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. I have formally retired, but have accepted a four-year appointment in a half time position. My office is in room 419 of the Mathematical and Physical Sciences Building. My phone number is (64)(9)9238697. Look at this map of the world to see where my office is.

To go back to the beginning, I was born in Inglewood, NZ, and attended the long since defunct Pukeho Primary School (it closed as a school one year after I left it and, while the building remains, the last time I visited I wondered whether it 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, from the big kids when I was little to the little kids when I was big; can you? Three of my high school years were at Inglewood High School: I hope the photo on the IHS website of the school with the mountain as background makes you jealous! IHS celebrated its 50th anniversary in 2007. I attended the official celebrations and was thrilled to meet quite a few others who, like me, attended IHS that hot summer's day it opened. Here's a photo of my fifth form class (I'm the smallest kid there). 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 (which has undergone a number of transitional names to its current Wanganui City 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. One of the signatures on my PhD certificate was of someone who went on to bigger things. 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 was also the founding secretary of the New Zealand Mathematics Research Institute, but retired from that position in 2011 after 13 years on the job. For many years I have been involved with the New Zealand Journal of Mathematics and its predecessor the Mathematical Chronicle: this journal, a joint publication of the Mathematics Department of the University of Auckland and the New Zealand Mathematical Society, has no page charges and is available free of subscription at the address above.

I have served in a number of administrative roles in the University. Most notable was as Head of the Department of Mathematics (Mathematics and Statistics a lot of that time). 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 eight 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. 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 and recently transferred to Nizwa University, also in 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 which he took up in 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 an NZIMA Programme awarded to Vaughan Jones, Roger Fenn and me. He completed his formalities in September 2009 and has returned to England. Qingxiang Zhang is another PhD student whom I co-supervised but Gaven Martin was her main supervisor and she completed her degree at Massey University. She was also supported by the NZIMA Programme. Afshin Mardani, originally from Iran, completed his PhD in 2014: he was supported for some of the time by a Marsden grant awarded to me. I was co-supervisor for Nazli Uresin but Sina Greenwood was her main supervisor. 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 a 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.

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 an 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).
• In 2008 these visitors came: Bob Raphael (Concordia University, Quebec), Grant Woods (University of Manitoba), Max Ganster (Graz University of Technology), Shura Arhangelskii (Moscow and Ohio Universities) and Peter Nyikos.
• Judy Kennedy has visited Sina annually for several years now.
• In 2011 Jan van Mill (Frei University, Amsterdam) and T. M. G. Ahsanullah (King Saud University, Saudi Arabia) visited.
• In 2012 Peter Nyikos paid us another visit.

  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.
  • A topological manifold is metrisable if and only if its group of homeomorphisms is metrisable.
  • There are surfaces which carry exotic differential structures. Indeed, the long plane carries '2 to the aleph 1' of them!

  Selected 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.

  Much more recently I have worked on a book about non-metrisable manifolds. It was published at the very end of 2014 by Springer as Non-metrisable Manifolds. As far as I am aware there is no other book on the topic, though Peter Nyikos has written several survey articles, including his long one in the Handbook of Set-Theoretic Topology. So I found it quite a challenge. The content is somewhat biassed towards work done by me and my colleagues, but includes a description of the early remarkable work by Mary Ellen Rudin and Phillip Zenor showing that the question whether a manifold is metrisable if it is perfectly normal is independent of the ZFC axioms as well as Peter Nyikos’s Bagpipe Theorem and his construction of many distinct differential structures on the long line.

  • '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 of Manifolds 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. (Don't get fooled by the date on the website the link takes you to; it wasn't published until 2006)
  • '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.
  • 'Intuitionistic Fuzzy Translation Invariant Spaces' (with V. Lakshmana Gomathi Nagayam, Geetha Sivaraman and G. Venkateshwari), 2008 IEEE International Conference on Fuzzy Systems (FUZZ2008), 2157-2161.
  • 'Strong Fuzzy Topological Groups' (with V. Lakshmana Gomathi Nagayam, Geetha Sivaraman and G. Venkateshwari), New Zealand Journal of Mathematics, 38(2009), 187--195.
  • 'The mapping class group of powers of the long ray and other non-metrisable spaces' (with Mathieu Baillif and Satya Deo), Topology and its Applications.
  • 'Eventually constant spaces and nonmetrizable homology spheres' (with Satya Deo), Journal of Indian Mathematical Society, Special Centenary Volume (1907--2007), 165--175.
  • 'Jordan and Schoenflies for non-metric surfaces' (with Alexandre Gabard), New Zealand Journal of Mathematics, 40(2010), 49–58.
  • 'Homeomorphisms of Bagpipes', Topology Proceedings, 40(2012), 109–120.
  • 'Homeomorphism Groups and Metrisation of Manifolds' (with Jan van Mill), New Zealand Journal of Mathematics, 42(2012), 37–43.
  • 'Foliations on non-metrisable manifolds: absorption by a Cantor black hole' (with Mathieu Baillif and Alexandre Gabard). A slightly abbreviated version appears in the Proceedings of the American Mathematical Society, 142(2014), 1057–1069.
  • 'Exotic Differential Structures in Dimension 2' (with Sunanda Dikshit), Topology and its Applications, 164(2014), 229–234.
  • 'Selections and metrisability of manifolds', Topology and its Applications, 160(2013), 2473–2481.
  • 'Enriched lattice-valued convergence groups' (with T. M. G. Ahsanullah, Jawaher Al-Mufarrij and Fawzi Al-Thukair), Fuzzy Sets and Systems, 238(2014), 71–88.
  • ‘Enriched lattice-valued topological groups’ (with T. M. G. Ahsanullah, Jawaher Al-Mufarrij and Fawzi Al-Thukair), New Mathematics and Natural Computation, 10(2014), 27–53.
  • 'Foliations on non-metrisable manifolds II: contrasted behaviours' (with Mathieu Baillif and Alexandre Gabard).

