# Research Grants

## Externally funded research projects commencing in 2023

Future Fellowship

• Associate Professor Kevin Coulembier: Categorical geometry and perfect group schemes. ($895,000) [FT220100125] Summary The aims of this project are to construct novel geometric theories based on newly discovered tensor categories, to apply the theories to solve open problems in representation theory, algebra and category theory, and to establish profitable new connections between the influential theories of affine group schemes and classifying spaces. The geometric theories will be developed in a universal way, generalising both classical algebraic geometry and super geometry from physics, and specialising to infinitely many new theories. This universality ensures a significantly broader basis for long term applications of geometry in many areas of science. Other benefits include enhanced international collaboration and scientific capacity in Australia. NHMRC Investigator Grant (leadership 2) • Professor Jean Yang: Statistical bioinformatics at single cell resolution. ($2,682,170) [App ID 2017023]
Summary

This program will generate a framework and innovative bioinformatics tools to extract disease-associated signatures from multiple-modality single cell platforms for large cohort studies. Our strategy is based on contextual focus machine learning methods and novel statistical models to identify stable relationships among clinical and molecular measurements with robust predictive power across multiple platforms. This will accelerate the process of translational significant biomedical research.

MRFF Grants

• Professor Jean Yang is a member of the Chief Investigator team for Professor Alex Brown's project Pathways to benefit for Indigenous Australians in Genomic Medicine. ($4,986,948) Summary Indigenous populations are not yet appropriately represented in genomic research. We have assembled a national consortium of Indigenous researchers, health services, institutions and industry to empower Indigenous leadership in genomics with a focus on 1) Governance; 2) Data Systems and Sovereignty; 3) Genomics Policy; and 4) Indigenous Genomics Capacity Development. Our network will enable equitable, culturally safe and responsive access to the benefits of genomic medicine for all Australians. • Associate Professor Pengyi Yang is a member of the Chief Investigator team for Doctor Anai Gonzalez Cordero's project Development of photoreceptor cell therapy to treat blindness. ($2,566,652)
Summary

This research lays the foundation of retinal cell therapy for the treatment of blinding eye diseases caused by the degeneration of the light-sensing cells in the eye. A renewable source of human stem cells will be created from which transplantable photoreceptor cells will be produced for regenerative therapy of the retina. This pre-clinical process development enables translational research in stem cell medicine and cell therapy and offers a path to clinical trials of retinal cell therapy.

• Dr Ellis Patrick is a member of the Chief Investigator team for Associate Professor Natasha Rogers's project REnal FactORs Modify HEART disease Study – REFORM HEARTS. ($865,396) Summary Approximately 10% of Australians have chronic kidney disease (CKD). As kidney function deteriorates, the risk of cardiovascular disease rises sharply. CKD is an independent risk factor for cardiovascular death, but few effective therapies are available. We will address this area of unmet need by identifying a new culprit that is contributing to the burden of cardiovascular disease in CKD patients. Our research will provide new knowledge and potential therapeutic opportunities for development. ## Externally funded research projects commencing in 2022 ARC Discovery Projects — at the University of Sydney • Associate Professor Florica-Corina Cîrstea and Professor Yihong Du (chief investigators): Singular solutions for nonlinear elliptic and parabolic equations. ($427,000) [DP220101816]
Summary

The analysis of many models fundamental to physical and biological sciences is obstructed by singularities. This project aims to discover and classify the singular solutions for two important types of nonlinear equations: elliptic and parabolic. The project expects to generate novel methods to decipher singularities by using innovative approaches from geometric analysis and dynamical systems. Expected outcomes of this project include new and powerful tools to advance a more general theory of singularities. This should provide significant benefits, such as new mathematical knowledge on key issues on singularities lying at the forefront of international research and enhanced expertise in an area of worldwide recognition for Australia.

• Professor Georg Gottwald (chief investigator): A dynamical systems theory approach to machine learning. ($356,000) [DP220100931] Summary Forecasting the future state of a high-dimensional complex multi-scale system is a challenge we face in areas ranging from climate science to epidemiology. Even when basic physical mechanisms have been identified, the actual evolution equations are often unknown. This project will develop a computationally cheap machine learning framework for forecasting. The proposed mathematical framework provides a forecast together with a quantification of its uncertainty. We will develop sophisticated mathematical theory underpinning the novel methodology, as well as applying it to the perennial problem of subgrid-scale parametrisation of tropical convection, a missing key element in current climate models. • Professor Mary Myerscough (chief investigator): Space, time and boundary conditions: Mathematics for evolving plaques. ($421,000) [DP220101454]
Summary

This project aims to create new mathematical theory to model the morphology of atherosclerotic plaques, which cause heart attacks and strokes, as plaques grow or regress. The project expects to devise new mathematical tools for formulating novel spatial models for cellular processes inside the plaque. These should give a new window into plaque growth and spatial structures . Expected outcomes include powerful and reliable mathematical models, new tools to understand plaque evolution, and national and international collaborations with scientists and mathematicians. This should provide significant benefits including increased capacity to use mathematical models in vascular biology and training young researchers in interdisciplinary methods.

