The Partnership of Pure and
Applied Math
David Mumford
January, 2013
Sanya City, Hainan, China
Before I begin, I want to
show a wheel depicting
how I see the
relationship of
pure and applied math:
What is applied math? It is the construction and
analysis of mathematical models to describe and
predict any mathematizable aspect of our world.
There can be great synergy between pure and
applied math – and but there is also divergence.
Outline
I. Some Personal History and the contemporary
threat to this partnership
II. Math in Early China was algorithmic and applied,
very much like computer code.
III. ‘Fourier’ Series were invented as an applied tool
for astronomy by Clairaut in the mid 18th century.
IV. Combining Markov random fields with Bayes’
Theorem is one of the keys to computer vision
V. Infinite dimensional differential geometry helps
the analysis of medical scans
VI. Words of Caution
1. My own path meandered between pure
and applied math
• My first venture in math -- a digital computer built in 1952
with my soldering gun and 100 WWII surplus relays.
• Then I went on to work at Westinghouse on an analog
computer that simulated fast and slow neutron flux in a
submarine reactor
Continuum model: D f s = csf s - c f f
f
f f = fast neutron flux; f s = slow
Discrete model:
R gird ×å
(Vb( g ) - Va( g ) ) = R grnd ×Va( g ) - R batt ×Va( b)
nbrs b of a
Va( g ) = volts at grid pt a ; Va( b ) = volts of battery at a
• Around 1956, listening to the lectures of George Mackey,
Lars Ahlfors and Oscar Zariski, I fell in love with abstraction,
with the idea of a secret garden containing amazing things
that could only be seen with the mind’s eye but never
touched: pure math.
• Again, in 1982, I read a book by David Marr and realized
that animal skills in as basic an area as seeing were
immensely subtle and cried out for new mathematical
models and was drawn back to applied math.
• Finally in 1998, I studied Grenander and Miller’s ideas on
warping the medical scan of a patient to a body template
and realized that a huge area of the differential geometry of
infinite dimensional manifolds was largely unexplored, so
got sucked into pure math again – with the help, soon after,
of Peter Michor.
But there are dark clouds
• I think I understand the very distinctive charms of
both pure and applied math, and how tightly they
are joined in a symbiotic relationship.
• Yet there is a huge tension between them, people
rarely cross over or even listen to each other’s
lectures. The morning talks all related to physics –
yet how many physicist’s talks can you understand?
• Pure math believes it is intellectually superior,
infinitely deeper; applied math has much more
money, support from industry, a sense of belonging
to contemporary culture.
• This is not healthy. Applied math has always
rejuvenated pure, theorems in pure unexpectedly
lead to new tools with vast applications.
2. Classical Chinese algebra was
algorithmic and applied
• The great classic in the Chinese tradition is the
“Nine Chapters of the Mathematical Art”, Jiǔ
Zhāng Suàn Shù, 九章算術
• Compiled in the Han Dynasty but based on work
from the Warring States or even the Zhou
Dynasty, destroyed in the massive book burning
of the Qin, the first “cultural revolution”.
• Of course, “Pythagoras’s” theorem is in the Nine
Chapters (it was also in India c.800 BCE, Babylon
c.1800 BCE) but I want to look at algebra.
Rectangular Arrays
• This is Chapter 8, 方程 Fang Cheng
• Here’s a typical problem (everything is done with
problems)
– Now sell 2 cows and 5 sheep, to buy 13 pigs. Surplus: 1000
cash. Sell 3 cows and 3 pigs to buy 9 sheep. There is exactly
enough cash. Sell 6 sheep and 8 pigs. Then buy 5 cows.
There is 600 coins deficit. Tell: what is the price of a cow, a
sheep and a pig respectively?
• This means the three equations:
2C + 5S −13P = 1000
3C − 9S + 3P = 0
−5C + 6S + 8P = −600
• The solution is found to be C = 1200, S = 500, P = 300.
How it was done
• An array of sticks, exactly what we call a matrix, was laid
out representing the coefficnts (red if pos, black if neg).
• These were manipulated exactly as in Gaussian
elimination.
• Equations were never used, no symbols for unknowns!
