-one correspondence with what I call strongly asymmetric incidence structures, which are related to asymmetric graphs—graphs with no non-trivial automorphisms. My aim is to establish the number of strongly asymmetric incidence structures. - 2017 – Dr Joshua Ching Dr Ching's research: (i) To submit a paper written jointly with my PhD supervisor on gradient estimates for solutions of a certain equation (1.2) with isolated or boundary singularity. This research originates from Chapter 3 of my PhD thesis. (ii) To present my thesis results in the geometric analysis seminar at McGill University, Montreal, Canada. (iii) To commence new research with the aim of extending the results in Ching and Cîrstea (Analysis & PDE, 2015) to the full case of (1.2). I aim to obtain sharp classification and corresponding existence results for solutions of (1.2) for general parameter values. Some effort has already been made in this direction in recent years (see for example the papers of Nguyen (Analysis & PDE, 2016)).
- 2017 – Dr Ting-Ying Chang Dr Chang’s research: I plan to complete and submit a paper based on material from my thesis, and pursue new research on a borderline case of the isolated singularity problem extending the results of my thesis.

My thesis includes four chapters of new results, three of which are published in the paper Chang, T.-Y. and Cîrstea, F.C., Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification, Annales de l'Institut Henri Poincaré/Analyse non lineaire} (2016), 10.1016/j.anihpc.2016.12.001. In press.

First, I am completing a paper based on the final chapter of my thesis on Singular solutions to weighted divergence operators. This paper complements and generalises existing*p*-Laplacian results in the literature by Satyanad Kichenassamy and Laurent Veron. The novelty of this paper is that we are classifying the solutions for the entire range of p and the presence of the weight requires careful consideration of critical cases.

For further research, I aim to extend the results of my thesis to classify the behaviour near zero of positive solutions with isolated singularities for a weighted p-Laplacian equation in borderline cases of regular variation theory. This involves the*p*-Laplacian equation with absorption terms, whence the difficulty of strong singularities arises. This work will generalise the results of Yves Richard and Laurent Veron from the framework of pure-power functions to regular variation. I will be presenting my results at the following conferences: Harmonic Analysis and PDE conference at Macquarie University from 17–21 July; the 2017 WIMSIG Conference at the University of South Australia on 24–26 September; and Elliptic PDEs of Second Order at the Matrix Institute on 16–28 October. - 2017 – Dr Joachim Worthington Dr Worthington's research: I plan to complete two papers based on material contained in my PhD thesis, and travel to an international conference. My thesis consists primarily of four chapters of new results. I have already had papers published based on two of these chapters: Instability of equilibria for the two-dimensional Euler equations on the torus with Holger Dullin and Robert Marangell, SIAM Journal on Applied Mathematics, 76(4), 1446–1470. Stability Results for Idealized Shear Flows on a Rectangular Periodic Domain with H. Dullin, to appear in Journal of Mathematical Fluid Mechanics DOI 10.1007/s00021-017-0329-2.

