A virtual interview with Jon Berrick
What is mathematics?
That's a tough first question! A few years ago I tried to answer
it in an essay "What do mathematicians do?".
Although most academic disciplines can be defined by what they study,
my claim is that mathematics is better described by how it proceeds,
that is, the kind of thinking it uses. If true, then this would help to
explain how it is that mathematics keeps appearing in lots of unexpected
places.
Such as?
Well, current applications of mathematics that few people would
have predicted twenty years ago include mathematical finance, risk management,
cryptography and data security, image processing, computational biology.
For a longer list, see the topics for recent research programs at Cambridge's
Newton Institute for Mathematical Sciences, or at Singapore's
IMS. This has been called the "unreasonable effectiveness of
mathematics", but maybe the real point is that if you think hard, precisely
and deeply enough about any problem, then eventually you find yourself
doing mathematics.
How about the next twenty years?
Fortunately, we don't know! We can only guess. The
only safe guess is that new technologies will lead to new knowledge and
new problems that will call upon mathematics for their solution.
In many cases, it's the most powerful mathematics that turns out to have
the widest applications. So, there's a strong case for anyone seeking
to apply mathematics to prepare for this by first acquiring some powerful
mathematics. To throw an object a long way, first climb to a high
place.
So, what's your "high place"? What sort of mathematics
do you study?
A broad name for my subject area is topology.
Meaning?
This is a mixture of two Greek words that literally means "the
study of shapes". Topology is the study of the shapes of geometric objects,
which in applications may be as small as knotted DNA or long-chain polymers,
or as large as the universe itself.
That sounds very broad - there are lots of shapes!
Yes indeed. So, there are different ways of attacking the problem.
Algebraic topologists like me try to distinguish such continuously varying
objects by associating to them discrete, algebraic invariants. The
process is comparable to capturing analogue data in digital format, and
is intended to produce something that is easier to analyse, without too
much loss of information along the way.
Can you give a simple example?
If we think about the letters of the (Roman) alphabet, then among
the lower case letters there are just two that cannot be written without
lifting pen from paper. We say that " i " and " j " have two components,
whereas all other letters have only one. If we turn to upper case letters,
then the number of "holes" distinguishes between " A ", " B " and " C ",
for example. A useful technique for more complicated objects, in more dimensions
than just the two-dimensional plane, is to count higher-dimensional holes.
This leads to topics like homology and homotopy.
More Greek words?
That's right. Many of the properties one is interested in are
retained when the object under investigation is subjected to a continuous
deformation or homotopy. (Continuity means that pushing, pulling,
twisting and squashing are allowed, tearing and gluing are not.) One example
is the degree of a continuous function or map: the degree (an integer)
is unchanged when the mapping is continuously deformed. One says that the
degree is invariant up to homotopy. The power of homotopy theory lies in
this invariance up to homotopy: often it allows one to replace complicated
objects by simple models of them. The strategy of homotopy theory is to
establish this invariance for as many properties as possible, and then
exploit the invariance to obtain a classification.
Classifying objects usually involves calculating homotopy groups. As
the direct approach is typically too difficult, one uses some kind of homology
to calculate these homotopy groups by indirect means.
I need another example.
Even in two dimensions, there can be very difficult questions.
One problem I've thought a bit about was posed by Henry Whitehead in 1941.
We can stretch and blow on soap film using a configuration of wires, to
make a complex of soap bubbles. Suppose we make such a complex that
can be deformed to a point (no bursting allowed). Is it possible that some
subcomplex can contain a two-dimensional hole?
Has algebraic topology only been useful in geometry?
Over the past fifty years, the richness of the agebraic structures
it's constructed has had a major impact on other areas of mathematics,
leading to new research areas such as category theory, homological algebra
and K-theory. Some of the tools have proved very powerful, and have solved
outstanding problems in geometry.
I think I know what soap bubbles look like. Are there any other
geometric objects you study that I might recognize?
Sure, another problem I look at concerns braids.
The notion of a braid as "anything plaited, interwoven, or entwined" goes
back many centuries, and braids have been used universally for decoration,
art and fastening purposes. Only recently have mathematicians tried to
describe braids by means of abstract theory. Amazingly, as the theory
has developed, it has enabled applications to outstanding problems in physics,
chemistry and biology.
How do you go from concrete braids to abstract theory?
In topology, braid theory is an abstract geometric theory studying
the everyday braid concept, and some generalizations. The idea is
that braids can be organized into groups, in which the group operation
is "do the first braid on a set of strings, and then follow it with a second
on the twisted strings". Such groups may be described by explicit presentations,
as was shown by E. Artin in 1925. A braid with n strands can also
be thought of as paths of n distinct particles moving through time,
with no collisions.
What results have you obtained in this theory?
In work done recently with Yan-Loi Wong and Jie Wue here at NUS (together
with Fred Cohen), we showed that encoded in those braids (called Brunnian)
that disentangle when any strand is removed, are all the homotopy types
of maps from high-dimensional spheres to the two-dimensional sphere.
This was a big surprise, and may have a number of consequences.
Such as?
We don't know yet. However, in 2007 we hosted a program
at the IMS here in Singapore. Possible applications discussed included
robotics, magnetohydrodynamics, molecular biology and cryptography.
Any other problems you've been working on that are easy to tell us
about?
Here's one that was originally posed (by Hyman Bass thirty years
ago) in an algebraic way that's fairly technical. However, recently Indira
Chatterji, Guido Mislin and I made substantial progress on it, and recast
it in a topological form. So the question considers a map from a
smooth geometric object (called a manifold) to itself. Such a mapping
is called homotopy idempotent if iterating it leads only to maps
homotopic to the original one. The question is whether every such map can
be deformed to another map with the property that it moves every point
of the manifold except for exactly one point that stays fixed.
So we deform the original map to a new one that has a single fixed
point?
That's the aim, yes.
And can we?
In dimensions 0, 1 and 3, the answer is Yes. However, in
dimension 2 the answer is No.
How about higher dimensions?
The answer is unknown in general, except that, whether it's Yes
or No, it's the same for every dimension! In many cases, if we know
more about the manifold, then we can answer the question for that particular
manifold.
The problems that you've mentioned in this interview - are they all
suitable for graduate students?
Definitely!
How can I find out more about your recent research?
For the more technically minded, there are abstracts
of some of my publications. There's also a complete
list of my publications.
Thank you for the virtual interview.
It’s been a virtual pleasure.
10 December 2005 (with later amendments)