# Staff Profile

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## Career Summary

### Biography

A continuing theme of George Willis's research since he received a PhD from the University of Newcastle upon Tyne in 1981 has been the interaction between algebra and topology. However, the focus of his research has shifted from analysis to algebra, and he has also moved from being primarily a problem solver to being a creator of new mathematics.

His early work was in functional analysis, which is an all-embracing framework for such fundamental techniques as Fourier and Laplace transforms and the spectral theory of linear operators. In a subject called `automatic continuity', he demonstrated that algebraic properties determine topological ones in many circumstances and he has also established the strongest known results that distinguish between approximate and exact factorization. Most notably, he showed in 1991 that the compact approximation property for Banach spaces does not imply the approximation property, thus answering a question that had been open for 20 years. This result sheds light on how well infinite-dimensional spaces may be approximated by finite-dimensional ones.

In his more recent work, since moving to Newcastle (Australia) in 1992, he has done much to create the modern structure theory of totally disconnected, locally compact groups by introducing the notions of scale, tidy subgroups and flatness. Some of this work has been surveyed in the book `Banach algebras and the general theory of *-algebras', Volume II, Cambridge University Press (2001) by T. W. Palmer, where it is called `Willis theory' and said to provide `a whole new level of insight'. It has also been described in Mathematical Reviews as offering `genuinely original insights' into a `notoriously intractable' subject. Links with ergodic theory, geometric group theory and with arithmetic groups are currently being explored, amongst others.

### Qualifications

- PhD, University of Newcastle upon Tyne - England
- Bachelor of Science (Honours), University of Adelaide
- Bachelor of Science, University of Adelaide

## Research

### Research keywords

- Algebra
- Functional analysis
- Group theory

### Research expertise

Totally disconnected locally compact groups:

A structure theory for these groups is being developed that is beginning to parallel the theory of Lie groups. Totally disconnected groups occur as symmetry groups of discrete structures and the theory has links with other areas of mathematics such as: number theory; harmonic analysis; graph theory; geometry of relational structure such as trees and buildings; CAT (0) and hyperbolic spaces; ergodic theory and dynamical systems; p-adic analysis; finite and profinite groups; and Lie and algebraic groups. The workshop Totally Disconnected Groups, Graphs and Geometry held in Blaubeuren, Germany, from May 7th-12th 2008, explored some of these links. The basics of the new structure theory are described at http://en.wikipedia.org/wiki/Totally_disconnected_group

Beyond the spectrum:

This research aims to develop methods for analyzing radical Banach algebras that would apply in realms beyond the reach of such widely applied spectral methods as: eigenvalues; the spectrum of linear operators; and the Fourier, Laplace and, more generally, Gel'fand transforms.

Topological algebra:

Other areas of topological algebra of interest include: approximate identities and factorization; random walks on groups; abstract harmonic analysis, cohomology of Banach algebras and amenability; operator algebras; and Banach space approximation properties

### Fields of Research

Code | Description | Percentage |
---|---|---|

010105 | Group Theory And Generalisations | 100 |

### Memberships

#### Editorial Board.

- Associate Editor - Australian Mathematical Society

### Appointments

ARC - Aust Research Fellowship (level B C)ARC (Australia) | 01/01/2006 - 01/12/2006 |

Fellowship APFAustralian Research Council (Australia) | 01/04/2009 |

### Invitations

Locally comnpact groups beyond Lie theoryUniversité Louvain la Neuve, Belgium (Conference Presentation - non published.) | 2013 |

Banach Algebras 2007Laval University, Quebec, Canada (Conference Presentation - non published.) | 2007 |

British Mathematical ColloquiumLondon Mathematical Society, United Kingdom (Conference Presentation - non published.) | 2003 |

`Lie Groups, Representations and Discrete Mathematics' WorkshopPrinceton Insitute for Advanced Study, United States (Invitations to speak at Princeton IAS) | 2006 |

Topological transitivity of group actionsLa Trobe University, Australia (PHD Examiner.) | 2007 |

London Mathematical Society LecturerLondon Mathematical Society, United Kingdom (
) | 2007 |

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## Highlighted Publications

For publications that are currently unpublished or in-press, details are shown in italics.