I have begun work with my supervisor Holger Dullin and James Meiss of the University of Colorado, Boulder on writing two additional papers based on the final two chapters of my thesis. The papers would be titled Stability Theory in the Three-Dimensional Euler Equations and Poisson Structure of the Three-Dimensional Euler Equations. We are very excited about these papers, as they extend the results of our previous papers to a three-dimensional setting, which is much less well-known. The core of the research is complete, but we plan to substantially rewrite the material to make it more appropriate for publication and more relevant to the dynamical systems community. - 2017 – Dr Stuart Johnston Dr Johnston's research: It is well-known that individuals that are isolated from the bulk of a population exhibit different behaviour to individuals that are part of the bulk population (Wang et al., Scientific Reports, 2016). I have previously investigated the long-time behaviour of a population where isolated individuals undergo lattice-based birth, death and movement events at different rates compared to non-isolated individuals (Johnston et al., Scientific Reports, 2017). In this paper, we demonstrate that 22 different classes of partial differential equations (PDEs) arise from this one underlying model, in different parameter regimes. From these PDEs, we obtain rich insight into differences in the long-time behaviour of the overall population, depending on the relative frequency of birth, death and movement events for isolated and non-isolated individuals. Mathematical models that allow for an individual to bias its movement direction according to the local environment have been presented previously, and have been implemented to study the homing navigation of green turtles (Painter and Hillen, Journal of the Royal Society Interface, 2015). I propose incorporating differences in navigational bias between isolated and non-isolated individuals to understand how belonging to the bulk population can either help or hinder successful navigation.
- 2017 – Dr Sangeeta Bhatia Dr Bhatia's research: My dissertation focuses on new algebraic approaches to questions in biology, in particular phylogenetics. I have published a paper in the Journal of Mathematical Biology. One of the problems that I worked on was developing a framework to determine the rearrangement distance between genomes where the rearrangement operators may be weighted in some way — for instance, by the length of the genome they disrupt. My work uses algebraic methods (rewriting systems and Knuth–Bendix algorithm), which are very little known outside the community of algebraists and to the best of my knowledge, is the first application of these techniques to questions in phylogenetics. This work has the potential to offer a flexible and practical solution to the problem of determining weighted rearrangement distance.
- 2017 – Dr Adrian Grantham Dr Grantham’s research: This project falls into the applied mathematics area of environmental modelling, specifically modelling of solar radiation. Our previous work addressed the need for probabilistic forecasting of solar radiation. We developed a new computationally efficient and data-driven method for constructing a full predictive density for short-term forecasting of solar radiation, using nonparametric bootstrapping and a map of sun positions. A probabilistic forecast allows one both to assess a wide range of uncertainties in solar energy and to facilitate decision-making. A paper describing our work has been published in Solar Energy.
- 2017 – Dr Jason M. Whyte Dr Whyte's research: Ideally, a parametric model for a biological system enables prediction of system behaviour for conditions where we lack observations. This necessitates first estimating parameters from data. Unresolvable uncertainty may arise when multiple estimates are equally valid. Suitable model scrutiny may anticipate this prior to data collection. We consider this matter for systems of interacting biomolecules studied with a particular apparatus. We conduct the first analysis of suitable models (continuous-time linear switching systems). Our use of symbolic algebra obtains definitive results.
- 2017 – Dr Zeinab Abolhasani
- 2016 – Dr Chen Chen
Dr Chen Chen’s research is in the area of modelling with Differential Equations.

In the first month of the fellowship, I looked at the two-periodic diffusion problem proposed by Bunder & Roberts. I extend the problem by assuming Dirichlet boundary conditions to replace the original periodic boundary conditions. Introducing the new boundary condition made the problem more complicated. I then modified the original published algorithm to accommodate the new boundary conditions. Then I compared and contrasted the simulation models given by both multiscale modelling methods. As the model was linear, I also compared the eigen spectrum of the macroscale models given by two methods. The difficulty I faced during comparing the eigen-spectrum came from that most of the eigenvalues are complex and sorting complex eigenvalues in the best order is a non-trivial task (sic). During this fellowship, I completed a manuscript titled “boundary conditions for macroscale waves in an elastic system with microscale heterogeneity”, which summarises my PhD research. I plan to submit this manuscript to an A-ranked journal. With the funding provided by AustMS, I also managed to participated in emac 2015. During the conference, I presented and exchanged my research ideas on multiscale modelling with other mathematicians in this field. I thankfully acknowledge the support of AustMS and the Lift-off Fellowship. It provided me a fantastic opportunity to broaden my knowledge in multiscale modelling. - 2016 – Dr Daniel Ladiges
Dr Daniel Ladiges research is in the area of Fluid Dynamics.

For my Australian Mathematical Society Lift-off Fellowship I was awarded $5,000 for use as a stipend while continuing work begun during my PhD. My principal goal during this period was to write a manuscript for publication, which covered and extended on work contained in the final chapter of my PhD thesis. The chapter describes a variational method for obtaining approximate solutions to the Boltzmann equation, which approach is ideal for simulating nano-scale gas flows. When this work was almost complete we decided to split the material into two separate papers. The first has been finished and will shortly be submitted to Physical Review Fluids, the second is nearing completion. In addition to the above, we have commenced collaborations with several experimental groups, using the techniques developed during my PhD to help interpret the results of experiments involving micro- and nano-scale gas flows. This includes work modelling the flows in nano-scale squeeze film pressure sensors in order to understand the effects of gas rarefaction, and simulating the flow through hollow core optical fibres in order to characterise the boundary conditions. We have collected a large amount of data and made several findings regarding each of these systems, which should form the basis for several papers in the near future. Finally, I am developing a general implementation of Monte Carlo algorithms from my PhD thesis, allowing simulation of 2D nano-scale gas flows. The program emphasises ease of use, and will be made freely available to researchers investigating nano-mechanical systems operating in a gaseous medium. This work has been significantly advanced since the beginning of my fellowship, and an initial version should be completed within the next month. I am shortly commencing a post-doctoral position, during which I will be able to see each of these projects through to completion. The Lift-off Fellowship was invaluable in starting my academic career, and I with to thank the selection committee, and my supervisor Prof. John Sader. - 2016 – Dr Ashish Goyal
Dr Ashish Goyal’s research concerns the Mathematical modeling of hepatitis B and hepatitis D viruses.

