## Grants

### Diagram categories and transformation semigroups. ARC Future Fellowship - 2019-2023

**Dr James East** - FT190100632

A structural understanding of diagram categories is essential in many branches of mathematics and science. Despite this, very few methods for studying such categories are available, a fact this pure mathematics project seeks to rectify. By building strong bridges between diagram categories and semigroup theory, a field of abstract algebra that models transformation and change, the structure of diagram categories may be unlocked with powerful semigroup tools developed by the applicant investigator. Diagrammatic insights will also yield new ways to study semigroups, and the many other mathematical structures they interact with. Outcomes will have a lasting impact on both theories as well as the many fields influenced by them. $785,823.00

### An efficient approach to the computation of bacterial evolutionary distance. ARC Discovery Grant 2018 - 2021(via University of Tasmania)

**Prof Andrew Francis - **DP180102215

This project aims to apply advanced mathematical tools to improve our understanding of bacterial evolution. Bacteria account for as much total Earth biomass as all plant species put together, and have an unparalleled ability to evolve quickly and adapt to changing environments. Unfortunately, the existing mathematical models used to model bacterial evolution are generally computationally intractable. This project will rectify this situation by using representation theory to transform combinatorial group theory into linear algebra, allowing for the application of advanced methods of numeric approximation. This will provide a better understanding of how bacteria evolve and improve our ability to manage their impact. Announced Funding: $338,178.00

### Deep Conceptors for Temporal Data Mining - DAAD funding Jan 2017 - Dec 2018.

**A/Prof Oliver Obst **and Dr Sangeeta Bhatia

A collaboration with Prof Frieder Stolzenburg at Harz University of Applied Sciences (in Wernigerode, Germany. Prof. Dr. Frieder Stolzenburg (opens in a new window)). Specifically, Frieder and his student Falk will come visit CRM here (in 2017 and 2018), but also Sangeeta and Oliver will visit Germany, over the duration of the project.

### Security and Privacy of Individual Data Used to Extract Public Information - ARC Discovery Grant 2016-2019 (via RMIT Uni)

**A/Prof Leanne Rylands**, Prof Xun Yi (RMIT), Prof Jennifer Seberry (Wollongong), Partner Investigator: Prof Dr Josep Domingo-Ferrer (Universitat Rovira i Virgili)

The project aims to contribute to the development of techniques to allow the harvesting of useful information without compromising personal privacy. Intelligent analysis of personal data can reveal valuable knowledge about a population but at a risk of invading an individual's privacy. This project aims to provide at least partial solutions to some of the problems associated with the protection of private data. In particular, it plans to work on the problem of security of statistical databases and privacy of streaming data. This would be underpinned by a study of anonymisation and homomorphic encryption. The expected outcomes are new theoretical results, new algorithms and protocols applicable to at least some of the current significant problems in information security

### Graded K-theory as invariants for path algebras - ARC Grant 2016-2019

**Roozbeh Hazrat,** Pere Ara (University of Autonoma Barcelona) and Gene Abrams (University of Colorado)

This pure mathematics project focuses on Leavitt path algebras, which are structures that naturally arise from movements on directed graphs. These algebras appear in di¬verse areas (eg analysis, noncommutative geometry, representation theory and group theory). The aim of this project is to understand the behaviour of Leavitt path algebras and to classify them completely by means of graded K-theory. The project is an algebraic counterpart to graph C*-algebras (analytic structures that originated in Australian universities); both subjects have become areas of intensive research globally. The expected outcomes are to classify Leavitt path algebras, and to find a bridge (via graded K-theory) to graph C*-algebras and symbolic dynamics.

### Groupoids as bridges between algebra and analysis - ARC Grant 2015-2017 (via Wollongong Uni)

**Roozbeh Hazrat**, Aidan Sims (Wollongong) and David Pask (Wollongong)

This project is in pure mathematics and focusses on the interplay between abstract algebra and operator algebras. Specifically it deals with generalisations of graph C*-algebras and of Leavitt path algebras. Over the last decade, researchers have discovered remarkable similarities between the areas, but have not found a unifying explanation that would allow us to capitalise on them to transfer information from one area to the other. The CIs have recently obtained preliminary results showing that groupoids may provide the crucial missing link. This project will determine the role of groupoids in the two theories, and analyse and exploit the resulting synnergies between abstract algebra and functional analysis.

### Algebraic algorithms for investigating the space of bacterial genomes - ARC Discovery Grant 2013-2015

Andrew Francis and Volker Gebhardt

The aim of this project is to develop algorithmic approaches to algebraic problems associated with bacterial evolution. Building realistic group-theoretic models of bacterial evolution based on the inversion process, this project will establish methods for determining the evolutionary distance between two genomes. It will also address the central problem of constructing a phylogeny relating several bacterial genomes from the point of view of geometric group theory and walks on the Cayley graph. The outcomes will be new methods for evolutionary biology, and new results and algorithms in computational, combinatorial and geometric group theory.

### Quantized identification of feedback control systems - ARC Discovery Grant 2012-2017

The theory of system identification with quantified data underpins frontier technologies that enable more efficient and sustainable telecommunications, automotive and biomedical industry. This project extends the fundamental framework of quantified system identification. The work will enhance Australia's international standing in the control field.

### Algebraic evolution and evolutionary algebra - ARC Future Fellowships 2010-2014

Mathematics has made numerous significant contributions to our understanding of biological systems. This project brings a new approach to mathematical biology by modelling evolutionary processes in bacteria using algebraic ideas. This will not only provide the answers to questions in bacterial evolution that are otherwise unsolved and provide new mathematical and computational tools for biologists, but identify important new areas of research for algebraists.

### Algorithmic approaches to braids and their generalisations - ARC Discovery Grant 2010-2012

Volker Gebhardt, Patrick Dehornoy, Juan Gonzalez-Meneses

This project combines theoretical methods from pure mathematics with computational experiments in order to gain new knowledge. The objects of interest, so-called braid groups and generalisations, are important for many fields of mathematics, but also have applications for data security.

Both the theoretical outcomes of this project and the algorithms developed will strengthen Australia as a centre of cutting-edge research in computational algebra. Moreover, the results can lead to new technologies for protecting confidential data, which are more efficient and hence cheaper to implement than existing alternatives. Secure identification of legitimate users in the context of online banking is one possible field of application.

### Mathematical models and bioinformatic analyses of bacterial genome evolution - ARC Discovery Grant 2009-2013

Mark Tanaka, Andrew Francis and Ruiting Lan

This project aims to understand the evolution of bacterial genome organisation. It seeks to explain: why genes of a common pathway are often clustered along chromosomes, how mobile genes can survive despite their damaging effects, and why there is wide genomic variation within some bacterial species. We will construct biologically grounded mathematical models describing the relevant processes, using them to analyse the abundant genome data. This will allow discrimination among hypotheses concerning the observed genome structures. This research will make progress towards a coherent theory of bacterial genome evolution, and hence a better understanding of bacterial pathogens.

### Development of Identification Methods for Nonlinear Dynamical Systems - ARC Discovery Grant 2007-2009

Wei Zheng, E.-W. Bai and Y. Zheng

It is widely recognized that nonlinear systems theory will mark a new era of control science in the coming decade, and will be used in various types of applications. Driven by such immense opportunities and needs, identification of nonlinear systems is emerging as a vital, active area of research. The success of this project will enhance Australia's leading role in the international control community. The training of the postdoctoral research associates will generate the expertise needed to maintain the involvement of the coming generation in cutting-edge technological advancement. The project will strengthen research activities in Australia through strong international collaborations.