#15 Abhishek Banerjee – Math as a window to the universe

Most technology is based on Mathematics. As Tony Crilly puts it in his book ‘50 mathematical ideas you really need to know’, “there is no longer any pride left in announcing to have been no good at it when at school.” However, in the presence of Indian Institute of Science Assistant Professor Abhishek Banerjee, one can’t help but feel a bit mathematically overwhelmed.  He is our Bangalore Global Icon #15 for inspiring us to look at Mathematics, not with fear and trepidation as some of us did in school, but with awe and wonder.

Before joining IISc in January 2014, Abhishek used to be a Maître de Conférences (Associé) at Collège de France in Paris. And prior to that for three years he was a post-doc at Institut des Hautes Études Scientifiques (IHÉS), Bures sur Yvette and a Zassenhaus Assistant Professor at Ohio State University. He received his PhD in 2009 from Johns Hopkins University in Baltimore, Maryland.

His research work is in the fields of Algebraic Geometry and Non-commutative Geometry. In the domain of Non-commutative Geometry, his main interest is in Hochschild and cyclic homology. In Algebraic Geometry, his main interest is in developing results for schemes over symmetric monoidal categories. Additionally, he studies bivariant Chow groups and also likes to dabble in K-theory and sometimes in arithmetic geometry.

One hears of different kinds of science – ancient Indian science and
Greek science. I wondered if in modern times, if we differ mathematically? Abhishek
says that modern mathematics is very international. “It is hard to identify any
particular subculture within mathematics based on geography. My
understanding is that at the college level, mathematics is taught in the same
way roughly everywhere; in India, in the US or in Europe.”

Here are excerpts from the interview:

Being
a Non-Mathematician, and for a long time at that, I asked Abhishek how the
mathematical theories he was working on could help the world we live in.

AB: My work is
mostly theoretical, in fields of mathematics known as 'non-commutative
geometry,' 'algebraic geometry,’ and 'category theory.' In terms of
applications, it is hard to know exactly. But sometimes, the applications are
not just more than we imagine, but more than we can imagine. For instance, non-commutative
geometry is one of the key tools used to understand the universe in terms of
what is called the ‘standard model.’  The standard model touches
everything from general relativity to the Higgs Boson and basic principles of
electromagnetic force.

Recently, I was invited to give a series of lectures to physicists at a
large meeting in IISER-Kolkata on Quantum Computing. I was rather
surprised to find out that category theory is of great interest to those
working on Quantum computing. As you know, quantum computing is one of
those technologies that is likely to change everything about our world.

Personally, I tend to think of mathematics more as a sport, such as
running a race. Stretching the mind just as an athlete would push their body,
searching for its limits.

Although I suspect I know the answer, I still ask Abhishek about his
role model in Mathematics. After all Mathematics though spoken as one subject,
is a vast ocean – and one can have heroes in pure and applied Mathematics,
abstract and concrete, and even in the common man who can solve a Sudoku in a
minute.

AB: That's easy. Srinivasa Ramanujan of course. Like I said, I think of
math more like a sport. And as with sport, we want to be like our heroes. At
one point, I really wanted to be a number theorist, which is the field that
Ramanujan worked in. But as I got more and more invested in the subject, I
realized there were other forms of mathematics that were better suited to my
temperament. 

I wanted to become a number theorist when I was young. Eventually, my
research went another way. So I married a number theorist. She is a faculty
member at IISER Pune.

From his ocean of research, I asked Abhishek to list his top three
research works out of a very long, mind-boggling list and their application in
layman terms.

AB: As I said before, my work is rather theoretical, but I can try to explain some of the themes. I already spoke about noncommutative geometry. In algebraic geometry, we often try to understand lines, curves and surfaces. A point has zero dimension, a line has one dimension and a surface has two. If you take two straight lines, they meet at one point, unless they are parallel. If you take two planes, they will meet in a line, unless they are
parallel. 

From a mathematical point, this is unpleasant. The fact that we have to
make an exception for parallel cases is ugly. In algebraic geometry, we find
ways to redefine what it means for lines to ‘meet’, so that two straight lines will
always meet in a point. Even the line will meet itself as in a single point.

A lot of questions in math begin like this. We identify something that
looks like an ugly exception and then we define something new and beautiful
that incorporates it into a single general rule. For example, we can add 10
given numbers or 100 given numbers. But how do you add infinitely many numbers
together? 