• The whole procedure resembles in an uncanny way what
goes on in a digital computer today.
The Chinese algorithmic orientation
• Nothing like this appears in Europe until 18th
cent. Fibonacci’s Liber Abaci (1202) had a
confused awkward sort of tabular notation.
• Remarkably, in the early Yuan Dynasty, Zhu Shijie
(朱世杰) proposed solving n simultaneous
polynomial equations in n unknowns, using yet
larger arrays for all their coefficients – but still
without using x,y, etc.
• Their algorithm is very similar to Bayer-Stillman’s
computer code ‘Macaulay’.
• The same story can be told with Chinese
astronomy: code-like algorithms, theory implicit
3. How ‘Fourier’ series were discovered: applied
math waiting for pure math to catch up.
• Two physical problems which pre-occupied
mathematicians in the first half of the 18th century:
the motion of a vibrating string with fixed endpts
and the perturbations of the moon by the sun.
• Both led them (in 1749/1754 resp.) to ‘Fourier’
series (including the formula for its coefficients),
but they were distracted by the relationship of a
physical variable with a mathematical formula, i.e.
not formalizing the concept “function”, so could
not make Fourier’s 1822 leap, (also based on an
application: heat conduction).
y ( x, t ) = vertical displacement of string at pt. x and time t
d2y
d2y
= c× 2
2
dt
dx
D'Alembert and Euler: if y ( x,0) = f ( x ), then
y = 12 ( f ( x + t ) + f ( x - t ))
Euler noted what he thought were special solutions:
y=
å
an sin(np x / a )
Soon after, Daniel Bernouilli -- thinking physically -- asserted
that this gives all functions with f (0) = f (a ) = 0. Then
Euler -- thinking formally -- wrote that he doubted whether
all "eellike" curves could be expressed like this.
Clairaut, in 1754, studying the three body
problem, explicitly asserted that all periodic
functions had a trigonometric expansion and
gave the inversion formula! So why aren’t
these called “Clairaut series”?
………………………………………
Pure math did catch up: here’s how I
learned it:
(Loomis, Abstract Harmonic Analysis)
But this is how engineers (and I) have taught it:
A spectrogram of a female voice singing
do, re, mi, fa, sol, la, ti, do
Why, when
Gowers won the
Fields Medal
proving the nonexistence of
“unconditional
bases” in various
Banach spaces,
did no one
explain bases to
the general
public using the
example of
musical notes?
4. Links of pure and applied math in
computer vision – image segmentation
• Shah and I introduced a free boundary variational
problem for segmenting the domain of a real
world image into pieces representing the distinct
objects in the scene. Its math (pursued by the
French school) is challenging -- still unknown if this
is well-posed! Vision-wise, it is a very crude model.
I ( x ) = a given function, e.g. value of image I at pixel x,
which we want to approximate by a piecewise smooth fcn.
K =bndaries of decomp. of domain, u( x )="cartoon" of I
Minimize : E (u, K ) =
ò (u R
I ) + l .len( K ) +
2
ò
R- K
Ñu
2
Below left: an example of the simplified Mumford-Shah
algorithm on an easy image. Below right: more complex
black/white images, human segmentations and one
attempt to segment them
(in false color).
Gibbs’ random fields and Bayes’ rule:
Energy = log(probability)
in non-physical settings
• These experiments and the work of S. and D.
Geman led to a general perspective which underlies
much of our present approach to vision.
I ( x ) = image, J ( x ) = 'hidden' variables describing scene near x
(e.g. object label, lighting, albedo, texture)
C = cliques of pixels whose variables are mutually constrained
Pr( J | I ) = Pr(1I ) Pr( I | J ) ×Pr( J )
æ
ç
- çç
ççè
C
Pr( I | J ) = Z1 e
1
å
ö
1
EC ( I |C , J |C )÷÷÷÷
÷
ø
æ
ç
- çç
èçç C
, Pr( J ) = Z1 e
2
å
ö
EC2 ( J |C )÷÷÷÷
÷
ø
• This link has been very productive. People
have gathered probabilities of many aspects of
natural images (how likely is an edge caused
by a shadow, a crease, foregrnd/backgrnd?)
and fed these into Gibbsian models.