• Dr Jonathan Spreer and Professor Francisco Santos (chief investigators): Triangulations: linking geometry and topology with combinatorics. ($429,000) [DP220102588] Summary Triangulations are the method of choice to represent geometric objects given by a finite sample of points. Prominent examples include the pictures produced by the finite element method, polytopes in optimisation, or surfaces in computer graphics. Knowledge about the triangulations of an object and how they relate to each other is essential for these applications. Seemingly canonical and straightforward methods perform well – or not at all, depending on intricate and highly involved mathematical properties. In this project we combine geometric and topological viewpoints to tackle high-profile questions about triangulations. This will unlock the full potential of combinatorial methods and practical algorithms in applications. • Associate Professor Leo Tzou and Dr Justin Tzou (chief investigators): Microlocal Analysis – A Unified Approach for Geometric Models in Biology. ($405,000) [DP220101808]
Summary

This project will use microlocal analysis to create a unified approach for predicting the outcome of a broad class of diffusion and reaction-diffusion models. This will replace the traditional theory which is no longer adequate for the level of geometric complexity demanded of current models arising in biology/ecology. This project will address the urgent need for a systematic theoretical underpinning of diffusion/reaction-diffusion in geometric settings whose scope of application is broader than the the existing patchwork of methods.

• Professor Martin Wechselberger (chief investigator): A coordinate-independent theory for multi-time-scales dynamical systems. ($432,000) [DP220101817] Summary Biochemical reaction networks operate inherently on many disparate timescales, and identifying this temporal hierarchy is key to understanding biological behaviour. Currently, the existing dynamical systems theory is not able to rigorously analyse many important biological systems and networks due to this inherent non-standard multi-time-scale splitting. This project aims to remove these stumbling blocks and develop a coordinate-independent mathematical theory that weaves together results from geometric singular perturbation theory, differential and algebraic geometry and reaction network theory to decompose and explain the structure in the dynamic hierarchy of events in non-standard multi-time-scale systems and networks. — at Monash University • Dr Julie Clutterbuck, Dr Daniel Hauer, Dr Paul Bryan and Professor Guofang Wei (chief investigators): Optimal shapes in geometry and physics: Isoperimetry in modern analysis. ($295,000) [DP220100067]
Summary

This project will find the best isoperimetric shapes in curved spaces: shapes that optimise geometric or analytic quantities, such as the volume enclosed by a surface of a given area, or the resonant frequency of a drum of given area. The optimal shapes lead to tools that are widely used in differential equations, geometric analysis, statistical physics, probability theory, and quantum computing. Through this work, we will forge connections between the geometry of curved spaces, and the physics of operators therein. The significant benefits of this project include increasing fundamental mathematical knowledge, building capacity in Australia’s world-class geometric analysis community, and strong links with international partners.

• Professor Fima Klebaner, Associate Professor Kais Hamza, Dr Jie Yen Fan, Professor Andrew Barbour and Professor Bohdan Maslowski (chief investigators): New universality in stochastic systems. ($410,000) [DP220100973] Summary This project aims to uncover new analyses and effects in the complex behaviour of non-linear systems with random noise. Many systems originate near an unstable equilibrium. This project will develop a new mathematical theory that establishes a universality in the way the long term effect of noise expresses itself as random initial conditions in the dynamics. It will fill gaps in Mathematics and make refinements to existing fundamental scientific laws by including random initial conditions as predicted by our theory. This will advance our understanding of complex systems subjected to noise and will provide significant benefits in the scientific discoveries in Biology, Ecology, Physics and other Sciences where such systems are frequently met. • Dr Ivan Guo, Dr Zhou Zhou, Dr Anna Aksamit Dr Kihun Nam and Professor Marek Rutkowski (chief investigators): Can green investors drive the transition to a low emissions economy? ($390,000) [DP220103106]
Summary

The project aims to develop a game-theoretical approach to model the impact of climate change on financial markets by studying the interactions between the government, companies and investors. Expected outcomes include novel solution concepts for stochastic games with heterogeneous beliefs, asymmetric information, and model uncertainty, as well as optimal investment and production strategies under climate driven economic transitions. Results will be used to validate and improve the recently launched Australian based climate transition index. The project should yield significant benefits for the financial industry and investors by providing novel insights into financial risks during the transition to a low emissions economy.

ARC Discovery Early Career Researchers

• Dr Shila Ghazanfar: Statistical approaches for spatial genomics at single cell resolution. (443,869) Summary Cells cooperate to form complex, dynamic and varied tissue structures. This project aims to develop statistical and computational approaches to analyse spatial genomics data, a novel technology that retains vital spatial information at single cell resolution while detecting RNA molecules for hundreds of genes. Observing the molecular activity of cells in their spatial context is critical for tackling key biological questions, such as how tumour cells behave during malignancy or how stem cells determine their fate. Expected outcomes also include techniques to fully harmonise spatial and non-spatial genomics datasets, and methods toward understanding the complex relationships among cells in their environment, revealing novel cell biology • Dr Ashish Goyal: Multiscale mathematical modelling to gain insights into hepatitis viruses. (444,068)
Summary

This project aims to use mathematical modelling to study hepatitis viruses at multiple levels. The project expects to develop complex yet analysable mathematical models to comprehend the fundamental biology of hepatitis viruses by elucidating longitudinal patterns in viral and immune markers at intracellular and cellular levels, and advance a new subfield in mathematical biology, i.e., modelling codependent human viruses. Expected outcomes of the project include new generalized mathematical tools, biological insights that may aid research beyond the scope of this project, and strong interdisciplinary collaborations. Expected benefits include an increased capacity of the research community in Australia to use mathematical models in virology.

