David Gauld

e-mail me at my email address

I am a professor in the Department of Mathematics of the University of Auckland, having been in the Department as a lecturer and so on since 1969. Well, to go back to the beginning, I was born in Inglewood, NZ, and attended the long since defunct Pukeho Primary School (last time I visited I wondered whether the school building will outlast the last of its former pupils. Challenge: I can name all of the children who went to my primary school during the whole of the time I attended; can you?). Three of my high school years were at Inglewood High School, which celebrated its 50th anniversary in 2007. Here's a photo of my fifth form class. This is the entire fifth form at IHS that year: why are there twice as many girls as boys? The year before IHS opened I attended Wanganui Technical College and my last high school year was at the New Plymouth Boys' High School. I did my BSc and MSc degrees at Auckland before moving to Los Angeles where I completed a PhD in topology. I am a member of the Allahabad, American, Australian and New Zealand Mathematical Societies, having been President of the last of these in 1981-2. I am also the founding secretary of the New Zealand Mathematics Research Institute. My office is in room 432 of the Mathematical and Physical Sciences Building. My phone number is (64)(9)3737599 ext 88697. Look at this map of the world to see where my office is.

I have served in a number of administrative roles in the University. Most notable is as Head of the Department of Mathematics or Mathematics and Statistics. My terms as Head were from 1/6/1981 to 31/1/1990, 1/2/1993 to 14/8/1994 and 21/5/2001 to 31/1/2005. From 15/8/1994 to 28/2/1997 I was the University's Assistant Vice-Chancellor (Research) and from 1/1/2001 until 31/7/2001 I was Associate Dean for Budgets for the Faculty of Science.

I have had five PhD students who have completed their degrees. Two of those, Sina Greenwood and Abdul Mohamad, graduated in 1999 and a third, Kerry Richardson, in 2000 but they are still working hard . All three were co-supervised by David McIntyre and I and all three have been awarded a New Zealand Science and Technology Post-Doctoral Fellowship. Sina remains in the Department having transferred to a lectureship in the Department on 29 February 2004 when her fellowship ended (they must have forgotten that 2004 was a leap year!), and is now a senior lecturer. Abdul took up his fellowship at the University of Auckland for a while but decided to accept a post at Sultan Qaboos University, Oman where he is now an associate professor. Kerry went to Japan late 2001 on a post-doctoral fellowship sponsored by the Japan Society for the Promotion of Science before being awarded his NZS&T Fellowship; he has recently returned from Boston. My fourth PhD student, Brian Van Dam, completed his formalities in June 2004 and went to teach at Wollongong University's Auckland branch. My fifth PhD student was Stevie Budden who was supported by the NZIMA Programme referred to below and who completed his formalities in September 2009: he has returned to England. I was also mentor for another New Zealand Science and Technology Post-Doctoral Fellow, Jiling Cao, who also completed his PhD in this Department (with Ivan Reilly and M K Vamanamurthy as supervisors) and held a JSPS Post-Doctoral Fellowship. Jiling is now an associate professor at the Auckland University of Technology. Look at a photo of Sina, Abdul and Jiling, who also graduated in 1999, along with David McIntyre and I.

Sunanda Dikshit, whose most recent degree is from Berhampur University in India, joined us as a PhD student in September, 2007. My most recent new PhD student is Afshin Mardani from Shiraz University in Iran. He arrived in May 2008.

For a month during January and February 2005 I visited Abdul Mohamad at Sultan Qaboos University in Oman where I did some work with both Abdul and Jiling Cao, who also visited SQU. Apart from the mathematical excitement I found the visit to Oman very interesting: the country is so vibrant, which is amazing considering the changes which have taken place there in the past 40 years. We met many wonderful people and saw some amazing sights. Abdul and I even got to climb the highest mountain in the Arabian Peninsula, Jabal Shams (Mountain of the Sun), 3009 metres. I shall return to Oman for a fortnight in January/February 2010, in part to attend the International Conference on Analysis and its Applications in Muscat.

