mathematics
20
February 2014
Monash University
Pure mathematicians
start by thinking about
the world, making
sense of the world,
so even after many
transformations, the
ideas can’t shake
loose their connection
to the world.
– Professor Nicholas Wormald
random
importance
Professor Nicholas Wormald is an adventurer, a mathematical frontiersman
probing for new ways to explain how our world works, or could work.
Words Brad Collis
It requires a refined curiosity to be
a pure mathematician, to be impelled
not only by solving problems but by also
creating them, by contriving abstract
challenges designed to push out the
boundaries of mathematical theory.
It can be an obsessive, esoteric pursuit,
seemingly removed from the applied
mathematics or engineering that deals with
real-world challenges, but it underpins just
about all scientific and industrial progress.
As Professor Nicholas Wormald will tell
you, pure mathematics requires a creative
bent and passion for mathematical intrigue.
You set, or are set by your peers, abstract
algebraic challenges that are beyond the
reach of known mathematical tools or
equations and which take researchers into
expansive and tantalising fields of discovery.
This exhaustive process of solving, or
at least understanding, these contrived
explorations delivers the theorems and
most innovative algorithms that science
and industry need to keep advancing.
“If, say, you want to build a bridge,
as an applied mathematician you
use all the mathematical tools at your
disposal to find answers to any problems
arising from the bridge’s design and
construction,” Professor Wormald says.
“The pure mathematician, on the other
hand, is the person who has explored,
even devised, the mathematical tools and
generated the knowledge and assurity
that the algorithms do actually work, as
well as what their limitations might be.”
Pure mathematics, he suggests, is
where new technological capability begins.
Professor Wormald applies his
skill and insights through a particular
interest in the field of combinatorics. This
branch of mathematics, which gained
prominence during the 1950s and 1960s,
deals with the possible arrangements
of, or connections between, sets of
objects (be they arrangements of atoms
Photo: Brad Collis
of
mathematics
Monash University
21
February 2014
in a chemical molecule, or connections
between users of the internet).
In combinatorics, the focus is on the
arrangement of what or who is connected,
not how long or slow the link is. The discipline
has risen with computer science and the need
to create or optimise complex algorithms
for computing and data management.
Professor Wormald, formerly
professor and Canada Research Chair
in the Department of Combinatorics
and Optimization at the University of
Waterloo, Canada, began an Australian
Laureate Fellowship at Monash University
last year to research new approaches to
“probabilistic combinatorics”, which explores
the effects of introducing randomness
into networks or arrangements.
network to avoid traffic congestion and, in
turn, facilitate increased network speed.
Similarly, random structures in
program algorithms are also used to
achieve better “load balancing” when
data is stored on discs and other storage
devices (see example illustration).
Load balancing, such as the data
storage example, also takes researchers
such as Professor Wormald into another
field of pure mathematics – graph theory.
This is a visual representation of a set of
objects, or “vertices”, that are connected
by links, or “edges”. In lay terms, it is
essentially a network schematic.
In computer science, graphs are used to
represent networks and their elements, such
as the computational devices being used, data
An example of random graphs modelling load balancing for
organisation
and storage,
information
computer
data storage.The
objectiveand
is tothe
avoid
any one
disk having
too greatThe
a load.
In this example
a user hasto
pathways.
development
of algorithms
World Wide Web
requested a set of files to be returned from the bank of
model
Professor Wormald points to the internet asdisks,
an and
eachsuch
file is increasingly
stored on twocomplex
disks. Sorelationships
the controller
needs touses
decide
from which
diskThis
to read
file. The first
random
graphs.
haswhich
become
example of why fields such as combinatorics
picture shows each file with arrows to the two disks it’s on.
fundamental
to
advances
in
computer
science.
have become critical. The internet has
To model the problem as a graph, look at the second
billions of web pages with tens of billions ofpicture. The two disks each file is on are linked by a line, an
'edge' of the graph. But the file symbols are irrelevant to the
links between them, and understanding theproblem,Making
ofpicture.
the world
so we getsense
the third
As an abstract graph it
matter how challenges
it is drawn because
doesn’t change
properties of abstract networks of pages doesn’t Academic
aside, itProfessor
the problem – for example, the fourth picture. The problem
and their links is necessary to creating real-is to choose
Wormald
emphasises
that
pure
one end
of each edge
– let’s
putmathematics
a marker on
the chosen
so that no disk
has too “You
many start
markers.