## Externally funded research projects commencing in 2021

Australian Laureate Fellowship

• Professor Helen Byrne: New approaches to understand how form and function shape complex. ($3,021,288.) Summary As biology and medicine transform into quantitative sciences, existing mathematical methods are often inadequate to explain the data they generate. This project aims to unlock the potential of such biomedical data through the development of new mathematical approaches that combine concepts from pure and applied mathematics, statistics and data science, and then to investigate their ability to generate mechanistic insight into fundamental biomedical processes. In this way, the project expects to affect a paradigm shift in mathematical biology while strengthening Australia’s reputation as a world-leader in mathematical biology. An outcome from this project could be new mathematical models that guide decision making in the clinic. ARC Discovery Projects — at the University of Sydney • Associate Professor Damian Birney and Professor Sally Cripps (chief investigators), with Prof Dr Jens Beckmann and A/Prof Rui Nouchi (partner investigators): A paradigm shift in understanding cognitive flexibility. ($493,123.)
Summary

The project aims to model cognitive flexibility as a dynamic process within people that varies across situations and occasions using advanced data analytics. Significance: The project intends to generate new knowledge in intelligence theory using recent advances that overcome known theory-testing limitations that have historically been ignored.

Expected Outcomes: An authentic account of cognitive flexibility and a new paradigm for developing and testing models of dynamic change within people. Benefits: Dynamic models are needed to understand authentic problem-solving and cognitive function. The advances benefit research and applied areas where dynamic processes are important, including education, work, and cognitive aging.

• Dr Kevin Coulembier (chief investigator) with Dr Pavel Etingof (partner investigator): New constructions and techniques for tensor categories. ($422,887.) Summary The goal of this project is to make fundamental advances in the structure theory of tensor categories. Such categories play crucial roles in numerous fields of mathematics, physics and beyond. New methods, theory and examples will be developed, inspired by algebra, representation theory and geometry. These will then be applied in the foundational study of tensor categories for (dis)proving several of the most important open conjectures in the field. This will open new perspectives for applications in other areas, most notably in representation theory. Other benefits include enhanced international collaboration and scientific capacity in Australia. • Professor Holger Dullin and Dr Robert Marangell (chief investigators), with Prof Dr Yuri Latushkin (partner investigator): Spectral Theory of Hamiltonian Dynamical Systems. ($310,000.)
Summary

Stability theory of steady states, travelling waves, periodic waves, and other coherent structures in nonlinear Hamiltonian partial differential equations is a cornerstone of modern dynamical systems. In particular it is of utmost importance to reliably compute eigenvalues, which determine the stability or instability of such structures. This project will develop methods to compute the spectrum of Hamiltonian operators in more than one spatial dimension. It will use the powerful geometric tools of the Maslov index and the Evans function. We will use these to simultaneously advance, and bring together the theories of the two dimensional Euler equations and Jacobi operators.

• Dr Alexander Fish (chief investigator): Additive combinatorics of infinite sets via ergodic theoretic approach. ($340,000.) Summary The proposed project will utilise innovative ergodic theoretic approaches to enable us to address important questions in Additive Combinatorics (Number Theory) and Fractal Geometry. In particular, we will resolve long-standing inverse additive problems for infinite sets, discover sum-product phenomena in Number Theory, and find a plethora of finite configurations in fractal sets. We will also extend the structure theory of one of the most popular mathematical models of quasi-crystals to a more extensive class of groups. This project will make significant contributions to Additive Combinatorics and Ergodic Theory and will bring the Australian research in these fields to ever greater heights • Professor Nalini Joshi (chief investigator): Dynamics on space-filling shapes. ($501,777.)
Summary

Modern science derives its power from mathematical models and tools that enable us to predict their behaviours. The project aims to construct new models given by dynamical systems that move consistently from one tile to another in a lattice of higher-dimensional shapes called polytopes. The construction is expected to lead to new functions with properties that will provide extensions of current models of growth processes. The intended outcomes of the projec include predictive tools that describe nonlinear special functions and information about their symmetry reductions. This should provide significant benefits, such as new mathematical knowledge innovative techniques, and enhanced scientific capacity in Australia.

• Associate Professor John Ormerod, Dr Garth Tarr and Professor Samuel Muller (chief investigators): Fast flexible feature selection for high dimensional challenging data. ($390,000.) Summary The project aims to provide new frameworks for fast flexible feature selection and appropriate modelling of heterogeneous data through structural varying-coefficient regression models. The outcomes will be a series of new statistical methods and concepts enabling more powerful modelling of complex bioscience data. The project will create the science for building reliable statistical models taking model uncertainty into account, impacting how results will be interpreted, and with accompanying software. This will be a significant improvement in the assessment of model confidence in the food and health research priority areas including areas such as meat science, Huntington’s disease, and kidney transplantation. — at La Trobe University • Dr Yuri Nikolayevsky and Professor Holger Dullin (chief investigators) with Prof Dr Vladimir Matveev (partner investigator): Finite dimensional integrable systems and differential geometry. ($390,000.)
Summary

Mathematical models of many processes in science (physics, engineering) and in the real world (nature, economics) are governed by complicated systems of differential equations. An important, distinguished class of such models is described by integrable systems, the systems for which one can provide a comprehensive qualitative picture, and in many cases, a complete solution. Using recently developed, powerful methods of integrable systems and differential geometry, this project will focus on a range of important, interconnected theoretical problems in both disciplines. The expected outcomes will provide new, deep, mathematically and physically significant results which will lead to applications and developments across a range of fields.