I took research leave from the beginning of July 2005 to the end of June 2006. I find travel disrupting and, given that I had a lot of work lying incomplete while I was Head of Department, I spent most of the leave working in Auckland. I did spend about 7 weeks from early September to late October 2005 overseas, giving talks at Queens University (Belfast), the National University of Ireland in Galway, Universite de Bourgogne and Universite de Geneve. Most of the time I was away I stayed in Dijon. I also attended the conference Manifolds at Melbourne in January 2006 and went to the Harish Chandra Institute in Allahabad, India, late February and early March to begin my Marsden funded research project with Professor Satya Deo.

As part of the NZIMA Programme, Roger Fenn of Sussex University visited until late April 2006. While he was here we organised three conferences (workshops) with Vaughan Jones. The wrap-up conference of the Programme takes place in Hahei in January 2010.

Making use of a welcome Marsden grant I was able to invite three visitors here in the middle of 2006: Mathieu Bailiff of Geneva, Abdul Mohamad of Oman and Peter Nyikos of Columbia, South Carolina.

I had another semester of research leave for the first half of 2007 but again I spent most of that time in Auckland. I had two visitors during the first half of 2007, their visits being funded at least in part by my Marsden award. Paul Gartside of Pittsburgh was here from February to April and Aisling McCluskey of Galway was here for April. Right at the end of that leave I visited Aisling in Galway and Belfast as well as Mathieu Baillif and Alexandre Gabard in Geneva.

At the end of 2007 I had a visit from V. Lakshmana Gomathi Nayagam of the National Institute of Technology, Tiruchirappalli, India. Together we worked on intuitionistic fuzzy spaces, a topic which is new to me. Peter Nyikos made a return visit in late 2008, at the same time as Alexandre Arhangelskii of Ohio and Moscow visited. During May and June, 2009 Satya Deo of the Harish Chandra Institute made a return visit, and Abdul Mohamad made a return visit in August 2009.

Summer Topology Conference

Research Interests

We have quite an active group in topology at the University of Auckland at the moment. Some indication of this activity may be seen by reading my final report to the Marsden Fund, which for 3 years supported part of this group. We also have a nice lot of visitors coming through.

• In 2004 we had the following visitors: Chris Good of Birmingham University, UK; Zbigniew Piotrowski of Youngstown State University, USA; Grant Woods of the University of Manitoba, Canada; Peter Nyikos of the University of South Carolina, USA and Abdul Mohamad of Sultan Qaboos University, Oman.
• In 2005 Aisling McCluskey (NUI, Galway, Ireland), Dima Shakhmatov (Ehime University, Japan) and Artur Tomita (University of Sao Paulo Brazil) visited.
• In 2006 we had many topologically minded visitors but either they were here just briefly for the NZ Japan Knot Theory Conference or they shot through Auckland on the way to Taipa. Roger Fenn of Sussex University stayed here for a few months as part of the NZIMA Programme. In May and June Peter Nyikos of Columbia, South Carolina, Abdul Mohamad of Sultan Qaboos University Oman and Mathieu Baillif of Geneva visited.
• In 2007 four visitors came: Brian Raines (Baylor University, Texas), Paul Gartside (University of Pittsburgh, Pennsylvania), Aisling McCluskey (National University of Ireland, Galway) and V. Lakshmana Gomathi Nayagam (National Institute of Technology, Tiruchirapalli).
• So far this year three visitors have come: Bob Raphael (Concordia University, Quebec), Grant Woods (University of Manitoba) and Max Ganster (Graz University of Technology).

  Here are some results we have discovered very recently.