Year | Citation | Altmetrics | Link |
---|---|---|---|

2004 | Willis GA, 'Tidy subgroups for commuting automorphisms of totally disconnected groups: An analogue of simultaneous triangularisation of Matrices', New York Journal of Mathematics, 10 1-35 (2004) [C1] | ||

2001 | Willis GA, 'Further properties of the scale function on a totally disconnected group', Journal of Algebra, 237 142-164 (2001) [C1] | ||

2004 | Baumgartner U, Willis GA, 'Contraction groups and scales of automorphisms of totally disconnected locally compact groups', Israel Journal of Mathematics, 142 221-248 (2004) [C1]
| ||

2001 | Willis GA, 'Factorization in finite-codimensional ideals of group algebras', Proc. London Math. Soc., 82 676-700 (2001) [C1] |

## Publications

For publications that are currently unpublished or in-press, details are shown in italics.

Click on a category title below to expand the list of citations for that specific category.

### Chapter (1 outputs)

Year | Citation | Altmetrics | Link |
---|---|---|---|

2003 | Willis GA, 'Harmonic Analysis and amenability', Introduction to Banach Algebras, Operators and Harmonic Analysis, CUP, Cambridge 321 (2003) [B2] |

### Journal article (50 outputs)

Year | Citation | Altmetrics | Link |
---|---|---|---|

2014 | Willis GA, 'The nub of an automorphism of a totally disconnected, locally compact group', Ergodic Theory and Dynamical Systems, 34 1365-1394 (2014) [C1] | ||

2014 | Caprace P-E, Reid CD, Willis GA, 'Limits of contraction groups and the tits core', Journal of Lie Theory, 24 957-967 (2014) [C1] | ||

2014 | Willis GA, 'The scale and tidy subgroups for endomorphisms of totally disconnected locally compact groups', Mathematische Annalen, (2014) | ||

2013 | Caprace P-E, Reid CD, Willis GA, 'Locally normal subgroups of simple locally compact groups', Comptes Rendus Mathematique, (2013) [C1] | ||

2013 | Shalom Y, Willis GA, 'Commensurated Subgroups of Arithmetic Groups, Totally Disconnected Groups and Adelic Rigidity', GEOMETRIC AND FUNCTIONAL ANALYSIS, 23 1631-1683 (2013) [C1] | ||

2012 | Baumgartner U, Moller RG, Willis GA, 'Hyperbolic groups have flat-rank at most 1', Israel Journal of Mathematics, 190 365-388 (2012) [C1] | ||

2012 | Willis GA, 'Probability measures on semigroups: Convolution products, random walks and random matrices [Book Review]', SIAM Review, 54 414-416 (2012) [C3] | ||

2010 | Baumgartner U, Schlichting G, Willis GA, 'Geometric characterization of flat groups of automorphisms', Groups, Geometry, and Dynamics, 4 1-13 (2010) [C1]
| ||

2010 | Glockner H, Willis GA, 'Classification of the simple factors appearing in composition series of totally disconnected contraction groups', Journal Fur Die Reine Und Angewandte Mathematik, 141-169 (2010) [C1] | ||

2010 | Ghahramani F, Read CJ, Willis GA, 'Closed ideal structure and cohomological properties of certain radical Banach algebras', Proceedings of the London Mathematical Society, 100 533-559 (2010) [C1]
| ||

2009 | Baumgartner U, Laca M, Ramagge J, Willis GA, 'Hecke algebras from groups acting on trees and HNN extensions', Journal of Algebra, 321 3065-3088 (2009) [C1] | ||

2007 | Baumgartner U, Remy B, Willis GA, 'Flat rank of automorphism groups of buildings', Transformation Groups, 12 413-436 (2007) [C1] | ||

2007 | Willis GA, 'Compact open subgroups in simple totally disconnected groups', Journal of Algebra, 312 405-417 (2007) [C1] | ||

2006 | Baumgartner U, Ramagge J, Willis GA, 'A compactly generated group whose Hecke algebras admit no bounds on their representations', Glasgow Mathematical Journal, 48 193-201 (2006) [C1] | ||