**Outcomes**: (1) The possibility of the existence of cell-to-cell transmission in the spread of hepatitis B virus was proposed and investigated for the first time using computational modelling. This project has recently been accepted for publication under the title of “Modelling the Impact of Cell-to-cell Transmission in Hepatitis B Virus” in PLOS ONE. (2) The impact of ignoring HDV presence in the population on socio-economic outcomes of policies aimed at eliminating HBV prevalence was studied and this study has been accepted for publication as “Recognizing the impact of endemic hepatitis D virus on hepatitis B virus eradication” in Theoretical Population Biology. (3) During the period of the fellowship, a new collaboration with Dr. Harel Dahari from Loyola University was established. We are currently investigating how HCV cures after a short term of direct antiviral agents related drugs. (4) I have additionally initiated a project aimed at explaining the correlated dynamics of HBsAg and HDV RNA levels in HBV/HDV coinfected patients. The model has been developed and initial testing of the model to the real world data has been performed successfully.

**Future work**: The research during this fellowship helped improve our understanding of hepatitis viruses and may lead to the development of a successful vaccine and/or therapy. The models proposed are novel and generic in nature that can be employed for the investigation of other viruses. This probably will be a profit to the scientific community as our research can be employed as a building tool in fields such as mathematical biology, epidemiology and public health policymaking. This fellowship will definitely improve my career opportunities in the future. - 2016 – Dr Chris Bourne
Dr Chris Bourne’s work is in the area of of Mathematical Physics known as Noncommutative Geometry and Index Theory.

**Summary of Research**. The purpose of the Fellowship was for funding to attend the workshop on “Refining*C*-algebraic invariants for dynamics using*KK*-theory”, that was held at the University of Melbourne, Creswick Campus on 18–29 July 2016. The Fellowship also funded a research visit to the University of Wollongong to work with A/Prof. Adam Rennie following the*KK*-theory workshop. During this period, I worked with A/Prof. Rennie on the problem of index theory of twisted crossed product*C*-algebras by ℝ^{d}. Our motivation comes from continuous models of topological quantum systems in condensed matter physics, whose algebra of observables is a twisted crossed product acting on*L*_{2}(ℝ^{d}). In particular we studied the semifinite index theory of such algebras from the perspective of Kasparov theory. Expressions for the semifinite index pairing were derived and can be considered as the continuous analogue of the “noncommutative Chern numbers” studied by Prodan and Schulz-Baldes (3). We were able to generalise Prodan and Schulz-Baldes results as well as expand on earlier work of ours on the problem (1). We expect to finalise and submit these results for publication soon (2).

**References**: (1) C. Bourne, J. Kellendonk and A. Rennie. The*K*-theoretic bulk-edge correspondence for topological insulators. arXiv:1604.02337, 2016.

(2) C. Bourne and A. Rennie. Chern numbers, localisation and the bulk-edge correspondence of continuous topological phases. To appear.

(3) E. Prodan and H. Schulz-Baldes. Bulk and Boundary Invariants for Complex Topological Insulators: From*K*-Theory to Physics. Springer, Berlin, 2016. - 2016 – Dr Joshua Howie
Dr Joshua Howie’s work is in the area of Topology, in particular Knot Theory.

In my PhD thesis I introduced the class of weakly generalised alternating (WGA) links, and proved that they have essential checkerboard surfaces. I used the Lift-off fellowship to work with Jessica Purcell at Monash University to study the hyperbolic geometry of WGA links. We used the essential checkerboard surfaces to detect when a WGA diagram represents a hyperbolic, satellite, or torus link, including a complete classification for diagrams on the torus, a project begun in my thesis. For the hyperbolic WGA links, we were able to give criteria for when the checkerboard surfaces are accidental, virtually-fibred, or quasi-fuchsian. We also proved volume bounds on the complements of hyperbolic WGA links in terms of properties of a WGA diagram. I also used the fellowship to attend conferences in Europe in July. In particular, I presented my work at a low-dimensional topology summer school run by Central European University and Alfréd Rényi Institute in Budapest, Hungary, and at Knots in Hellas in Olympia, Greece. - 2016 – Dr Hayden Tronnlone
Dr Hayden Tronnlone’s research is in the area of Fluid Dynamics.