  • A topological manifold is metrisable if and only if the space of real-valued functions with the compact-open topology is Volterra. Volterra spaces are something Zbiggie Piotrowski and I introduced in the 1990s and developed with Sina Greenwood. It is a generalisation of the Baire property. Of course a manifold is also metrisable if and only if the space of real-valued functions with the compact-open topology is Baire.
  • Up to isotopy homeomorphisms of powers of the long ray and line behave just like homeomorphisms which permute the axes (and possibly reverse directions in the case of the line).
  • There is a foliation of the plane such that any leaf-preserving homeomorphism of the plane is rigid in the sense that it must map each leaf to itself.
  • Any (dimension 1) foliation of the semi-long cylinder (ie product of a circle and the long ray) must either have circles appearing as leaves unboundedly often or must be trivial from some point on the long ray factor.
  • Unlike the usual plane which has uncountably many inequivalent foliations, the long plane (ie square of the long line) has only two inequivalent dimension 1 foliations.

  Selected Earlier Publications

  A long time ago I wrote a book, "Differential Topology: an introduction," published by Marcel Dekker. It sold only about 1000 copies and apparently they decided to put it out of print. In 2006 Dover reprinted it so it is available again.

  At the time of original publication I also prepared solutions to the exercises. Of course it is better for one to attempt to solve exercises oneself but sometimes it is useful to have worked solutions available too. In that spirit I have scanned the originals and put them on my website. They are in four sections: Chapters 1-2, Chapters 3-5, Chapters 6-10 and Chapters 11-15.

  • 'Mersions of topological manifolds', Trans. Amer. Math. Soc., 149(1970), 539-560.
  • 'Calculating homology groups using submersions', Math. Chronicle, 4(1976), 90-93.
  • 'Local contractibility of spaces of homeomorphisms', Comp. Math., 32(1976), 3-11.
  • 'Quasiconformal extensions of mappings' (with M K Vamanamurthy), Ann. Acad. Sci. Fenn., 3(1977), 229-246.
  • 'Lipschitz and quasiconformal flattening of spheres and cells' (with Jussi Vaisala), Ann. Acad. Sci. Fenn., 4(1979), 371-382.
  • 'Something for nothing: some profound topological consequences of a trivial algebraic theory', Austral. Math. Soc. Gazette, 7(1980), 1-5.
  • 'On normal subspaces' (with Ivan Reilly and M K Vamanamurthy), Bull. Austral. Math. Soc., 23(1981), 1-4.
  • 'Continuity properties of functions' (with Mila Mrsevic, Ivan Reilly and M.K.Vamanamurthy), Coll. Math. Soc. Janos Bolyai, 41(1984). Topology and its Applications, Eger(Hungary), 311-322.
  • 'Variation of fixed-point and coincidence sets', Jour. Austral. Math. Soc. (Series A), 44(1988), 214-224.
  • 'A strongly hereditarily separable non-metrisable manifold}, Topology and its Applications, 51(1993), 221-228.
  • 'Essential singularities of quasimeromorphic mappings' (with Gaven Martin), Math. Scand. 73(1993), 36-40.
  • 'On Volterra spaces II' (with Sina Greenwood and Zbigniew Piotrowski), Papers on General Topology and Applications, Ann. New York Acad. Sci., 806(1996), 169-173.
  • 'Microbundles revisited'(with Sina Greenwood), Proceedings of Prague TOPOSYM 1996, Topology Atlas, (1997), 114-119.
  • 'On variations of continuity' (with Sina Greenwood and Ivan Reilly), Topology Atlas Invited Contribution.
  • 'Manifolds At and Beyond the Limit of Metrisability' Proceedings of the Second Galway Colloquium, Paul Gartside ed., Topology Atlas, (1998).
  • 'Manifolds at and beyond the limit of metrisability', Geometry and Topology Monographs, 2(1999), 125-133. This issue of Geometry and Topology Monographs is in honour of my PhD advisor Robion Kirby. Despite the same name as the paper above it is actually quite different.
  • 'Microbundles, Manifolds and Metrisability' (with Sina Greenwood), Proc. Amer. Math.Soc. 128(2000), 2801-2807.
  • 'On Volterra spaces III' (with Sina Greenwood and Zbigniew Piotrowski), Topology Proceedings, 23(Spring 1998), 167-182.
  • 'Covering Properties and Metrisation of Manifolds', Topology Proceedings 23(Summer 1998), 127-140.
  • 'Covering Properties and Metrisation ofManifolds 2' (with MK Vamanamurthy). Topology Proceedings 24(Summer 1999), 173-185.
  • 'The Torsion of the Group of Homeomorphisms of Powers of the Long Line' (with Satya Deo), J. Austral. Math. Soc., 70(2001), 311-322.
  • 'Addendum to "The Torsion of the Group of Homeomorphisms of Powers of the Long Line" ' (with Satya Deo) (This addendum corrects some minor errors in the previous paper.)
  • 'A Baire product theorem for separately open sets and separate continuity' (with Sina Greenwood and Zbigniew Piotrowski), Topology Proceedings 25(Summer 2000), 129-144.
  • 'Metrisability of Manifolds in Terms of Function Spaces' (with Frederic Mynard), Houston Journal of Mathematics, 31(2005), 199-214.
  • 'Volterra spaces revisited' (with Jiling Cao), Journal of the Australian Mathematical Society, 79(2005), 61-76.
  • 'Differentiability as continuity' (with Frederic Mynard), Real Analysis Exchange, 31(2006), 425-430.
  • 'Spaces with property pp' (with Paul Gartside and Abdul Mohamad), Topology and its Applications, 153/15(September 2006), 3029-3037.
  • 'Irregularity' (with Szymon Dolecki), Topology and its Applications, 154/8(April 2007), 1565-1580.
  • 'Homogeneous and Inhomogeneous Manifolds' (with Paul Gartside and Sina Greenwood), Proceedings of the American Mathematical Society, 136/9(September 2008), 3363-3373.
  • 'Games and metrisability of manifolds' (with Jiling Cao, Sina Greenwood and Abdul Mohamad), New Zealand Journal of Mathematics, 37(2008), 1-8.
  • (with V. Lakshmana Gomathi Nagayam, Geetha Sivaraman and G. Venkateshwari), 'Intuitionistic Fuzzy Translation Invariant Spaces', 2008 IEEE International Conference on Fuzzy Systems (FUZZ2008), 2157-2161.
  • (with V. Lakshmana Gomathi Nagayam, Geetha Sivaraman and G. Venkateshwari), 'Strong Fuzzy Topological Groups', New Zealand Journal of Mathematics, 38(2009), 187--195.
  • (with Mathieu Baillif and Satya Deo), 'The mapping class group of powers of the long ray and other non-metrisable spaces', Topology and its Applications.
  • (with Satya Deo), 'Eventually constant spaces and nonmetrizable homology spheres', Journal of Indian Mathematical Society, Special Centenary Volume (1907--2007), 165--175.

