NEWSLETTER

Interview
Do you understand everything you are doing in mathematics?
I mean, what is important about a theorem
is not its precise formulation but the idea of the proof
which can be used to prove many different theorems.
If you just use the exact formulation, you are actually
treating the theorem as a black box.
I used to understand all the mathematics that I do, from
the very basics all the way up. There were no black boxes.
But in some recent collaborations just in the last year, I
have used some theorems I did not understand from first
principles but took on trust, using them as black boxes
as you say. I must admit that at first it made me quite uncomfortable.
It did not seem to bother my collaborators
as much though, who were already used to it after many
years of it! This is a modern phenomenon in doing mathematics,
building on years and years – and pages and
pages – of our predecessors’ work, to the point where it
is not possible for any one person to know every detail.
But one must get used to it. In some sense, it is also aweedge
of the pyramid. This is impressive. When did you
encounter, say, complex analysis?
I always enjoyed doing such things on my own for fun.
But, as I said, I never skipped any grades. For example, I
did not learn complex analysis until I was a sophomore in
college and took a course. So I was never that much more
advanced than my peers, particularly when I reached college
at Harvard, where so many students were really advanced.
Did you ever read a math book cover to cover?
Yes, I often did in graduate school. Princeton did not offer
many courses for beginning graduate students so this
was the way I learned much mathematics.
You always were a good boy?
I guess so. I’m told I was quite hyper and naughty when
I was a 3- or 4- or 5-year-old but it seems I outgrew that!
Did you have other interests? Reading? The classical
Indian epics such as Ramayana and Mahabharata constitute
a treasure trove which must be very fascinating
to a child.
Yes, these epics, as well as various other works of literature,
were frequently the topics of conversation over
family dinners at home. In fact, my uncle often made
these conversations into contests, as to who knew the
most details about various aspects of these works. These
contests were a lot of fun and added – for my cousins and
me – a further motivation to read!
Later on, did you consider any alternative careers?
I did often toy with the idea of becoming a musician and
tabla player as a profession. But at some point, I realized
that if I became a mathematician, I’d still find time to
keep up music as well but if I became a full-time musician,
I’d probably have a tough time keeping up mathematics!
So you were all set to be a mathematician from an early
age?
I always loved math and because my mother was a mathematician,
I was always aware of that pleasant career option.
But I did have brief periods when I changed and
thought maybe I’d like to do computer science, or economics,
or physics, or linguistics, as I liked these subjects a
lot too. But I always realized that mathematics was what
was bringing these subjects together, and mathematics
was what I liked about them.
How do you do mathematics?
I usually like thinking about mathematics while walking,
or jogging, or pacing, or working with someone at the
blackboard. I quite rarely sit down with a pen and paper
and do mathematics; that usually comes after something
has been worked out already.
So you cannot think when you are standing still.
I guess that’s a good way of putting it! The reason I stay
away from paper is that often when I think about math-
ematics, it is not in words or in terms of things that can
be written down yet. That usually comes later when I try
to translate these thoughts into more usual mathematical
language.
What about collaboration?
I love working with others. Thinking and talking about
and working on mathematics together with others is one
of the joys of mathematics. It can also be more productive.
Collaboration forces you to think things through
by explaining them, and the other person can then give
invaluable input, feedback and new perspectives, as
you bounce ideas off one another. Sometimes, different
collaborators can bring different areas of expertise to
a problem, which can be very valuable and help bring
some cross-pollination between areas.
Does working with someone not interfere with your
thinking process? When you are thinking hard about
something, you do not want to be interfered with.
Sometimes. But, in collaboration, one generally decides
which parts are more productive or fun when discussed
together and which parts each should go home and think
about alone. In the latter case, collaborators can go home,
think separately and then bring their ideas the next time.
So, even in collaboration, there is usually a lot of individual
thinking. Actually, sometimes you’ll be working
with a collaborator where no one speaks for over an hour
– individual thinking is going on! But still it’s fun to do
it together, knowing that there is someone there in case
you are ready to share an idea, or problem, or general
confusion.
Nothing beats direct human conversation, or should
I say interaction. Are there problems you have solved
in collaboration that you would not have been able to
solve by yourself?
Almost certainly! This is, of course, impossible to know
for sure. But I definitely think so, and I suspect this is
likely the case in most of my collaborations!
40 EMS Newsletter December 2015