2006 | Dani SG, Shah NA, Willis GA, 'Locally compact groups with dense orbits under Z(d)-actions by automorphisms', Ergodic Theory and Dynamical Systems, 26 1443-1465 (2006) [C1] | ||

2006 | Baumgartner U, Willis GA, 'The direction of an automorphism of a totally disconnected locally compact group', Mathematische Zeitschrift, 252 393-428 (2006) [C1] | ||

2004 | Willis GA, 'Tidy subgroups for commuting automorphisms of totally disconnected groups: An analogue of simultaneous triangularisation of Matrices', New York Journal of Mathematics, 10 1-35 (2004) [C1] | ||

2004 | Willis GA, 'A noncommutative half-angle formula', Bulletin of Australian Mathematical Society, 69 369-382 (2004) [C1] | ||

2004 | Baumgartner U, Willis GA, 'Contraction groups and scales of automorphisms of totally disconnected locally compact groups', Israel Journal of Mathematics, 142 221-248 (2004) [C1]
| ||

2004 | Abdollahi A, Rejali A, Willis GA, 'Group Properties characterised by configurations', Illinois Journal of Mathematics, 48 861-873 (2004) [C1] | ||

2001 | Kepert AG, Willis G, 'Scale Functions and Tree Ends', Journal of Australian Mathematics Society, 70 273-292 (2001) [C1]
| ||

2001 | Willis GA, 'Factorization in finite-codimensional ideals of group algebras', Proc. London Math. Soc., 82 676-700 (2001) [C1] | ||

2001 | Willis GA, 'Further properties of the scale function on a totally disconnected group', Journal of Algebra, 237 142-164 (2001) [C1] | ||

2001 | Glockner H, Willis GA, '$p$-adic Lie groups', Forum Math, 13 273-292 (2001) [C1] | ||

2001 | Rosenblatt JM, Willis GA, 'Weak convergence is not strong convergence for amenable groups', Canadian Math. Bull., 44 231-241 (2001) [C1] | ||

2001 | Willis GA, 'The number of prime factors of the scale function on a compactly generated group is finite', Bulletin London Mathematical Society, 33 168-174 (2001) [C1] | ||

2001 | Glockner H, Willis GA, 'Uniscalar p-adic Lie groups', FORUM MATHEMATICUM, 13 413-421 (2001)
| ||

2000 | Willis GA, Loy RJ, Read CJ, Runde V, 'Amenable and weakly amenable Banach algebras with compact multiplication', Journal of Functional Analysis, 171 78-114 (2000) [C1] | ||

2000 | Willis GA, 'An extension of a non-commutative Choquet-Deny Theorem', Proceedings of the American Mathematical Society, 128 111-118 (2000) [C1] | ||

2000 | Willis GA, Ghahramani F, Runde V, 'Derivations on Group Algebras', Proceedings of the London Mathematical Society, 80 360-390 (2000) [C1] | ||

Show 47 more |

### Review (2 outputs)

Year | Citation | Altmetrics | Link |
---|---|---|---|

2002 | Willis GA, 'Banach algebras and the general theory of *-algebras: Vol II, *-algebras', Bulletin of London Mathematical Society (2002) [D1] | ||

2002 | Willis GA, 'The Honors Class: Hilbert's Problems and Their Solvers; Benjamin H. Yandell', Australian Mathematical Society Gazette (2002) [D1] |

### Conference (3 outputs)

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## Grants and Funding

### Summary

Number of grants | 16 |
---|---|

Total funding | $1,741,987 |

Click on a grant title below to expand the full details for that specific grant.

### 2012 (1 grants)

#### Theory and applications of symmetries of relational structures$325,000

**Funding Body: **ARC (Australian Research Council)

Project Team | ||
---|---|---|

Professor George Willis, Doctor Murray Elder | ||

Scheme | Role | |

Discovery Projects | Chief Investigator | |

Total Amount | Funding Start | Funding Finish |

$325,000 | 2012 | 2014 |

GNo:G1100080 |

### 2009 (2 grants)

#### Totally disconnected groups in algebra and geometry$381,868

**Funding Body: **ARC (Australian Research Council)

Project Team | ||
---|---|---|

Professor George Willis, Professor Jacqui Ramagge | ||

Scheme | Role | |

Discovery Projects | Chief Investigator | |

Total Amount | Funding Start | Funding Finish |

$381,868 | 2009 | 2013 |

GNo:G0188711 |

#### Computer Assisted Research Mathematics and its Applications$267,361

**Funding Body: **University of Newcastle

Project Team | ||
---|---|---|

Laureate Professor Jon Borwein, Professor Natashia Boland, Associate Professor Brailey Sims, Professor George Willis | ||