## Externally funded research projects commencing in 2020

ARC Future Fellowship

• A/Prof Peter Kim: Mathematical modelling unravels the impact of social dynamics on evolution. (Four years, $1,028,533.) Summary This project aims to mathematically model human evolution as a dynamical process. The anticipated goal is to quantitatively analyse theories of human origins. The project expects to develop innovative mathematical models, improve our understanding of the evolutionary process, and advance a unique area of interdisciplinary collaboration: applied mathematics and anthropology. Expected outcomes include refined methods for mathematical modelling of human evolution and improved techniques for analysing such models. It should provide benefits, such as increasing research in mathematical biology, an important growth area of science in Australia, and advancing mathematical approaches to engaging questions arising from anthropology. ARC Discovery Projects — at the University of Sydney • Dr Lamiae Azizi, with Professor Margaret Barbour, Dr Daniel Tholen, Professor John Evans, Assistant Professor Craig Brodersen, Dr Andrew McElrone and Dr Thomas Buckley. (Four years,$500,000.)
Summary

This project aims to develop leaf anatomical ideotypes with improved photosynthesis and water-use efficiency for wheat, rice, chickpea and cotton using novel three dimensional imaging and modelling techniques. This project expects to generate new understanding of the role of leaf anatomy on leaf function. Expected outcomes of this project include the world's first 3D spatially-explicit, anatomically accurate model of leaves of crop plants to allow virtual experiments identifying optimized anatomy for improved photosynthetic performance. Benefits to the agricultural industry include increased crop productivity and water-use efficiency to meet future global food demand and to make the most of Australia's limited water resources.

• Professor Beniamin Goldys, with Associate Professor Thanh Tran and Dr Kim Ngan Le: Mathematics for breaking limits of speed and density in magnetic memories. (Three years, $525,000.) Summary The aim of this project is to develop a mathematical theory and numerical models of stochastic partial differential equations for magnetic nano-structures. Such materials will yield next-generation magnetic memories with up to three orders of magnitude faster switching speeds and dramatically increased data storage density. New mathematical theories will help understand their sensitivity to small random fluctuations that can destroy stored information. This project aims to revolutionise mathematical modelling of magnetic memories and put Australia at the forefront of international research. Technological advances to create much smaller and faster memory devices are expected to enable groundbreaking ways of managing and mining big data. • Professor Nalini Joshi and Dr Milena Radnović: Geometric analysis of nonlinear systems. (Three years,$426,000.)
Summary

Modern science derives its power from mathematics. The project aims to capture, identify and describe pivotal, transcendental solutions of nonlinear systems that are universal in science, in the sense that they always arise as mathematical models under certain physical limits. The project expects to produce new mathematical methods to describe such functions by using a newly discovered geometric framework. Expected outcomes include the description of elusive solutions of discrete and higher-dimensional nonlinear systems. This should provide significant benefits, such as new mathematical knowledge, innovative techniques, enhanced scientific capacity in Australia.

• Professor Andrew Mathas: Graded semisimple deformations. (Three years, $474,000.) Summary Recent advances in representation theory have revealed beautiful new structures in the classical representation theory of the symmetric groups and Hecke algebras. These discoveries have provided us with new algebras, the cyclotomic KLR algebras, that encode deep properties of fundamental objects in algebraic combinatorics and geometric representation theory. The cyclotomic quiver Hecke algebras are central to several open problems in mathematics but they are still poorly understood, with even basic properties like their dimensions being unknown. This project will establish a new framework for studying these algebras that will remove the current obstacles in this field and alllow us to prove substantial new results that advance the theory. • Professor Mary Myerscough, with Dr Christina Bursill and Professor Helen Byrne: New mathematics for lipids and cells: structured models for atherosclerosis. (Three years,$500,000.)
Summary

The project aims to create new mathematical theory for immune cell behaviour which leads to heart attacks and strokes. This includes formulation and analysis of new types of mathematical models for atherosclerotic plaque development, leading to the creation of new mathematical tools to investigate cell fate in plaques and to generate new hypotheses for experimental research. Expected outcomes of this project include powerful and reliable mathematical models ready for application, and national and international collaborations with scientists and mathematicians. This should provide significant benefits including increased capacity to use mathematical models in vascular biology and training young researchers in interdisciplinary methods.

• Dr James Parkinson, with Dr Jérémie Guilhot and Professor Bernhard Mühlherr: Lusztig's conjectures for Hecke algebras with unequal parameters. (Three years, $453,000.) Summary The goal of this project is to make fundamental advances in representation theory, a powerful branch of mathematics focused on taking abstract mathematical structures and "representing" them in a concrete and useful way. In particular we aim to prove a series of long standing and influential conjectures by George Lusztig concerning the representation theory of Hecke algebras, objects which are ubiquitous in modern algebra. Our work will lead to new discoveries, a fundamentally deeper understanding of Kazhdan-Lusztig theory, and will drive future research. Benefits include enhanced international collaboration and increasing capacity in pure mathematics, especially in the cutting-edge area of representation theory. • Professor Marek Rutkowski, with Associate Professor Shige Peng: Fair pricing of superannuation guaranteed benefits with downturn risk. (Three years,$390,000.)
Summary

Australia is the fourth largest holder of pension fund assets worldwide. Hence the impact of market fluctuations on financial well-being of retirees can be detrimental, especially during market downturns associated with economic crises. The finance industry addresses this issue by complementing variable annuities with riders designed to protect the income stream of retirees. This project aims to develop a novel approach to fair pricing and optimal withdrawals and surrender policies for superannuation guaranteed benefit products through a comprehensive analysis of complex optimisation problems in stochastic models of financial markets with downturn risk.