  Papers in Preparation

  • 'Metrisability of Manifolds'. This lists over 100 topological conditions equivalent to metrisability of a manifold. The paper linked to above was put on arXiv several years ago so is already out of date. For recent additions to the list see the paper 'Homeomorphism Groups and Metrisation of Manifolds' with Jan van Mill and 'Selections and metrisability of manifolds' already referred to as well as the book Non-metrisable manifolds.
  • 'Boundedly Metacompact or Finitistic Spaces and the Star Order of Covers' (with Satya Deo).

  Outside Diversions

  I like to spend my spare time wandering around the forested hills near Auckland or further afield: the highest recent climb was Humphreys Peak, at 3851 metres the highest point in the US State of Arizona, and prone to monsoons at the time of year we were there (early July, 2014) but fortunately we had great views. Near Auckland, there are some great streams to enjoy in the Waitakere Ranges as well as inland from Thames. Bird life was very rich centuries ago and is slowly recovering from the devastation caused by humans and their predatory companions. A particular thrill is to hear a kokako singing in the wild in the Hunua Ranges to the east of Auckland: their population got down to only about 6 pairs of this rare bird in those ranges but hopefully it is building up again. Not surprisingly I am a member of The Royal Forest and Bird Protection Society of New Zealand. I am also involved in Ark-in-the-Park, which is an amazing project set in the Waitakere Ranges to the west of Auckland. The aim is to reduce the number of non-native mammalian predators to the point where the native bird life can flourish once more. Many birds which have been locally extinct in the Waitakeres for decades are once again breeding there. One such is the kokako: many native birds lost the ability to fly because of the absence of mammalian predators and the kokako is one of these but unlike the others it can still glide from high up a tree to not so high up another tree. Here is another address that you might like to look at for lots of attractions. A few years ago with my wife I climbed Mt Euclid, a remote peak in the Paparoa Range on the west coast of the South Island (most of the other peaks in the Paparoas are named for scientists, too). So now I can claim to have viewed euclidean space!

  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. To show how we are interconnected, years ago I 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. This list includes only topologists, not colleagues or visitors working in other branches of mathematics who descended from Newton.

  Photos

  If you really need to see a photo of me then look at this one, or this one, or this one, or this one. They were taken in 1963, 1982, 1999 and 2007: guess which was taken when. The group photo shows a bunch of us over 30 years ago. The occasion was the centenary of the University of Auckland when many from the Maths Dept (plus a few former members) went to the memorial wall at the west end of Grafton Bridge to mark also the 100th anniversary of the death of the University's first Professor of Mathematics. Professor George Walker was one of the founding four professors of the University but he was drowned in a boating mishap almost immediately after his arrival in Auckland and was buried in the cemetery below the bridge. Eighteen of the originals returned 25 years later in 2008. We were accompanied by other colleagues in this 125th anniversary photo.

  Kaimai Plane Crash

  When I was a student New Zealand's worst ever domestic air disaster took place when a plane travelling from Auckland to Tauranga and carrying 23 people crashed into the Kaimai Range on Wednesday 3 July, 1963. I was one of those helping search for the plane and the image of the burnt out plane and worse is seared on my brain. Following the crash it was over 40 years before I returned to the site. On the 50th anniversary of the crash, Wednesday 3 July, 2013, I returned to the site and stayed there for an hour around the time of the crash. Here are some photos I took 50 years on. In early July 1963 the weather was atrocious but in early July 2013 the sun shone on the now peaceful scene.

  The site of the crash. Down the main valley to the Hauraki Plains. A video showing the terrain.

  Last Modified: 2 January, 2015