world tools such as web-search algorithms.
is aend
very– grounded
discipline:
by
One solution is shown in the last picture. Of course, the load
“The network keeps changing
thinking
about
the
world,
making
sense
of
balancing problem is not just to solve such small examples,
but to quickly
make so
theeven
best choices
when there many
are
so we need some abstract way of
the world,
after undergoing
thousands of disk servers and files involved.
modelling this vast structure,” he says.
transformations, the ideas can’t shake loose
“It’s a combinatorial problem and we
their connection to the world,” he says.
often use randomness to attack it.
“It’s never surprising that after
“The broad objective of my research
following up abstract ideas, and coming
fellowship is to develop more interesting
up with questions in pure mathematics
insights into random structures. This has
that seemingly don’t relate to anything
impacts for algorithms for the internet,
else in reality, somewhere down the track
algorithms in applications to create more
someone finds an application for these very
efficient methods of data storage and
same ideas. This is because the original
retrieval, and even algorithms that model the
abstract ideas were usually abstractions
growth of social networks. The applications
of what is around us in the world.”
are not just related to computers.”
For the pure mathematician, making
The study of random structures
sense of our world means stating
addresses the probability of events
hypotheses about relationships and, by
(wanted or unwanted) occurring within a
proving them, turning them into theorems.
hypothetical random network. And there
“People who are attracted to pure
is seemingly no limit to the number of
mathematics are people who want to
ways that random structures can help to
discover how things work; to understand
understand the nature of networks – those
the mathematical patterns and behaviour
occurring naturally around us, or those
inherent in almost every activity,” he says.
we can create for specific purposes.
To this end, the pure mathematician
As an example of the value of
is not looking to create an application,
randomness, Professor Wormald points
but is seeking knowledge and insight.
out that with any network carrying traffic
“One begins finding intrinsic beauty
– whether the internet, a communications
in exceptional results, and outstanding
network or a large transport system – the
proofs take on a quality of pure
default model tends to program traffic
elegance,” Professor Wormald says.
to get from A to B by the shortest route.
It is a pursuit that can evoke great
But if every “package” in a crowded
passion because it is a shared quest,
network was programmed to do this,
“working with other researchers and students
there would inevitably be points where
who have the same interest in accumulating
the network would become congested.
knowledge … in challenging each other to
So randomness can be introduced by
explore further how our universe works at the
programming random waypoints in the

most fundamental level”.
Files
Disks
Files =
Edges
Disks =
Vertices
Edges
Vertices
An example of random graphs modelling load balancing for computer
data storage. The objective is to avoid any one disk having too great
a load. In this example a user has requested a set of files to be read
by the bank of disks, and each file is stored on two disks. So the
controller needs to decide from which disk to return which file. The first
picture shows each file with arrows to the two disks it is on. To model
the problem as a graph, look at the second picture. The two disks
each file is on are linked by an edge. But the file dots are irrelevant to
the problem – see the third picture. As an abstract graph it does not
matter how it is drawn because it does not change the problem – for
example, the fourth picture. The problem is to choose one end of
each edge – for example, put a square on the chosen end – so that no
disk has too many squares. One solution becomes clear in this last
picture. However, the load balancing problem is not just to solve such
small examples, but to quickly make the best choices when there are
thousands of disk servers and files involved.
Professor Nicholas Wormald’s interest
in mathematics and combinatorics
was triggered in early secondary
school, in particular by a graph theory challenge posed in
a mathematics competition: “If in a finite graph (network)
each vertex has k edges coming out of it, show that there’s
a cycle of length that is at least k+1.” It is an example of the
language, and mental agility, of the pure mathematician.
In this instance the challenge was to find a path, or trail,
that joins the vertices without repeating until it returns to
the original vertex and uses at least k+1 vertices; at first
glance a simple sounding task for someone with a basic
notion of algebra, but in fact one requiring the insight
of a mathematics prodigy to find a convincing proof.
classics
22
February 2014
Monash University
Dr Jane Griffiths: Arts that explore
interpretation have a crucial role in questioning
other absolutisms in society, particularly
corrosive influences such as fundamentalism.