• Professor Martin Wechselberger and Dr Robert Marangell, with Dr Bronwyn Hajek and Dr Petrus van Heijster: A novel geometry approach to shocks in reaction-nonlinear diffusion models. (Three years, $480,000.) Summary Reaction-nonlinear diffusion models play a vital role in the study of cell migration and population dynamics. However, the presence of aggregation, or backward diffusion, leads to the formation of shock waves - distinct, sharp interfaces between different populations of densities of cells - and the breakdown of the model. This project will develop new geometric methods to explain the formation and temporal evolution of these shock waves, while simultaneously unifying existing regularisation techniques under a single, geometric banner. It will devise innovative tools in singular perturbation theory and stability analysis that will identify key parameters in the creation of shock waves, as well as their dynamic behaviour. — at Macquarie University • Dr Daniel Hauer, with Associate Professor Adam Sikora, Associate Professor Zihua Guo and Dr Melissa Tacy: Nonlinear harmonic analysis and dispersive partial differential equations. (Three years,$420,000.)
Summary

This proposal is devoted to linear and nonlinear harmonic analysis. It aims to unify the most significant attributes of harmonic analysis such as restriction estimates, dispersive properties of differential operators, spectral multipliers, uniform Sobolev estimates and sharp Weyl formula. Such unification will strongly improve tools for mathematical modelling in all areas of technology and science. Notable applications include medical imaging, fluid dynamics and subatomic modelling using quantum interpretation. It will solve several important open problems in spectral analysis of partial differential operators and develop new cutting-edge techniques in harmonic analysis with application to nonlinear partial differential equations.

— at the University of Wollongong

• Professor Jacqui Ramagge and Dr Nathan Brownlowe, with Professor Aidan Sims, Associate Professor David Pask and Associate Professor Lisa Clark: There and back again: operator algebras, algebras and dynamical systems. (Three years, $461,000.) Summary The aim of this project is to develop mathematics that enables us to transfer information back and forth between dynamical systems and algebras, including operator algebras. Dynamical systems - systems that change over time - are ubiquitous, and central to modern mathematics and its applications. In mathematics, dualities allow us to translate questions from one context to another in which they are easier to solve and then translate the answer back again. Expected outcomes include increased understanding of the relationship between operator algebras and the dynamical systems that they represent. Benefits include enhanced international collaboration, and increased Australian capacity in pure mathematics, particularly operator algebras. ARC Discovery Early Career Researchers • Dr Anna Aksamit: How to beat model uncertainty with more information. (Three years,$427,008)
Summary

Experience of the 2008 financial crisis exposed a weakness in our over-reliance on mathematical models. The main aim of this project is to develop mathematical tools to investigate the role of information in reducing model uncertainty. The project will undertake pressing research in robust finance, which is now one of the most active and dynamic topics in financial mathematics. It expects to quantify the value of information under uncertainty in mathematical modelling. It will generate new knowledge in probability theory and stochastic processes providing a significant mathematical contribution in its own right.

• Dr Ellis Patrick: Statistical frameworks for high-parameter imaging cytometry data. (Three years, $427,068) Summary The project aims to develop statistical and bioinformatics methodology for characterising the complex interactions between cells in their native environment. Recent advances in imaging cytometry technologies have made it possible to observe the behaviour of multiple cell-types in tissue concurrently. The intended outcome is a suite of statistical methodologies that are crucial for addressing a variety of biological problems with these state-of-the-art technologies. This work will advance knowledge in bioinformatics, statistics and image analysis, providing benefits to scientists studying the fundamental behaviour of cells and underlying disease mechanisms. ## Externally funded research projects commencing in 2019 ARC Discovery Projects — at the University of Sydney ARC Discovery Early Career Researchers • Dr Wenshuai Jiang: Singularity analysis for manifolds with Ricci curvature bounds. ($345,000)
• Dr Ulrich Thiel: Representation theory: studies of symmetry shadows. ($420,256) • Dr Guangbo Xu: Gauged sigma model, mirror symmetry, and related topics. ($350,000)

## Externally funded research projects commencing in 2018

ARC Discovery Projects

— at the University of Sydney

• Dr Oded Yacobi, Dr Kevin Coulembier and Professor Julia Pevtsova: New Dualities in Modular Representation Theory. (2018–2020: $273,485.) • Dr Anne Thomas, Dr Elizabeth Townsend Milicevic and Assistant Professor Petra Schwer: Affine flags, Euclidean reflection groups and folded galleries. (2018–2020:$317,288.)
• Prof Georg Gottwald: Mathematical model reduction for complex networks. (2018–2020: $357,072.) • Prof Alexander Molev: Quantum vertex algebras. (2018–2020:$411,584.)
• A/Prof Peter Kim, Dr Federico Frascoli, A/Prof Adelle Coster and Prof Chae-Ok Yun: Dynamical systems theory and mathematical modelling of viral infections. (2018–2020: $401,706.) • Prof Martin Wechselberger: A geometric theory for nonstandard relaxation oscillators. (2018–2020:$401,706.)

— at Monash University

— at the Australian National University

• A/Prof Samuel Müller, Prof Alan Welsh, Dr Francis Hui and Prof Yanyuan Ma: Dimension reduction and model selection for statistically challenging data. (2018–2020: $359,083.) ARC Discovery Early Career Researchers • Dr Alistair Senior: Diet, Variance and Individual Variability in Life-History. (2018–2020:$365,058.)
• Dr Haotian Wu: Singularity Analysis for Ricci Flow and Mean Curvature Flow. (2018–2020: $328,075.) • Dr Rachel Wang: Statistical theory and algorithms for joint inference of complex networks. (2018–2020:$343,450.)

