# Mathematics at the Victoria University of Manchester

The Victoria University of Manchester (VUM) merged with the University of Manchester Institute of Science and Technology (UMIST) in 2004 to form The University of Manchester. The two former Departments of Mathematics joined together to form the School of Mathematics. In 2007 the School moved into a new building at the heart of the campus: the Alan Turing Building. The School is one of the largest integrated schools of mathematics in the UK, with around 75 permanent lecturing staff and over 1000 undergraduates.

As the School moves ahead it is important to keep an eye on the past, and to maintain valuable historical information about the predecessor departments. I know from emails I receive and contact with alumni (most recently at a reception in London last summer) that former students and staff like to look at photos and documents relating to their time here.

I have previously made available various documents and photos concerning the VUM Mathematics Tower on Oxford Road.

Now I have scanned five documents that provided information for prospective and current VUM mathematics undergraduates.

David Broomhead passed away on July 24th, 2014 after a long illness. David was a Professor of Applied Mathematics in the School of Mathematics at the University of Manchester. I got to know him in 2004 when the Victoria University of Manchester merged with UMIST and the two mathematics departments, his at UMIST and mine at VUM, became one.

David was a truly interdisciplinary mathematician and led the CICADA (Centre for Interdisciplinary Computational and Dynamical Analysis) project (2007-2011), a £3M centre funded by the University of Manchester and EPSRC, which explored new mathematical and computational methods for analyzing hybrid systems and asynchronous systems and developed adaptive control methods for these systems. The centre involved academics from the Schools of Mathematics, Computer Science, and Electrical and Electronic Engineering, along with four PhD students and six postdocs, all brought together by David’s inspirational leadership.

One of the legacies of CICADA is the burgeoning activity in Tropical Mathematics, which straddles the pure and applied mathematics groups in Manchester, and whose weekly seminars David managed to attend regularly until shortly before his death. Indeed one of David’s last papers is his Algebraic approach to time borrowing (2013), with Steve Furber and Marianne Johnson, which uses max-plus algebra to study an algorithmic approach to time borrowing in digital hardware.

Among the other things that David pioneered in the School, two stand out for me. First, he ran one of the EPSRC creativity workshop pilots in 2010 under the Creativity@Home banner, for the CICADA project team. The report from that workshop contains a limerick, which I remember David composing and reading out on the first morning:

One who works on Project CICADA

Has to be a conceptual trader

Who needs the theory of Morse

To tap into the Force –

The workshop was influential in guiding the subsequent activities of CICADA and its success encouraged me to organize two further creativity workshops, for the numerical analysis group and for the EPSRC NA-HPC Network.

The second idea that David introduced to the School was the role of a technology translator. He had organized (with David Abrahams) a European Study Group with Industry in Manchester in 2005 and saw first-hand the important role played by technology translators in providing two-way communication between mathematicians and industry. David secured funding from the University’s EPSRC Knowledge Transfer Account and combined this with CICADA funds to create a technology translator post in the School of Mathematics. That role was very successful and the holder (Dr Geoff Evatt) is now a permanent lecturer in the School.

I’ve touched on just a few of David’s many contributions. I am sure other tributes to David will appear, and I will try to keep a record at the end of this post.

Photo credits: Nick Higham (1), Dennis Sherwood (2).

# Workshop on Matrix Functions and Matrix Equations

Last month we (Stefan Guettel, Nick Higham and Lijing Lin) organized a 2.5 day workshop Advances in Matrix Functions and Matrix Equations We had 57 attendees from around the world (see group photo): UK (19), Italy (7), USA (7), Germany (6), Canada (2), France (2), Portugal (2), South Africa(2), Saudi Arabia(2), Austria (1), Belgium (1), India (1), Ireland (1), Poland (1), Russia (1), Sweden (1), Switzerland (1).

We last organized a workshop on matrix functions in Manchester in 2008 (MIMS New Directions Workshop Functions of Matrices). The field has advanced significantly since then. Some emerging themes of this year’s workshop were as follows.

Krylov methods: Several speakers presented new results on this class of methods for the approximation of large-scale matrix functions, including a convergence analysis by Grimm of the extended Krylov subspace method taking into account smoothness properties of the starting vector, black-box parameter selection for the rational Krylov approximation of Markov matrix functions by Guettel and Knizhnerman, and an adaptive tangential interpolation strategy for MIMO model order reduction by Simoncini and Druskin.

Matrix exponential: Research continues to focus on this, the most important of all matrix functions (the inverse is excluded as being too special). We were delighted that Charlie Van Loan opened the workshop with a talk “What Isn’t There To Learn from the Matrix Exponential?”. Charlie wrote some of the key early papers on exp(A). Indeed his work on exp(A) began when he was a postdoc at Manchester in the early 1970s, and his 1975 Manchester technical report A Study of the Matrix Exponential contains ideas that later appeared in his papers and his book (with Golub) Matrix Computations. In particular, it makes the case that “anything that the Jordan decomposition can do, the Schur decomposition can do better”, and is still worth reading.