Photo: Paul Jones
classics
Monash University
23
February 2014
History making
Research into how an ancient Greek poet
has been interpreted over the ages also
offers insights into how knowledge and myth
can be entwined to create a history.
Words Brad Collis
… by the light of
the silvery moon …
These words were inked onto
papyrus more than two thousand
years ago. It is just a fragment of verse,
but as classicist Dr Jane Griffiths
notes: “It’s pure Sappho.”
With this observation, Dr Griffiths raises
the curtain on her own exploration of this
famed lyric poet, a literary enigma who has
tantalised scholars for centuries. And in
crafting her own bridge across time,
Dr Griffiths turned a research project sifting
myth and defining knowledge into sell-out
theatre performances in five countries.
Dr Griffiths is among a cohort of
international scholars who have long been
intrigued by Sappho, who lived on the
island of Lesbos in the sixth century BCE.
Sappho was esteemed in her own lifetime
as the first great love poet, and the passage
VIDEO: see more at http://monash.edu/monashmag
of two millennia has only amplified this
bouquet. Her poems have been reprised by
lyricists and poets in every culture whose
language and sensibilities draw at least
partially on the literature of ancient Greece.
Sappho’s poetry is firmly embodied in
the lyric tradition in Western culture, with
just about every poem or expression of
sexual desire in literature regarded as having
a Sapphic lineage. But the challenge for
classics scholars such as Dr Griffiths is to
ascertain – among the layers of interpretation
and appropriation – exactly what Sappho
wrote, as the sum total of her known writing
is still just one complete poem, three semicomplete poems and about 200 fragments.
So on one level, the Sappho story
offers insights into how different ages
and cultures have related to eros in the
ancient world. But Dr Griffiths has also
sought to explore the broader issue of
knowledge-creation – how myth-making
shapes knowledge and how belief in texts
that are essentially inventions can still be
unshakable. This, she points out, has realworld effects when trying to understand
societal pressures such as fundamentalism.
Knowledge construction
What is fascinating about the Sappho story
as an example of history building, Dr Griffiths
says, is that almost every poem has been
constructed by a scholar who studied the
fragments, decided what the complete text
would have been, and filled in the gaps.
Continued page 24
Portrait of a poet
Sappho lived on the island of
Lesbos, off the coast of what
is now Turkey, at the end of the
seventh and beginning of the sixth
century BCE. Author of The Sappho
History, Dr Margaret Reynolds, from the
University of London, says much of Sappho’s writing was
composed for women and girls, possibly for celebrating rites
of passage such as betrothal, marriage and motherhood.
Dr Reynolds, who wrote the program introduction to
Sappho … in 9 fragments, says that so little of Sappho’s
original work has survived because she wrote on fragile
papyrus rolls and pottery tablets at a time when the oral
traditions of the Mediterranean were just giving way to a
literary practice. Much of what has survived was copied
at the time by admirers, including Roman scholars.
classics
24
How these gaps were filled reflects the
culture and society of the time and this
is what Dr Griffiths was keen to explore
when she embarked on her own analysis
of the conjectures that fill the gaps.
Dr Griffiths is head of the Centre for
Theatre and Performance at Monash
University and has combined a distinguished
academic career with professional
theatre production. In 1998, she was
the University of Cambridge’s Judith E.
Wilson Visiting Lecturer in Drama.
For Sappho, she made a significant
departure from philology, the preferred
method of classicists. Philology is the
systematic search for original form, original
meaning and authenticity through an
intense analysis of written records.
Instead, Dr Griffiths combined several
disciplines – classical scholarship, theatre
studies and theories of cultural transmission
– into a theatre production, Sappho … in
9 fragments. In this way she could widely
share the exploration; her audience now
exceeds 100,000 theatre patrons across
Melbourne, London, Edinburgh, New
York, Ottawa, Montreal and Toronto, and
a radio audience through the Australian
Broadcasting Corporation’s Radio National.
By comparison, “conventional philology
would have resulted in an academic paper
that maybe five people would have read”.