  Papers in Preparation

  • 'Metrisability of Manifolds'.
  • 'Boundedly Metacompact or Finitistic Spaces and the Star Order of Covers' (with Satya Deo).
  • 'Intuitionistic fuzzy translation invariant spaces' (with V. Lakshmana Gomathi Nayagam, Geetha Sivaraman and V. Venkateshwari)
  • 'Foliations on non-metrizable manifolds: absorption by a Cantor black hole' (with Mathieu Baillif and Alxeandre Gabard).
  • 'Homeomorphisms of Bagpipes'.

  Mathematics 750

  For several years up to 2005 in the first semester I taught the course Maths 750, Topology. All of the electronically prepared handouts may be found at '750-05'.

Non-mathematical Interests

If you are into genealogy then you might inspect the mathematical genealogy web-site. Like lots of others I can trace my ancestry on that site to Newton (HA, not Sir I!) then via a range of illustrious folk including Leibniz (this time the famous one!) to Elissaeus Judaeus who supervised Georgios Gemistos's thesis entitled 'Nomoi (Book of Laws)' in 1380. I have put together a brief genealogy which lists descendants of H A Newton who either worked here or else visited us during 2000 as well as their academic ancestors back to Newton. Those who were at Auckland in 2000 include all those at the end of a branch as well as Reilly, McIntyre and myself.

Photos

Last Modified: 11 November, 2009.