• But it’s been hard to find algorithms to
compute the modes of Gibbs distributions –
so engineers rejected this approach for a
decade. Now some better algorithms
appeared and the approach is popular!
• Behind this, there is a big issue of modeling
images on a discrete lattice or on R2. I will talk
about this in my next lecture.
5. A mathematical model for comparing
shapes: use geodesics in a space of
submanifolds
• Riemann’s Habilitationschift proposed this idea:
“There are however manifolds in which the fixing
of position requires not a finite number but either
an infinite series or a continuous manifold of
determinations of quantity. Such manifolds are
constituted for example by … the possible shapes
of a figure in space, etc.”
• Fixing an ambient manifold M and some topological invariants, let us define the Chow manifolds
of M to be the manifolds of all submanifolds of M
of the prescribed topological type.
How Riemann’s vision evolved
• Arnold made the first step studying a metric
on the infinite dimensional manifold of all
volume preserving diffeomorphisms of Rn,
showing its geodesics were solutions of Euler’s
equantion for fluid flow and calculating its
curvature.
• But it was the application to warping medical
images, body parts that has driven the study
of a huge variety of other metrics on
diffeomorphisms and submanifolds. Marsden
and his school picked this up.
A Sobolev-type Riemannian metric and its
geodesics – here is an example of Younes
M = {simple closed plane curves}mod translations
TC M = {normal vector fields a ( s)nC ( s)}
2
Define a = min b ò Ñ (an + bt ds
2
C
Two geodesics:
Now a major industry in analyzing medical scans
comparing the full cortex of a patient with a ‘template’
cortex using sulci, gyri, surface, MRI)
•Du, Younes, Qiu,
Neuroimage 2011
Whole brain
diffeomorphic
metric mapping
• a = healthy brain
• d = senile brain
with shrunken
white matter,
enlarged
ventricles
• c = warp of a,
matching d
• Kriegl and Michor’s theory of convenient
manifolds allows one to use freely C
submanifolds and do standard differential
geometry. One can take Cauchy completions
in Sobolev-type Riemannian metrics if
preferred.
• Strange new beasts occur: e.g. manifolds with
local geodesic spray but conjugate points
dense and inf(path length) 0.
• I will discuss some of this theory in my 3rd
lecture and apply it to the Yu Ji Tu 禹迹图 a
Song Dynasty map of China.
6. Words of Caution
• Almost all mathematicians, up to 1920 or so,
worked on physics and applied as well as pure math
• Then Bourbaki on the one hand and the rise of
computers on the other pulled them apart.
• In 1952 SIAM was founded in the US, followed by
ICIAM and all the other “IAM” societies.
• Mechanics was dropped as a course in math depts.
• CS and Statistics began to form separate
departments. Applied math is not one field but the
heavy use of computer experiments unites them all.
• Physicists dropped all pretense of using rigorous
math.
The negative side
• Explosion of specialized languages fragmented pure
math as well as applied.
• How many people can define all of these:
–
–
–
–
perverse sheaves,
the monster group,
barreled spaces,
inaccessible cardinals
• Try explaining a relevant piece of math to a biologist
(e.g. who doesn’t know what a group is)!
• We have allowed ourselves to live in a tower of Babel.
Each field, each sub-sub-field, is going its own way.
There is a culture of modeling as well as its
pitfalls
Develop
pure math
theory that
is
increasingly
irrelevant
Analyze
improved
model
Create
modifications
of math model
Run algorithms
to compute
with model
Fall in love
with code
Tweak
endlessly
Analyze errors
to
where results
improve
differ from data ROC curve
Some words to new students
• In my experience, this is a very hard paradigm to
teach – students are either by nature more mathy or
more engineery and veer off to left or right.
• One must not forget that all models are inaccurate
simplifications of nature so there will always be
errors. Don’t cling to your favorite model!
• But all models also have theoretical properties that
shed light on what they can and cannot capture and
these must not be neglected. Don’t ignore the math.
• One must go back and forth between
math/experiment.
• Thank you for listening.