To ensure that the project – formally
titled “Staging Sappho: Towards a New
Methodology of Performance Reception” –
was not just an academic exercise, it was
supported by an Australian Research Council
Linkage Project grant that teamed Monash
with Melbourne’s Malthouse Theatre.
The play charts how Sappho has
been interpreted, and misinterpreted,
through a storyline drawing on her
portrayal of the intensity of love.
One actor performs the play’s two
characters – Sappho and a contemporary
young woman in love for the first time with an
older woman. Dr Griffiths performed in the
opening Melbourne production. Others have
taken on the role elsewhere. Dr Griffiths has
combined acting with academia throughout
her professional life and has performed
with internationally renowned troupes
including the Bell Shakespeare company
and the Cambridge Theatre Company. Her
Sappho project entwined several layers of
investigation, including connections between
theatre performance and creativity, and
how the modern world views the ancient
through the study of performance.
Academic challenge
Dr Griffiths says that part of the academic
challenge was that, as a trained classicist
and a philologist, she had been educated
February 2014
Monash University
scholarship went into developing the play
as she would have put into a monograph.
It was this intellectual rigour that
Malthouse Theatre’s artistic director
Marion Potts says attracted the company
to the Monash collaboration.
“Theatre allows audiences to test
assumptions about the world and to play out
different versions of themselves and the society
we live in and want to shape,” she says.
“It follows that an even deeper and
more knowledgeable foundation for the
work will enrich that audience experience.
An academic foundation by no means
makes the work exclusive or elitist –
it just makes it better informed and
therefore more meaningful and satisfying
for artists and audiences alike.”
Photo: Jeff Busby
Sappho was the world’s
first love poet who we know
wrote nine volumes but only
fragments of text remain,
giving us somebody who we
really only know through acts
of imagination.
to believe in the primacy of text. “But in
performance studies we celebrate the
diversity and polyvalence of the text: that it
is wonderfully endless in its variation. So the
added challenge was how to bring together
these two very different and rigorous
intellectual disciplines,” she explains.
“Sappho provided the ideal platform: the
world’s first love poet who we know wrote
nine volumes, but only fragments of text
remain, giving us somebody who we really
only know through acts of imagination.
“And seeking to understand people’s
interpretations helps us to understand the
Zeitgeist, the spirit of the time, and the
societies that were interpreting these texts in
different ways. It becomes an archaeology
of cultural knowledge,” she says.
To this she then adds dissent: Who
creates the knowledge? Is there a
knowledge we can interpret? Is knowledge
created in the eye of the beholder?
“Searching back through time, you
don’t find a definite truth. Rather, you
find holes that have been filled. The
‘truth’ is how those holes were ‘filled’.”
Dr Griffiths admits it can be difficult
to define when a theatre performance is
also academic research, but says as much
– Dr Jane Griffiths
An invented truth
Dr Griffiths in particular wanted audiences
to think about parallels to the manner
in which Sappho has been interpreted
and reinterpreted over the ages to create
a perceived, largely invented, truth.
“Arts that explore interpretation
have a crucial role in questioning other
absolutisms in society, particularly corrosive
influences such as fundamentalism.”
Dr Griffiths hopes that the Sappho
experiment shows how performance
underpinned by a serious academic
rigour has a real, solid role in nurturing
social cohesion and cultural tolerance.
“There is nothing airy-fairy about
research by practice, by performance,
because it is able to probe the very core of
what makes humans human,” she says.
In an illustration of the real-world basis
to performance studies, she points to the
work of colleague Dr Yana Taylor, who is
researching and practising verbatim theatre
– a form of documentary theatre in which
plays are constructed from the precise
words spoken by people interviewed
about a particular event or topic.
Dr Taylor has been working with Israelis
and Palestinians and is using performance to
bring together their contradictory dialogues.
“Of course this is a conflict that cannot be
resolved by a piece of theatre. But what it
can do, by listening to both sides and by
bringing together the theorising of both sides,
is create new perspectives, understanding
and knowledge,” Dr Griffiths says.
For Dr Griffiths, theatre allows life,
including cultural diversity, antagonism and
integration, to be tested, not just observed.
This is how theatre developed. It was the
forum in which the ancient world’s great
political debates were enacted, and the
forum in which a lyric poet was free to
celebrate the most fundamental of all human
emotions, love. 