## Externally funded research projects commencing in 2017

NHMRC Project

• Professor David James, Dr Jacky Stoeckli and Professor Jean Yang: Dissecting Rapamycin sensitive and insensitive effects of mTOR. (2017–2020: $1,100,00.) ARC Discovery Projects • Professor Anthony Henderson and Associate Professor Pramod Achar: Modular character sheaves. (2017–2019:$345,00.)
Summary

This project aims to complete the fundamental mathematical theory of modular group representations, the algebraic description of symmetry over finite number systems. Group representation theory can be applied to any linear problem involving symmetry. However, the modular case, where the characteristic of the underlying field is a prime number, is less understood than real or complex scalars, and this lack of understanding blocks potential applications. This project will use geometric methods to answer questions about modular representations of the finite groups of Lie type, the most important class of finite groups. This project could make modular representation theory essential for computations, enabling faster solutions to problems of linear algebra and allowing future applications in such areas as data transmission technology.

• Professor Jacqui Ramagge, Dr Nathan Brownlowe, Professor Iain Raeburn and Professor Marcelo Laca: From actions to operator algebras and their equilibrium states. (2017–2019: $286,000.) • Associate Professor Qiying Wang, Professor Shiqing Ling and Professor Weidong Liu: Non-linear cointegrating regression with endogeneity. (2017–2019:$288,471.)
• Professor Jean Yang, Dr John Ormerod, Associate Professor Samuel Müller, Dr Pengyi Yang and Professor Graham Mann: Prognosis based network-type feature extraction for complex biological data. (2017–2019: $354,500.) • Professor Ruibin Zhang: Geometric themes in the theory of Lie supergroups and their quantisations. (2017–2019:$416,500.)

ARC Discovery Early Career Researchers

• Dr Kevin Coulembier: Quasi-hereditary categories in Lie theory. (2017–2019: $360,000.) • Dr Zsuzsanna Dancso: Homological methods in combinatorics, algebra and geometry. (2017–2019:$360,000.)
• Dr Pengyi Yang: Trans-omic networks. (2017–2019: $372,000.) ## Externally funded research projects commencing in 2016 NHMRC Project • A/Prof Jean Yang: Statistical bioinformatics for network based prognostic and precision therapy in complex disease. (2016–2019:$463,652.)

ARC Future Fellowship

• Dr Zhou Zhang: Comprehensive Study of Kahler-Ricci Flows. (2016–2019: $764,960.) ARC Discovery Projects • Prof John Cannon and Prof Derek Holt: Composition algorithms for large matrix groups. (2016–2018:$305,500.)
• Prof Nalini Joshi and Prof Kenji Kajiwara: Reflection groups and discrete dynamical systems. (2016–2018: $495,700.) • Dr Peter Kim and Prof Kristen Hawkes: Human longevity: Modelling social changes that propelled its evolution. (2016–2018:$396,338.)
• Prof Gus Lehrer, Dr Anthony Henderson and Dr Geordie Williamson: Algebraic Schubert geometry and unitary reflection groups. (2016–2018: $519,300.) • A/Prof Mary Myerscough, A/Prof Charlie Macaskill and Dr Christina Bursill: Dynamics of atherosclerotic plaque formation, growth and regression. (2016–2018:$342,200.)
• Dr Stephan Tillmann, Prof Joachim Rubinstein and A/Prof Craig Hodgson: Invariants, geometric and discrete structures on manifolds. (2016–2018: $334,000.) • Prof Beniamin Goldys, Prof Thanh Tran, Prof Zdzisław Brzeźniak, Prof Andreas Prohl, Prof Ernst Stephan and A/Prof Salim Meddahi: Novel Approaches for Problems with Uncertainties. (2016–2018:$329,377.)

ARC Discovery Early Career Researchers

• Dr Andrew Papanicolaou: Solving non-Markov optimistisation problems using forward-backward stochastic differential equations. (2016–2018: $295,020.) ## Externally funded research projects commencing in 2015 ARC Discovery Projects • Prof Gus Lehrer and Prof Ruibin Zhang: Symmetry via braiding, diagrammatics and cellularity. (2015:$130,000; 2016: $124,700; 2017:$130,000. 2018: $130,000. 2019:$130,000.)
• Prof Andrew Mathas: The dimension problem for Hecke algebras. (2015: $135,000; 2016:$129,500; 2017: $135,000.) • Prof Alex Molev: Classical and affine $$W$$-algebras. (2015:$115,000; 2016: $114,000; 2017:$124,000.)

ARC Discovery Early Career Researchers

• Dr Peter McNamara: Higher Representation Theory. (2015: $120,000; 2016:$120,000; 2017: $120,000.) • Dr Oded Yacobi: Quantum Groups and Categorification in Geometric Representation Theory. (2015:$110,000; 2016: $110,000; 2017:$110,000.)

## Externally funded research projects commencing in 2014

ARC Discovery Projects

• Prof Alan H Welsh and Dr Samuel Müller: Prediction, inference and their application to modelling correlated data. (2014: $117,000; 2015:$117,000; 2016: $117,000.) • Dr Stephan Tillmann: Moduli Spaces of Geometric Structures. (2014:$90,000; 2015: $110,000; 2016:$70,000.)
• Ruibin Zhang: Super Duality and Deformations in the Representation Theory of Lie Superalgebras. (2014: $120,000; 2015:$120,000; 2016: $120,000.) ARC Discovery Early Career Researchers • Dr Geoff Vasil: Computational geophysical and astrophysical fluid dynamics at the petascale. (2014:$110,540; 2015: $111,140; 2016:$111,140.)