In a recent talk for CSP Abhishek spoke about a stereotypical Math
teacher whom all of us have encountered in our school days. So how do we get
children to love maths nonetheless?

AB: The way mathematics is taught at the ground level, in school, needs
serious reform. Not just in India, but all over the world. By the time students
are in college, many have already internalized the idea that math is ‘difficult’.
This is a cumulative failure of parents, teachers and popular culture, which
work together to create this impression. And really sad, because math is in
many ways the easiest subject. It requires almost zero rote memorization.
There are so few rules that they would fit on the back of an envelope and never
any exceptions to those rules. 

So the first thing would be to stop telling kids that math is ‘difficult,’
so that it does not turn into a self-fulfilling prophesy. Let children
judge for themselves. Give them questions involving numbers and challenge them
to find a solution. Once they have a solution, ask them if there's any other
way they could have solved the same problem. 

Often times, the solutions to standard problems are presented as boring algorithms
to be memorized, starting with addition, carry overs and long division. This
should be avoided. Help kids figure out the place value system, not just in
terms of powers of 10, but powers of any number. Use the binary system to
explain powers of 2. When a kid understands how to take powers, ask them how
they would raise 2 to the power of pi. Would you multiply 2 to itself pi number
of times? What does that even mean? Don't be afraid to think laterally or
ask questions that may be beyond the level.  It's okay to have some
questions unanswered, left for later as fodder for curiosity. 

So what drives Abhishek -the most complex of numbers or how they relate
to life?

AB: I said that I often think of mathematics as a sport. I try to push
my mind constantly. Every day is an adventure. I fail constantly, but you have
to keep thinking harder and harder. A lot of mathematics is about seeking
fundamental patterns. Often times, I think I have a pattern but it turns out to
be a mirage. Remember that the person we can fool most easily is ourselves. So
first you think you have something and next moment you are disappointed. But
you think again, look for different fundamental properties and then maybe you
will find something. I would say what drives me is failure. No tonic is better
to keep going.

The beauty of maths lies in truths that have universal and absolute
validity. Abhishek Banerjee gives the example of certain kinds of cicada, known
as magicicadas, which have a 13 year life cycle, i.e. they hatch en masse once
every 13 years.

AB: “Now, why is that so clever? Notice that 13 is a prime number. In
nature, the life cycles of a predator will synchronize with prey to maximize
availability of food. If Magicicadas emerged say every 6 years, a predator
with a 2, 3, or 6 year life cycle would be able to feed on them. But the
number 13 makes it that much harder for predators. So we are all thinking
mathematically: even cicadas. If tomorrow we were to try to talk to an alien
species, how would we communicate? Hard to think of any language more universal
than Math.”

https://www.youtube.com/watch?v=AJJlE8VTEhk

In the competitive world we live in the ‘Two-person, zero-sum games’ are
very common. A game played by all of us, involving two sides, where one wins
and the other loses. Finding the cheapest route, sending coded secret messages,
mathematics is all pervasive. I ask Abhishek about his conversations with a
biologist on the meaning of life as understood through mathematics.

One of my most interesting scientific interactions ever was with an
Israeli professor who was a biologist. We would meet for lunch every single day
and ask simple but perhaps unlikely questions. And try to find rational
answers. The first one we had was ‘why do we die?’ After a few days we agreed
that an organism that never dies would have to be able to survive without food,
water, be fire proof ... not even susceptible to being crushed by a rock. The complexity
that would require is simply unachievable in one single evolutionary
step. 

Another question we had was - why are there two genders? Why not three
or five? We eventually settled on the conclusion that having sexual
reproduction by bringing together two genders is hard enough. Think about how complicated
cross pollination is with two genders already. Can you imagine how unlikely
pollination would be if a bee had to visit three different flowers of the same
species but of three different genders? As for genetic diversity, powers of 2
grow fast enough to produce healthy offspring with variety of traits. No need
to go for powers of 3 or more.

The first thing to realize is that things like these are scientific
questions that deserve rational answers. We wouldn't claim that our answers
above were the correct ones. As I said, this was some informal chatting between
us, purely for entertainment purposes. I think the important thing here is the
focus on rationalism.