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CMI Profi le
Interview with Research Fellow Terence Tao
Terence Tao (b. 1975), a native of Adelaide, Australia,
graduated from Flinders University at the age of 16 with
a B.Sc. in Mathematics. He received his Ph.D. from
Princeton University in June 1996 under the direction of
Elias Stein. Tao then took a teaching position at UCLA
where he was assistant professor until 2000, when he
was appointed full professor. Since July 2003, Tao has
also held a professorship at the Mathematical Sciences
Institute Australian National University, Canberra.
Tao began a three-year appointment as a Clay Research
Fellow (Long-Term Prize Fellow) in 2001. In 2003,
CMI awarded Tao the Clay Research Award for his contributions to classical analysis and partial differential
equations, as well as his solution with Alan Knutson of Horn’s conjecture, a fundamental problem about the
eigenvalues of Hermitian matrices. Tao is the author of eighty papers, concentrated in classical analysis and partial
differential equations, but ranging as far as dynamical systems, combinatorics, representation theory, number theory,
algebraic geometry, and ring theory. Three-quarters of his papers have been written with one or more of his thirty-
three collaborators.
Interview
From an early age, you clearly possessed a gift for math- who were willing to spend time with me just to discuss
ematics. What stimulated your interest in the subject, mathematics at a leisurely pace. For instance, there was a
and when did you discover your talent for mathematical retired mathematics professor, Basil Rennie (who sadly
research? Which persons inß uenced you the most? died a few years ago), whom I would visit each weekend
to talk about recreational mathematics over tea and
Ever since I can remember, I have enjoyed mathematics; cakes. At the local university, Garth Gaudry also spent
I recall being fascinated by numbers even at age three, a lot of time with me and eventually became my masters
and viewed their manipulation as a kind of game. It thesis advisor. He was the one who got me working in
was only much later, in high school, that I started to analysis, where I Ever since I can
realize that mathematics is not just about symbolic still do most of my remember, I have enjoyed
manipulation, but has useful things to say about the real mathematics, and
world; then, of course, I enjoyed it even more, though at who encouraged mathematics; I remember
a different level. me to study in being fascinated by
the US. Once in numbers even at age three.
My parents were the ones who noticed my mathematical graduate school,
ability, and sought the advice of several teachers, I benefi tted from
professors, and education experts; I myself didn’t feel interaction with many other mathematicians, such as my
anything out of the ordinary in what I was doing. I didn’t advisor Eli Stein. But the same would be true of any other
really have any other experience to compare it to, so it graduate student in mathematics.
felt natural to me. I was fortunate enough to have several
good mentors during my high-school and college years
10 CMI ANNUAL REPORT
What is the primary focus of your research today? Can you I work in a number count this as one of my
comment on the results of which you are most fond? favorite areas to work
of areas, but I don’t in. This is because of
I work in a number of areas, but I don’t view them as view them as being all the unexpected
being disconnected; I tend to view mathematics as a disconnected; I tend structure and algebraic
unifi ed subject and am particularly happy when I get the to view mathematics ªmiraclesº that occur in
opportunity to work on a project that involves several these problems, and also
fi elds at once. Perhaps the largest ªconnected componentº as a unifi ed subject because it is so tech-
of my research ranges from arithmetic and geometric and am particularly nically and conceptually
combinatorics at one end (the study of arrangements of happy when I get challenging. Of course,
geometric objects such as lines and circles, including one the opportunity to I also enjoy my work
of my favorite conjectures, the Kakeya conjecture, or the in analysis, but for a
combinatorics of addition, subtraction and multiplication work on a project that different reason. There
of sets), through harmonic analysis (especially the study involves several fi elds are fewer miracles, but
of oscillatory integrals, maximal functions, and solutions at once. instead there is lots of
to the linear wave and Schrödinger equations), and ends intuition coming from
up in nonlinear PDE (especially nonlinear wave and physics and from geometry. The challenge is to quantify
dispersive equations). and exploit as much of this intuition as possible.