Much of my thesis was devoted to the development of models for the stretching under gravity of fluid cylinders with internal air channels, leaving limited scope for a thorough comparison of the model with extruded preforms. Owing to this, I applied for an AustMS Lift-Off Fellowship that would provide financial support while I extended the brief analysis from my thesis into a thorough study of three representative preform designs, in each case comparing the behaviour observed in experiments with the predictions made by the model.

This work showed that surface tension contributes significantly to deformation during preform extrusion and can explain some of the features observed in experiments; however, surface tension alone cannot explain all of the deformation. We concluded that surface tension and a second effect, extrudate swell, work in concert to deform the leading to a greater interaction between holes and hence greater surface-tension-driven deformation. The results of this investigation have been submitted for publication.

I would like to offer my thanks to AustMS and the Lift-Off Fellowship committee for providing me with the opportunity to continue this research. Through this Fellowship I not only received financial support between the completion of my PhD and the commencement of a postdoctoral position but was able to continue my work and identify avenues for future research and collaboration. I highly encourage all graduating PhD students to consider applying to this worthy scheme.

**Reference**: Hayden Tronnolone, Yvonne M. Stokes, and Heike Ebendorff-Heidepriem.*Extrusion of fluid cylinders of arbitrary shape with surface tension and gravity.*Submitted, 2016. - 2016 – Dr Alex Amenta
Dr Alex Amenta’s research is in the area of differential equations.

During my Lift-off Fellowship, working with Pascal Auscher, I improved the results of my thesis on adapted Besov–Hardy–Sobolev spaces and boundary value problems and added some new ones, with the intention of eventually publishing them as a research monograph. We finally finished this in July 2016. Recently, Chen, Coulhon, Feneuil, and Russ showed that for Vicsek manifolds (whose large-scale geometry resembles the global geometry of a Vicsek fractal) of any dimension, (*R*_{p}) holds if and only if*p*≤ 2. Their proof of*L*_{p}-unboundedness of the Riesz transform for*p*> 2 is quite simple, and during my Lift-off Fellowship I recognised that the combinatorial structure underlying this proof is that of a “spinal graph” (*G*, ∑): an infinite graph*G*with a subset*V*(*G*) such that*G*is a collection of finite graphs attached only along ∑. For any spinal graph(*G*, ∑) with certain volume and dimension assumption–roughly speaking, ∑ should be 1-dimensional, while*G*should have dimension greater than 1 – (*R*_{p}) fails for*p*> 2 for all manifolds with geometry “close to” that of*G*. This gives a mechanism for constructing manifolds which fail (*R*_{p}) for all*p*> 2, and which need not have any fractal-like structure (unlike Vicsek manifolds).

Following my Lift-off Fellowship I identified concrete examples of spinal graphs (*G*,∑) satisfying the relevant volume and dimension conditions: it is not obvious that many examples other the Vicsek graph exist, but indeed they do. However, the question of whether the associated manifolds satisfy (*R*_{p}) for 1 <*p*< 2 remains open.

I thank the AustMS for their generous support, which allowed me to keep working after submitting my thesis and thus maintain momentum before starting a postdoc. Thanks! - 2016 – Dr Elena Tartaglia
Dr Elena Tartaglia’s research is in the area Mathematical Physics.

We are interested in studying two-dimensional lattice models in statistical physics that satisfy the Yang–Baxter equation and are, therefore, exactly solvable. At criticality, these models exhibit universal behaviour that can be characterised by a corresponding conformal field theory. It is the aim of our research to study exactly solvable models and discover their corresponding conformal field theory. This AustMS Lift-off Scholarship has allowed me to conduct research further to my recent paper with collaborators Jean-Emile Bourgine and Paul A. Pearce titled “Logarithmic minimal models with Robin boundary conditions.” In this paper we applied Robin boundary conditions to the logarithmic minimal lattice models and showed that the conformal dimensions of the related conformal field theory, the logarithmic minimal models, has Kac label*s*taking half-integer values, instead of the standard integer values. The work during this scholarship began the search for boundary conditions corresponding to half-integer values for the other Kac label,*r*. We began by modifying parameters in the previous Robin boundary conditions, but this did not lead to the discovery of any new conformal dimensions. We are currently looking for a new solution to the boundary Yang–Baxter equation which gives conformal dimensions with half-integer Kac label*r*. - 2016 – Dr Matthew Tam
Dr Matthew Tam’s research is in the area of optimisation.