Exotic matrix functions: Two talks focused on newer, more “exotic” matrix functions and had links to Rob Corless, who was in the audience. Bruno Iannazzo discussed how to compute the Lambert W function of a matrix, which is any solution of the matrix equation $X e^X = A$. The scalar Lambert W function was named and popularized in a 1996 paper by Corless, Gonnet, Hare, Jeffrey and Knuth, On the Lambert W Function; it has many applications, including in delay differential equations. Bruno finished with a striking photo of the equation written in sand. Mary Aprahamian presented a new matrix function called the matrix unwinding function, defined as $U(A) = (A - \log e^A )/(2\pi i)$, which arises from the scalar unwinding number introduced by Corless, Hare and Jeffrey in 1996. She showed that it is useful as a means for obtaining correct identities involving multivalued functions at matrix arguments, as well as being useful for argument reduction in evaluating the matrix exponential.

A special afternoon session celebrated the 70th birthday of Krystyna Zietak, who has made many contributions to numerical linear algebra and approximation theory. Krystyna gave the opening talk in which she described some highlights of her international travels and of hosting visitors in Wroclaw, well illustrated by photos.

Following the session we had a reception in the Living Worlds gallery of the Manchester Museum, followed by a dinner in the Fossil gallery, with Stan the Tyrannosaurus Rex looking over us.

Financial support for the workshop came from the European Research Council and book displays were kindly provided by Cambridge University Press, Oxford University Press, Princeton University Press and SIAM.

Most of the talks are available in PDF format from the workshop programme page.

A gallery of photos from the workshop has been produced, combining the efforts of several photographers.

# Arthur Buchheim (1859-1888)

The new second edition of Horn and Johnson’s Matrix Analysis, about which I wrote in a previous post, includes in Problem 2.4.P2 a proof of the Cayley-Hamilton theorem that is valid for matrices with elements from a commutative ring and does not rely on the existence of eigenvalues. The proof is attributed to an 1883 paper by Arthur Buchheim.

A few years ago Arthur Buchheim’s work came up in my own investigations into the history of matrix functions and I discovered that he was a mathematics teacher at Manchester Grammar School, which is located a couple of miles south of the University of Manchester, where I work.

In 1884 Buchheim gave a derivation of Sylvester’s polynomial interpolation formula for matrix functions. The original formula was valid only for matrices with distinct eigenvalues, but in 1886 Buchheim generalized it to handle multiple eigenvalues using Hermite interpolation.

Appropriately, Rinehart, in his 1955 paper The Equivalence of Definitions of a Matric Function, cited Buchheim when he wrote

“there have been proposed in the literature since 1880 eight distinct definitions of a matric function, by Weyr, Sylvester and Buchheim, Giorgi, Cartan, Fantappiè;, Cipolla, Schwerdtfeger and Richter … All of the definitions except those of Weyr and Cipolla are essentially equivalent.”

Buchheim studied at New College, Oxford, under the Savilian Professor of Geometry, Henry Smith, and then at Leipzig under Felix Klein. Then he spent five years at Manchester Grammar School, from which he resigned due to ill-health the year before his death.

In addition to his work on matrix functions and the Cayley-Hamilton theorem, Buchheim published a series of papers promoting Grassmann’s methods. In his A History of Mathematics (1909), Cajori notes that

“Arthur Buchheim of Manchester (1859-1888), showed that Grassmann’s Ausdehnungslehre supplies all the necessary materials for a simple calculus of screws in elliptic space.”

He goes on to say that

“Horace Lamb applied the theory of screws to the question of the steady motion of any solid in a fluid.”

thus bringing in another, much more famous, Manchester mathematician about whom I recently wrote.

Sylvester wrote an obituary in Nature in which he stated “I … know and value highly his contributions to the great subject which engaged the principal part of my own attention during the transition period between my residence in Baltimore and at Oxford”.

The best source of information on Buchheim is an article

Jim Tattersall, Arthur Buchheim: Mathematician of Great Promise, in Proceedings of the Canadian Society for History and Philosophy of Mathematics Thirty-first Annual Meeting, Antonella Cupillari, ed, 18 (2005), 200-208.

which lists lists 24 papers that Buchheim published in his short life of 29 years.

# Horace Lamb Portrait in Alan Turing Building

A portrait of Sir Horace Lamb (1849-1934), FRS, Beyer Professor of Pure and Applied Mathematics from 1888 to 1920, is on display on the Atrium bridge of the Alan Turing building in the School of Mathematics at the University of Manchester.

This is the School’s common room, where we meet for morning coffee and lunch and which is the focal point of the School.

The 1913 portrait, approximately, 4 feet by 4 feet, is by Lamb’s son, Henry Lamb, a distinguished painter, and was presented to the University by Ernest Rutherford. It’s difficult to photograph due to reflections on the glass, so I took the photo from an angle.

Lamb made important contributions to many topics in applied mathematics, including waves, acoustics, elasticity, fluid dynamics, with applications to areassuch as seismology and the theory of tides. He is perhaps best known for his book Hydrodynamics, first published in 1879 (under the original title “Treatise on the Mathematical Theory of the Motion of Fluids”), which went through six editions. The second edition (1895) has been digitized by Google and can be downloaded from The Internet Archive.

The School’s main meeting room is named the Horace Lamb Room and contains an ornate writing desk and display cabinets presented to Lamb by the University of Adelaide, where he worked for nine year before moving to Manchester. The cabinets contain the engravings pictured below.

The interesting story of how Lamb, born in Stockport near Manchester, came to take a chair in Adelaide, and why he subsequently returned to Manchester, is told in Horace Lamb and the Circumstances of His Appointment at Owens College by Brian Launder (2013).

For more about Lamb see The MacTutor History of Mathematics archive.