ARC Future Fellowship

• Leo Tzou, Inverse Problems for Partial Differential Equations—A Geometric Analysis Perspective. (2013: $85,240; 2014:$159,905; 2015: $148,085; 2016:$144,085; 2017: $70,665.) ## Externally funded research projects commencing in 2013 ARC Discovery Projects • Prof JJ Cannon and Prof DF Holt: Constructive Representation Theory. (2013:$130,000; 2014: $137,000; 2015:$145,000.)
• A/Prof S Yan, Prof EN Dancer and Prof Y Du: Singularity, degeneracy and related problems in nonlinear partial differential equations. (2013: $100,000; 2014:$100,000; 2015: $100,000.) • Prof N Joshi: Critical solutions of nonlinear systems. (2013:$110,000; 2014: $110,000; 2015:$110,000.)
• Prof JH Rubinstein, A/Prof CD Hodgson and Dr S Tillmann: Triangulations in dimensions 3 and 4: discrete and geometric structures. (2013: $120,000; 2014:$120,000; 2015: $120,000.) • Prof A Ward, A/Prof MR Myerscough, Prof J Kruse and Dr J Buhl: Leadership matters: the emergence of informed leaders and their influence on group movement. (2013:$120,000; 2014: $130,000; 2015:$110,000.)
• A/Prof Q Wang, Prof W Liu and Prof C Tudor: Asymptotics in non-linear cointegrating regression: theory and applications. (2013: $50,000; 2014:$60,000; 2015: $70,000.) • A/Prof J Yang, A/Prof S Müller and Prof GJ Mann: Vertically integrated statistical modelling in multi-layered omics studies. (2013:$130,000; 2014: $130,000; 2015:$130,000.)

ARC Discovery Early Career Researchers

• Dr Sheehan OlverA new class of fast and reliable spectral methods for partial differential equations. (2013: $108,221; 2014:$106,655; 2015: $100,764.) • Dr John OrmerodScalable Bayesian model selection for massive data sets. (2013:$124,969; 2014: $121,789; 2015:$123,652.)

Australian Laureate Fellowship

• Prof N Joshi: Geometric construction of critical solutions of nonlinear systems. (2012: $313,558; 2013:$632,847; 2014: $643,960; 2015:$653,644; 2016: $634,810; 2017:$305,837.)

ARC Future Fellowship

• Dr Martin WechselbergerGeometric methods in mathematical physiology. (2012: $83,957; 2013:$167,914; 2014: $167,914; 2015:$167,914; 2016: $83,957.) ## Externally funded research projects commencing in 2012 ARC Discovery Projects • Prof Maria Byrne, Dr Jean Y Yang, Prof Gregory A Wray: Heads or tails – which did echinoderms lose in the evolution of radial symmetry? (2012:$120,000; 2013: $110,000; 2014:$110,000.)
• Dr Florica C CîrsteaAnalysis of nonlinear partial differential equations describing singular phenomena. (2012: $30,000; 2013:$30,000; 2014: $30,000.) • A/Prof Georg Gottwald, A/Prof Gary A Froyland: Extracting macroscopic variables and their dynamics in multiscale systems with metastable states. (2012:$90,000; 2013: $90,000; 2014:$90,000.)
• Prof Gus Lehrer, Prof Ruibin Zhang: Quantised algebras, supersymmetry and invariant theory. (2012: $110,000; 2013:$110,000; 2014: $105,000.) • Prof Marek Rutkowski: Multi-person stochastic games with idiosyncratic information flows. (2012:$108,000; 2013: $117,000; 2014:$125,000.)

ARC Future Fellowship

• Dr Anthony HendersonSpringer fibres, nilpotent cones and representation theory. (2011: $81,672; 2012:$166,281; 2013: $169,046; 2014:$164,106; 2015: $79,669.) ARC Discovery Early Career Researcher • Dr Peter S KimMathematical modelling of breast cancer immunity: guiding the development of preventative breast cancer vaccines. (2012:$125,000; 2013: $125,000; 2014:$125,000.)

Go8 – Germany Joint Research Co-operation Scheme

• Dr Benjamin Burton (Qld), Prof Michael Joswig (TU Darmstadt), Dr Andreas Paffenholz (TU Darmstadt), Dr Stephan TillmannAlgorithmic methods in combinatorial topology. (2012: $14,000; 2013:$14,000.)

Marsden Fund (NZ)

• V.Kirk, J. Sneyd, H. Osinga, Dr Martin WechselbergerApplications of multiscale excitable systems to calcium dynamics and neuroscience. (2012–2014: $605,000.) WUN Research Development Fund • B. Krauskopf, H. Osinga,V. Kirk, J. Sneyd, R. Bogacz, A. Randall, K. Tsaneva-Atanasova, Dr Martin WechselbergerMathematics of non-communicable diseases: understanding failure of cell signalling. (2012–2013:$50,000.)