Scheme | Role | |

Priority Research Centre | Investigator | |

Total Amount | Funding Start | Funding Finish |

$267,361 | 2009 | 2012 |

GNo:G0189866 |

### 2008 (1 grants)

#### New Applications of Totally Disconnected Groups$20,000

**Funding Body: **University of Newcastle

Project Team | ||
---|---|---|

Professor George Willis, Dr Jacqueline Ramagge | ||

Scheme | Role | |

Near Miss Grant | Chief Investigator | |

Total Amount | Funding Start | Funding Finish |

$20,000 | 2008 | 2008 |

GNo:G0188399 |

### 2007 (1 grants)

#### Totally disconnected groups, graphs and geometry$73,787

**Funding Body: **Volkswagen Foundation

Scheme | Role | |
---|---|---|

Conference funding | Chief Investigator | |

Total Amount | Funding Start | Funding Finish |

$73,787 | 2007 | 2007 |

GNo:5656980 |

### 2006 (1 grants)

#### Totally disconnected groups, representations and discrete mathematics.$60,000

**Funding Body: **ARC (Australian Research Council)

Project Team | ||
---|---|---|

Professor George Willis | ||

Scheme | Role | |

Linkage International Fellowship | Chief Investigator | |

Total Amount | Funding Start | Funding Finish |

$60,000 | 2006 | 2006 |

GNo:G0185350 |

### 2005 (2 grants)

#### Geometric representation of small-rank totally disconnected groups$234,000

**Funding Body: **ARC (Australian Research Council)

Project Team | ||
---|---|---|

Professor George Willis, Dr Jacqueline Ramagge | ||

Scheme | Role | |

Discovery Projects | Chief Investigator | |

Total Amount | Funding Start | Funding Finish |

$234,000 | 2005 | 2007 |

GNo:G0184324 |

#### International Conference in Mathematics and Applications, 15-17 December 2005$2,070

**Funding Body: **University of Newcastle

Project Team | ||
---|---|---|

Professor George Willis | ||

Scheme | Role | |

Travel Grant | Chief Investigator | |

Total Amount | Funding Start | Funding Finish |

$2,070 | 2005 | 2005 |

GNo:G0185950 |

### 2003 (1 grants)

#### Lie-type methods for totally disconnected groups.$28,800

**Funding Body: **ARC (Australian Research Council)

Project Team | ||
---|---|---|

Professor George Willis | ||

Scheme | Role | |

Linkage International Award | Chief Investigator | |

Total Amount | Funding Start | Funding Finish |

$28,800 | 2003 | 2005 |

GNo:G0182680 |

### 2002 (2 grants)

#### Totally disconnected groups and their algebras$185,000

**Funding Body: **ARC (Australian Research Council)

Project Team | ||
---|---|---|

Professor George Willis, Dr Jacqueline Ramagge | ||

Scheme | Role | |

Discovery Projects | Chief Investigator | |

Total Amount | Funding Start | Funding Finish |

$185,000 | 2002 | 2004 |

GNo:G0181137 |

#### Visit of Dr Niels Laustsen, from 11 May 2002 to 21 June 2002$2,400

**Funding Body: **University of Newcastle

Project Team | ||
---|---|---|

Professor George Willis | ||

Scheme | Role | |

Visitor Grant | Chief Investigator | |

Total Amount | Funding Start | Funding Finish |

$2,400 | 2002 | 2002 |

GNo:G0181911 |

### 2001 (2 grants)

#### 1. International Conf on Banach Algebras & Cohomology, England, 25-28 June. 2. Random Walks & Geometry, Austria, June/July. 3. BALTICON 2001, Denmark, 5-18 Aug$2,500