Stated abstractly, the*feasibility problem*asks for a point contained in the intersection of a finite family of constraint sets. Many fundamental problems arising in mathematics, science and engineering can be phrased in this language. For instance: systems of linear equations and inequations, imaging reconstruction (phase retrieval, ptychograpy), combinatorial optimisation (knapsack feasibility), and various matrix completion problems (correlation and distance matrix reconstruction).*Iterative projection methods*are a class of general purpose algorithms which can be applied to these problems and remain popular due to their relative simplicity, easy-of-implementation, and experimentally observed good performance. These features are particularly appealing to the practitioner.

Whilst there are a number of books which touch on the topic of projection methods, there exist few which are dedicated to their in-depth treatment and, those which do, offer little on current state-of-the-art. In this direction, two examples of recent progress in the area include the discovery the cyclic Douglas–Rachford method (1) and the use of regularity assumptions on the local geometry to analyse the methods in non-convex settings (2).

I was fortunate enough to receive a Lift-off fellowship to fund a three week visit at the beginning of 2016 to the University of Alicante (Spain) where I was a visitor of Dr Francisco Aragón Artacho. During this visit, we commenced our ongoing preparation of a book, which we hope will provide an up-to-date account of projection methods covering aspects of both theory and applications. It is our hope that the book will be of interest to those working in optimisation theory as well as practitioners interested in using projection-type algorithms. I wish to thank the Lift-off fellowship for their generous support as well as Dr Aragón Artacho and the University of Alicante for their warm hospitality.

**References**: (1) Borwein, J. M., & Tam, M. K. (2014).*A cyclic Douglas–Rachford iteration scheme.*Journal of Optimization Theory and Applications, 160, 1–29. (2) Lewis, A. S., Luke, D. R., & Malick, J. (2009).*Local linear convergence for alternating and averaged nonconvex projections.*Foundations of Computational Mathematics, 9, 485–513. - 2016 – Dr Narjess Afzaly
Dr Narjess Afzaly’s work is in the area of Graph Theory.

The focus of my PhD studies was on the efficient isomorph-free generation of different classes of graphs. Our results were considerably in excess of the previous results of the many people who worked on the same problems. I have been awarded a Lift-off Fellowship that facilitates my visit to Professor Brendan McKay at the Australian National University where I continue my research and prepare journal publications as follows:

•*Employing advanced generation algorithms*, we have identified and catalogued the set of small Turán graphs for collections of short cycles. Currently we are preparing 4 manuscripts based on these results.

• We have introduced a new method of canonical labeling and a modified version of the Orderly Generation based on the new method. Our aim is to expand the application of this novel method to other classes of combinatorial objects such as Latin rectangles.