## Externally funded research projects commencing in 2011

ARC Discovery Projects

• Dr James Atkinson: Algebraic interpretations of discrete integrable equations. (2011: $82,000; 2012:$82,000; 2013: $82,000.) • Prof E Norman Dancer: Stable and Finite Morse index solutions and peak solutions of nonlinear elliptic equations. (2011:$120,000; 2012: $120,000; 2013:$110,000.)
• A/Prof Holger R Dullin, Prof Nalini Joshi: Geometry and analysis of discrete integrable systems. (2011: $100,000; 2012:$100,000; 2013: $100,000.) • Prof Gus Lehrer: Flag varieties and configuration spaces in algebra. (2011:$120,000; 2012: $120,000; 2013:$120,000.)
• A/Prof Andrew Mathas, Prof Jonathan Brundan: Graded representations of Hecke algebras. (2011: $149,000; 2012:$130,000; 2013: $130,000.) • A/Prof Alexander I Molev: Vertex algebras and representations of quantum groups. (2011:$110,000; 2012: $110,000; 2013:$110,000.)
• Dr James Parkinson, Asst Prof Joel Kamnitzer: The geometry and combinatorics of loop groups. (2011: $56,200.00 2012:$44,000; 2013: $43,000.) • Dr Anne C Thomas, Dr Inna A Capdeboscq: Lattices in locally compact groups. (2011:$73,185; 2012: $67,885; 2013:$74,285; 2014: $61,385.) Australian Postdoctoral Fellowship awarded to Dr Anne C Thomas. • Dr Benjamin A Burton, Dr Murray J Elder, Dr Stephan Tillmann: Generic complexity in computational topology: breaking through the bottlenecks. (2011:$85,000; 2012: $85,000; 2013:$85,000.)
• Dr Martin Wechselberger, Prof Graeme J Pettet, Prof Christopher K Jones: A geometric theory for travelling waves in advection-reaction-diffusion models. (2011: $85,000; 2012:$85,000; 2013: $85,000.) • Dr Zhou Zhang: Topological and analytic aspects of the Kaehler-Ricci flow. (2011:$22,560.00; 2012: $22,560.00; 2013:$22,560.00.)
• Prof Matt P Wand, Dr John T Ormerod, Prof Yongmei Michelle Wang: Fast approximate inference methods for flexible regression. (2011: $110,000; 2012:$110,000; 2013: $110,000.) • Prof Alan H Welsh, Dr Samuel Müller Building models for complex data. (2011:$130,000; 2012: $110,000; 2013:$110,000.)

Human Frontiers of Science Program

• Dr Guy Lyons, Dr Silvio Gutkind, A/Prof Mary Myerscough: Cell cooperation in cancer. (2011: US$300,000; 2012: US$300,000; 2013: US$300,000.) ## Externally funded research projects commencing in 2010 ARC Discovery Projects • Prof JJ Cannon, Dr DF Holt, Prof J Carlson: Constructive Module Theory for Algebras. (2010:$100,000; 2011: $100,000; 2012:$100,000.)
• Dr C Ewald: Quantitative and qualitative aspects of Asian and Australian options. (2010: $50,000; 2011:$50,000; 2012: $50,000.) • A/Prof G Gottwald: Stochastic methods in mathematical geophysical fluid dynamics. (2010:$90,000; 2011: $100,000; 2012:$100,000; 2013: $55,000; 2014:$55,000.)
Australian Research Fellowship awarded to Georg Gottwald.
• Dr V Jayaswal: Statistical methods for analysing multi-source microarray data and building gene regulatory networks. (2010: $85,000; 2011:$83,000; 2012: $83,000.) Australian Postdoctoral Fellowship awarded to Dr V Jayaswal. • Dr L Paunescu: The canonical stratification of jet spaces. (2010:$50,000; 2011: $50,000; 2012:$50,000.)
• Prof JH Rubinstein, A/Prof CD Hodgson, Dr S Tillmann: Triangulations in dimension three: algorithms and geometric structures. (2010: $120,000; 2011:$125,000; 2012: $100,000.) ARC Linkage Project • A/Prof Holger Dullin, Damien Sinclair O'Meara and Peter Singh Surya: Bodies in Space. (2010:$27,500.00; 2011: $55,000.00; 2012:$55,000.00; 2013: $27,500.00. New South Wales Institute of Sport.) ## Externally funded research projects commencing in 2009 ARC Discovery Projects • Dr A Henderson: The geometry of exotic nilpotent cones. (2009:$20,000; 2010: $20,000.) • Prof N Joshi: Integrable Lattice Equations. (2009:$108,000; 2010: $80,000; 2011:$90,000.)
• A/Prof A Mathas; Dr A Henderson: Pyramids and decomposition numbers for the symmetric and general linear groups. (2009: $98,000; 2010:$85,000; 2011: $78,000; 2012:$80,000; 2013: $87,000.) Australian Professorial Fellowship awarded to A/Prof Andrew Mathas. • Dr S Santra: Blow-up phenomena in semilinear elliptic partial differential equations. (2009:$69,000; 2010: $67,000; 2011:$67,000; 2012: $67,000.) Australian Postdoctoral Fellowship awarded to Sanjiban Santra. • A/Prof R Zhang: Noncommutative geometry in representation theory and quantum physics. (2009:$90,000; 2010: $90,000; 2011:$80,000; 2012: $90,000; 2013:$100,000.)
Australian Professorial Fellowship awarded to A/Prof Ruibin Zhang.

ARC Future Fellowships

• Dr Georg Gottwald: Stochastic Methods in Mathematical Geophysical Fluid Dynamics. (2009: $95,250; 2010:$185,650; 2011: $185,650; 2012:$185,650; 2013: $90,400.) • Dr Jean Yang: New statistical methods for identifying micro-ribonucleic acid (miRNA) regulatory networks. (2009:$80,800; 2010: $151,600; 2011:$151,600; 2012: $151,600; 2013:$70,800.)

• Prof J George, Dr Jean Yang, Dr FC McKay, Dr V Suppiah; Dr DR Booth, Prof G Stewart, Dr M Bahlo: Functional Genomics to Predict and Enhance Response to Interferon. (2009: $145,000; 2010:$110,000; 2011: \$115,000.)