**Funding Body: **University of Newcastle

Project Team | ||
---|---|---|

Professor George Willis | ||

Scheme | Role | |

Travel Grant | Chief Investigator | |

Total Amount | Funding Start | Funding Finish |

$2,500 | 2001 | 2001 |

GNo:G0180986 |

#### Prof Fereidoun Ghahramani, From 18 Feb to 21 April 2001$2,055

**Funding Body: **University of Newcastle

Project Team | ||
---|---|---|

Professor George Willis | ||

Scheme | Role | |

Visitor Grant | Chief Investigator | |

Total Amount | Funding Start | Funding Finish |

$2,055 | 2001 | 2001 |

GNo:G0180587 |

### 2000 (1 grants)

#### Analysis of radical algebras.$5,000

**Funding Body: **University of Newcastle

Project Team | ||
---|---|---|

Professor George Willis | ||

Scheme | Role | |

Project Grant | Chief Investigator | |

Total Amount | Funding Start | Funding Finish |

$5,000 | 2000 | 2000 |

GNo:G0178919 |

### 1997 (2 grants)

#### Symmetry and a structure theory for totally disconnected, locally compact groups.$150,000

**Funding Body: **ARC (Australian Research Council)

Project Team | ||
---|---|---|

Professor George Willis | ||

Scheme | Role | |

Large Grant | Chief Investigator | |

Total Amount | Funding Start | Funding Finish |

$150,000 | 1997 | 1999 |

GNo:G0176189 |

#### Random Walks and Discrete Potential Theory, Cortona, Italy, 22-28 June 1997$2,146

**Funding Body: **University of Newcastle

Project Team | ||
---|---|---|

Professor George Willis | ||

Scheme | Role | |

Travel Grant | Chief Investigator | |

Total Amount | Funding Start | Funding Finish |

$2,146 | 1997 | 1997 |

GNo:G0179608 |

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## Research Supervision

Number of current supervisions | 6 |
---|---|

Total current UoN PhD EFTSL | 2.7 |

For supervisions undertaken at an institution other that the University of Newcastle, the institution name is listed below the program name.

### Current Supervision

Commenced | Proposed Completion | Program | Supervisor Type | Research Title |
---|---|---|---|---|

2014 | 2018 | PhD (Mathematics) | Principal Supervisor | Classification of Simple Totally Disconnected Groups. |

2014 | 2016 | PhD (Mathematics) | Principal Supervisor | Topological Algebra |

2013 | 2017 | PhD (Mathematics) | Principal Supervisor | Automorphisms of Totally Disconnected, Locally Compact Groups and the Continuity of the Scale |

2011 | 2015 | PhD (Mathematics) | Principal Supervisor | Structure of Discrete and Non-discrete Locally Primitive Groups Acting on Trees |

2006 | 2010 | PhD (Mathematics) | Principal Supervisor | Tree Actions Associated with the Scale of an Automorphism of a Totally Disconnected Locally Compact Group |

2003 | 2012 | PhD (Mathematics) | Co-Supervisor | Nonlinear Analysis in Geodesic Metric Spaces |

### Past Supervision

Year | Program | Supervisor Type | Research Title |
---|---|---|---|

2014 | PhD (Mathematics) | Co-Supervisor | Fourier and Wavelet Analysis of Clifford-Valued Functions |

2011 | PhD (Mathematics) | Co-Supervisor | Connections Between Geometrical and Fixed Point Properties |

2011 | PhD (Mathematics) | Principal Supervisor | The Categorical Properties of Higher Rank Graphs and Applications to their Graph Algebras |

2005 | PhD (Mathematics) | Co-Supervisor | A New Structure for Generalized Metric Spaces - with Applications to Fixed Point Theory |

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## Professor George Willis

Work Phone | (02) 4921 5666 |
---|---|

Fax | (02) 4921 6898 |

George.Willis@newcastle.edu.au | |

Position | ProfessorSchool of Mathematical and Physical Sciences Faculty of Science and Information Technology |

Focus Area | Mathematics |

Office | V131, Mathematics, Callaghan University Drive Callaghan NSW 2308 Australia |

URL: | www.newcastle.edu.au/profile/george-willis |