• During my PhD studies, in collaboration with C. Menon, we have developed an efficient algorithm to generate 4-regular graphs. It is the most efficient known software for graphs up to 18 vertices. We are currently preparing a manuscript on this topic for publication. - 2016 – Dr Ravinda Pethiyagoda
- 2016 – Dr Ioannis Tsartsaflis
- 2016 – Dr Alexandra Hogan
- 2016 – Dr Alexander CampbellI was supported by an AustMS Lift-Off Fellowship from November 2016 until my assumption of a Postdoctoral Research Fellowship at Macquarie University in May 2017. My research in this period was concerned with a certain problem in three-dimensional category theory which I encountered when writing my PhD thesis. This research has, at the time of writing, resulted in two papers (one published, one under review) and other work in progress.
- 2015 – Dr Koenraad van den Dungen Dr van den Dungen's research is in the field of noncommutative geometry applied to gauge theory and relativity.
He has developed two definitions for Lorentzian spectral triples,
shown that they both preserve the link with analytic
*K*-homology, and identified the conditions under which Lorentzian or pseudo-Riemannian manifolds satisfy these definitions. He has also defined Krein spectral triples, proposed a Lorentzian alternative for the fermionic action, and shown that this action recovers exactly the correct physcial Lagrangian. Dr van den Dungen will use his Lift-Off Fellowship to conduct research with Associate Professor Adam Rennie at the University of Wollongong. He intends to develop a symmetric version of a theorem of Kaad and Lesch, in the context of indefinite Kasparov modules, and also to analyse the properties of the Dirac operator on the standard example of Schwarzschild spacetime. - 2015 – Dr Caley Finn
Dr Finn's research area is statistical mechanics, with a focus on the asymmetric exclusion process (ASEP), a one-dimensional
stochastic model. He has calculated the relaxation rate in the reverse bias regime, by numerically solving the Bethe ansatz equations.
He has studied the Prioritising Exclusion Process, a queueing model with high and low priority customer;
he found the exact stationary state in some phases, and calculated approximate average waiting times when the queue length is finite.
He has also found an integral form for some components of the solution of a
*q*-deformed Knizhnik–Zamolodchikov (qKZ) equation with mixed boundaries, along with developing a combinatorial algorithm that determines a factorised expression for a generalised sum rule. Dr Finn will use his fellowship to attend the three-week workshop “Statistical Mechanics, Integrability and Combinatorics”, at the Galileo Galilei Institute in Florence, Italy. He will also visit his future collaborator, Professor Eric Ragoucy, for research discussions at the Laboratoire d'Annecy-le-Vieux de Physique Théorique (LAPTh) in Annecy, France, where he will take up a post-doctoral position in October 2015. - 2015 – Dr Kristen Harley Dr Harley's research focuses on advection–reaction–diffusion equations and travelling wave solutions arising in models from mathematical biology, and uses methods from dynamical systems theory such as geometric singular perturbation theory and canard solutions. She has proved the existence of travelling-wave solutions for a malignant tumour model with small diffusion, and the existence of a shock-fronted travelling wave solution for a wound-healing model. Dr Harley will use her Lift-Off Fellowship to participate in the 2015 SIAM Conference on Applications of Dynamical Systems (DS15) in Utah, USA, and for research collaboration at the University of Oxford with Dr Frits Veerman. She will also continue collaborations on models of solid tumour growth, on the stability of travelling waves, and on periodic solutions to a three-component FitzHugh–Nagumo model, with collaborators including Dr Petrus van Heijster of the Queensland University of Technology, Associate Professor Sanjeeva Balasuriya of the University of Adelaide, Ms Lotte Sewalt of the University of Leiden, and Dr Robert Marangell and Professor Martin Wechselberger of the University of Sydney.
- 2015 – Dr Andrew Holder Dr Holder's research area is the mathematical modelling of tumour invasion, using differential equations, dynamical systems, spatio-temporal modelling, asymptotic analysis and numerical programming. He has introduced an algorithm for reduction of a dynamic system to a set of algebraic equations, from which parameter estimates can be deduced. He has applied this algorithm to three areas within mathematical biology. Dr Holder will use his Fellowship to participate in the 2015 ANZIAM Conference on the Gold Coast, and to write up a paper drawing on research from his thesis on a system of reaction–diffusion equations used for modelling the acid-mediation hypothesis with chemotherapy intervention. He also intends to develop a new model that examines the interaction between a tumour, its metabolism and the host immune response.