Currently my focus is more at the nonlinear PDE end In analysis, many research programs do not conclude
of this range, especially with regard to the global and in a defi nitive paper, but rather form a progression of
asymptotic behavior of evolution equations, and also steadily improving partial results. Much of my work has
with the hope of combining been of this type (especially with
the analytical tools of nonlinear regard to the Kakeya problem
PDE with the more algebraic and its relatives, still one of my
tools of completely integrable primary foci of research). But I
systems at some point. In do have two or three results of
addition, I work in a number of a more conclusive nature with
areas adjacent to one of the above which I feel particularly satisfi ed.
fi elds; for instance I have begun The fi rst is my original paper
to be interested in arithmetic with Allen Knutson, in which
progressions and connections we characterize the eigenvalues
with number theory, as well as of a sum of two Hermitian
with other aspects of harmonic matrices, fi rst by reducing it to
analysis such as multilinear a purely geometric combinatorial
integrals, and other aspects of © 1999-2004 by Brian S. Kissinger, licensed for use question (that of understanding a
PDE, such as the spectral theory of Schrödinger operators certain geometric confi guration called a ªhoneycombº),
with potentials or of integrable systems. and then by solving that question by a combinatorial
argument. (There have since been a number of other
Finally, with Allen Knutson, I have a rather different proofs and conceptual clarifi cations, although the exact
line of research: the algebraic combinatorics of several role of honeycombs remains partly mysterious.) The
related problems, including the sum of Hermitian second is my paper on the small energy global regularity
matrices problem, the tensor product muliplicities of of wave maps to the sphere in two dimensions, in
representations, and intersections of Schubert varieties. which I introduce a new ªmicrolocalº renormalization
Though we only have a few papers in this fi eld, I still in order to turn this rather nonlinear problem into a
THE YEAR 2003 11
more manageable semilinear evolution equation. While My work on Horn’s conjecture stemmed from discussions
the result in itself is not yet defi nitive (the equation of I had with Allen Knutson in graduate school. Back then
general target manifolds other than the sphere was done we were not completely decided as to which fi eld to
afterward, and the large energy case remains open, and specialize in and had (rather naively) searched around
very interesting), it did remove a psychological stumbling for interesting research problems to attack together.
block by showing that these critical wave equations were Most of these ended up being discarded, but the sum of
not intractable. As a result there has been a resurgence Hermitian matrices problem (which we ended up working
on as a simplifi ed
model of another
question posed
by another graduate
student) was a lucky
one to work on,
as it had so much
unexpected structure.
For instance, it can be
phrased as a moment
map problem in
symplectic geometry,
UCLA Spotlight Feature from the UCLA Website, Courtesy of Reed Hutchinson, UCLA Photographic Services and later we realized
of interest in these equations. Finally, I have had a it could also be quantized as a multiplicity problem in
very productive and enjoyable collaboration with Jim representation theory. The problem has the advantage
Colliander, Markus Keel, Gigliola Staffi lani, and Hideo of being elementary enough that one can make a fair
Takaoka, culminating this year in the establishment of bit of progress without too much machinery ± we had
global regularity and scattering for a critical nonlinear begun deriving various inequalities and other results,
Schrödinger equation (for large energy data); this although we eventually were a bit disappointed to learn
appears to be the fi rst unconditional global existence Collaboration is very important for me,
result for this type of critical dispersive equation. The
result required assembling and then refi ning several as it allows me to learn about other fi elds,
recent techniques developed in this fi eld, including an and, conversely to share what I have
induction-on-energy approach pioneered by Bourgain, learnt about my own fi elds with others.
and a certain interaction Morawetz inequality we had It broadens my experience, not just in a
discovered a few years earlier. The result seems to reveal
some new insights into the dynamics of such equations. technical mathematical sense, but also in
It is still in its very early days, but I feel confi dent that being exposed to other philosophies of
the ideas developed here will have further application research and exposition.
to understanding the large energy behavior of other
nonlinear evolution equations. This is a topic I am still that we had rediscovered some very old results of Weyl,
immensely interested in. Gelfand, Horn, and others). By the time we fi nished
graduate school, we had gotten to the point where we
You have worked on problems quite far from the main had discovered the role of honeycombs in the problem.