- 2015 – Dr Naghmana Tehseen Dr Tehseen's research areas are partial differential equations, differential geometry and mathematical physics. She has developed a new approach to constructing solutions of differential equations, which is complementary to the symmetry reduction methods initiated by Sophus Lie. She has used her approach to construct solutions to the Boyer–Finley and heavenly equations from general relativity. She has also analysed the monotonicity of the Shannon entropy on solution surfaces of the second-order evolution equation. Dr Tehseen will use her fellowship to collaborate with Professor Peter Vassiliou of the University of Canberra, and Associate Professor Jose de Dona of the University of Newcastle. She plans to use the geometry of differential systems on manifolds to attack the geometric control theory problem of finding computable necessary and sufficient conditions under which a control system is flat. Further, she will investigate higher-order solvable structures, in particular the properties of solutions found by her methods for partial differential equations, in more than two variables, that have infinite-dimensional symmetry algebras. Also, with Professor Geoff Prince of the Australian Mathematical Sciences Institute, she will apply geometric techniques to analyse the monotonic entropy behaviour of fourth-order evolution equations on solution surfaces.
- 2015 – Dr Ying Wan Yap Dr Yap's research area is the kinetic theory of rarefied gases. She has developed high accuracy numerical solutions for rarefied oscillatory nanoscale gas flows, specifically the Boltzmann–BGK equation for unidirectional steady and unsteady Couette flow. She has also found solutions to the 3D flow generated by an oscillating sphere, and has contributed to understanding the validity of the lattice Boltzmann method for non-continuum flows. Dr Yap's Fellowship will enable her to participate in the 2015 ANZIAM Conference. In addition, she intends to prepare further manuscripts stemming from her thesis work, in collaboration with Professor John Sader of the University of Melbourne and Dr Jason Nassios of Victoria University, and to initiate a study of heat transfer using the lattice Boltzmann method.
- 2014 – Dr Yi Huang
Dr Huang’s research focuses on genus
*g*hyperbolic surfaces with*n*cusps and their Teichmüller and moduli spaces. He has generalised known coordinate systems on the moduli space of hyperbolic surfaces with cusps or geodesic boundaries or a combination of both. He has also produced various new McShane identities on hyperbolic surfaces which generalise known identities. He has generated a simple length spectrum of a projective plane with three punctures to prove various results, including calculating the maximum systole on the moduli space of hyperbolic projective planes with three cusps. Dr Huang will use his fellowship to visit A/Prof Hideki Miyachi at Osaka University to develop a quasiconformal maps-based approach to the Teichmüller theory of crowned hyperbolic surfaces. He will also visit Dr Hengnan Hu and Prof Ser Peow Tan at National University of Singapore to determine a geometric interpretation of Hu and Tan’s recently derived new McShane identity. - 2014 – Dr Stephen Maher Dr Maher’s research area is mathematical optimisation with a focus on airline applications. He has investigated such problems using column-and-row generation and found that the implementation of this approach to solve an airline recovery problem required further development to reduce solution runtimes. To address this issue, Dr Maher will use his fellowship to visit Francois Vanderbeck at the University of Bordeaux to establish collaboration in order to improve the column-and-row generation solution approach to significantly improve the solution runtimes of large scale optimisation problems. The fellowship will assist Dr Maher to develop improved row generation strategies, analysing the formulations of problems successfully solved by column-and-row generation and improving the integration of column-and-row generation with branch-and-price.
- 2014 – Dr Hui Li
Dr Li’s research area is operator algebra. He has developed a structure theory for the associated Cuntz–Pimnser algebra and used his theory to show that a result in the literature is an example of a twisted topological graph C*-algebra. Dr Li will use his fellowship to visit Universidade Federal de Santa Catarina (UFSC), Florianópolis, Brazil, in order to collaborate with Professor Daniel Gonçalves and Professor Danilo Royer. At UFSC, Dr Li plans to study a representation of each
*k*-graph algebra on the Hilbert space*L*_{2}(**R**^{k}), where*k*is the dimension of the*k*-graph. For both directed graphs and higher-rank graphs, he intends to find conditions such that these representations are irreducible; conditions which are important in the representation theory of Lie groups and harmonic analysis. - 2013 – Dr Brian Corr Dr Corr is a computational group theorist with a particular interest in matrix groups and so-called black box algorithms for recognising groups. His thesis contains a generalisation of the famous Kung–Stong Cycle Index Theorem as well as developing several new algoithms for matrix group recognition. He will use his Lift-Off fellowship to visit researchers in Germany, the USA and New Zealand.
- 2013 – Dr Natalie Aisbett Dr Aisbett is a combinatorial topologist with particular interest in simplicial spheres. She has made significant contributions towards solving some of the major conjectures in this field. In particular, Dr Aisbett has proved the Nevo-Petersen conjecture for the class of simplicial spheres that are obtained from the boundary of a cross-polytope by an arbitrary sequence of edge subdivisions. She will use her Lift-Off Fellowship to continue her work on simplicial spheres and to study combinatorial complexes for complex reflection groups.