focus of your research, e.g., HornÕs conjecture. Could We could not rigorously prove the connection between
you comment on the motivation for this work and the honeycombs and the Hermitian matrices problem,
challenges it presented? On your collaborations and the and were otherwise stuck. But then Allen learned
idea of collaboration in general? Can a mathematician in of more recent work on this problem by algebraic
this day of specialization hope to contribute to more than combinatorialists and algebraic geometers, including
one area? Klyachko, Totaro, Bernstein, Zelevinsky, and others.
With the more recent results from those authors we were
12 CMI ANNUAL REPORT
able to plug the missing pieces in our argument and subfi eld of mathematics has a better chance of staying
eventually settle the Horn conjecture. dynamic, fruitful, and exciting if people in the area do
make an effort to write good surveys and expository
Collaboration is very important for me, articles that try to reach
as it allows me to learn about other fi elds, In fact, I believe that a subfi eld out to other people in
and, conversely, to share what I have of mathematics has a better neighboring disciplines and
learned about my own fi elds with others. invite them to lend their
It broadens my experience, not just in a chance of staying dynamic, own insights and expertise
technical mathematical sense but also in fruitful, and exciting if people to attack the problems
being exposed to other philosophies of in the area do make an effort in the area. The need to
research, of exposition, and so forth. Also, to write good surveys and develop fearsome and
it is considerably more fun to work in groups impenetrable machinery
than by oneself. Ideally, a collaborator should expository articles... in a fi eld is a necessary
be close enough to one’s own strengths that evil, unfortunately, but as
one can communicate ideas and strategies back and understanding progresses it should not be a permanent
forth with ease, but far enough apart that one’s skills evil. If it serves to keep away other skilled mathematicians
complement rather than replicate each other. who might otherwise have useful contributions to make,
then that is a loss for mathematics. Also, counterbalancing
It is true that the trend toward increasing complexity and specialization
mathematics is at the cutting edge of mathematics is the deepening insight
more specialized and simplifi cation of mathematics at its common core.
than at any time Harmonic analysis, for instance, is a far more organized
in its past, but and intuitive subject than it was in, say, the days of Hardy
I don’t believe and Littlewood; results and arguments are not isolated
that any fi eld technical feats but instead are put into a wider context
of mathematics of interaction between oscillation, singularity, geometry,
should ever get and so forth. PDE also appears to be undergoing a
so technical and similar conceptual organization, with less emphasis on
complicated specifi c techniques such as estimates and choices of
that it could function spaces, and instead sharing more in common
not (at least in with the underlying geometric and physical intuition.
principle) be In some ways, the accumulated rules of thumb, folklore,
accessible to a and even just some very good choices of notation can
general mathe- Godfrey Harold Hardy (1877±1947) make it easier to get into a fi eld nowadays. (It depends
reproduction from Remarkable Mathematicians by Ioan
matician after James, © Ioan James 2002, University Press, Cambridge. on the fi eld, of course; some have made far more progress
some patient work (and with a good exposition by an with conceptual simplifi cation than others).
expert in the fi eld). Even if the rigorous machinery is
very complicated, the ideas and goals of a fi eld are often How has your Clay fellowship made a difference for you?
so simple, elegant, and natural that I feel it
is frequently more than worth one’s while Also, counterbalancing the The Clay Fellowship
to invest the time and effort to learn about trend towards increasing has been very useful in
other fi elds. Of course, this task is helped complexity and specialization at granting a large amount
immeasurably if you can talk at length with of fl exibility in my
someone who is already expert in those areas; the cutting edge of mathematics travel and visiting plans,
but again, this is why collaboration is so is the deepening insight and especially since I was also
useful. Even just attending conferences and simplifi cations of mathematics subject to certain visa
seminars that are just a little bit outside your at its common core. restrictions at the time.
own fi eld is useful. In fact, I believe that a For instance, it has made
THE YEAR 2003 13
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