- 2013 – Dr Nahid Banihashemi Dr Banihashemi uses numerical methods to solve optimal control problems with an emphasis on real world applications. She has developed novel algorithms for solving a general class of box-constrained optimal control problems building on inexact restoration methods and Euler discretization. Dr Banihashemi will use her Lift-Off Fellowship to develop a computational model of the co-metabolism of carbon and nitrogen in collaboration with Professor John Crawford at the Charles Perkins Obesity Centre at the University of Sydney.
- 2012 – Dr Nicholas Beaton Dr Beaton works on solvable lattice models, polymer absorption and self-avoiding walks. With coauthors he has given precise estimates for the surface fugacity of random walks on several planar lattices. He will use his fellowship to study self-avoiding walks on a rotated honeycomb lattice and prudent walks on lattices.
- 2012 – Dr James Caffrey Dr Caffrey is a mathematical biologist who has developed models of interacting populations to understand experimental observation using a mixture of analytical tools and simulations. He will use his Lift-Off Fellowship to continue his work and to attend the 12th International Symposium on Mathematical and Computational Biology in Arizona.
- 2012 – Dr Bao Ho Dr Ho is a combinatorial game theorist with expertise in analyzing the game of NIM and its many variations, some of which he introduced. He will use his Lift-off Fellowship to investigate periodicity in subtraction games and, more generally, in octal games.
- 2012 – Dr Anthony Mays Dr Mays works in mathematical physics and random matrix theory, having particular expertise in studying the distributions of the eigenvalues of random matrices. Dr Mays will use his Lift-Off Fellowship to attend the “Random matrices” workshop at the University of Bonn in May 2012 and to visit Professor Akemann at the University of Bielefeld, one of the leaders in his field.
- 2011 – Dr Tyson Ritter Dr Ritter is a complex geometer with particular expertise in elliptic manifolds and Oka theory. He has shown that every Riemann surface with abelian fundamental group can be acyclically embedded into a two dimensional elliptic Stein manifold. Using very different techniques, he constructed some interesting new examples of elliptic manifolds which he then used to prove similar results for open Riemann surfaces. He will use his Lift-Off Fellowships to study related questions in Oka theory.
- 2011 – Dr Joanne Hall Dr Hall works on the interface between coding theory, combinatorics and geometry. She gave the first explicit construction of mutually orthogonal Latin squares from mutually unbiased bases. Her lift-off fellowship will allow her to continue her work relating these structures and Hjelmslev geometries. In addition, it will partially support her collaboration with Metod Saniga at the Slovak Academy of Sciences.
- 2011 – Dr Wendy Baratta Dr Baratta’s research is concerned with the non-symmetric Macdonald polynomials. She will use her Lift-off fellowship to continue working with Forrester at the University of Melbourne and to finishing writing up her work and making it readily available to other mathematicians via Mathematica.
- 2011 – Dr Michael Pauley Dr Pauley works on generalisations of cubic polynomials to Riemannian manifolds and Lie group. He will use his lift-off fellowship to investigate higher order variational problems with Prof Óscar Garay and his colleagues at the University of the Basque Country, Spain, and to visit Imperial College.
- 2010 – Dr Neil Saunders Dr Saunders' has obtained substantial new results about the minimal embeddings of finite groups inside symmetric groups. He will use his Lift-Off fellowship to continue his study of the minimal embedding problem for Coxeter groups and complex reflection groups and to help foster collaborations with groups in Cornell and Melbourne.
- 2010 – Dr Philip Brooker Dr Brooker has obtained significant results in the theory of operator ideals and Banach spaces where he has used Szlenk indices to define a class of operators ideals which, by his work, have very favourable algebraic and geometric properties. Philip will use his Lift-Off Fellowship to compute the Szlenk indices of the Banach spaces of continuous scalar valued functions on a compact space.
- 2010 – Dr Judith Egan Dr Egan has obtained significant results in the area of orthogonal partitions of latin squares. With her Lift-Off Felllowship Judith will continue her collaboration with Ian Wanless (Monash) and Brendan McKay (ANU).
- 2010 – Dr Michael Haythorpe Dr Haythorpe works in numerical optimisation and computational mathematics with particular expertise in numerical algorithms. He was awarded the TM Cherry Prize Winner for Best Student Presentation at the ANZIAM conference in 2008 and he was runner up in 2009 and 2010. Using his Lift-Off Fellowship Michael initiated a collaboration with Walter Murray at Stanford University.
- 2010 – Dr Roslyn Hickson
Dr Hickson is an applied mathematician with particular expertise in
modelling and industrial applications. She has wide ranging interests from
diffusion and heat transfer to disease modelling. Roslyn used her Lift-Off Fellowship to
participate in a course on
*Mathematical Modelling of Infectious Diseases*at the University of Utrecht.

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