- Since the beginning of humanity,
scientist have been exploring
the mysteries of space.
Today, 96% of the matter in the universe
remains an unknown mystery.
Astronomers have called this
mystery substance dark matter.
So, now, the question
remains, how do we prove
that this abstract
quantity definitely exists
or how do we show for
certain that it cannot exist?
This notion of proving or
disproving the existence
of an abstract idea is at
the core of pure mathematics.
As an algebraist,
I work in an area called
representation theory
where we take complex algebraic structures
and represent them as simpler ones
that are easier to understand.
For over 150 years,
mathematicians worldwide
have worked to understand
the representations
of algebraic structures
and this has united so many
different areas of math
that the theory has been called
the great romantic story
of mathematics (laughs)
so, to answer the question
on everyone's mind,
why should you care about
representation theory?
In fact, we can see
representations of abstract
math almost everywhere.
The Fibonacci sequence of numbers
shows up in our own Milky Way galaxy
represented as something
called the golden spiral,
which you can see here.
This actually shows up
overwhelmingly in nature and space.
It can even be found in
Leonardo da Vinci's art.
My work in this area looks at
the part of a representation
called an invariant and I
try to establish patterns
for how these structures
are related to each other.
An invariant is something
that does not change
after we do change
the mathematical environment that it's in.
An example that you might be familiar with
is in the quadratic formula.
The part under the square
root, b squared minus four ac,
is actually an example of an invariant.
So, now, there's classical theory
that gives us the relationships
between these invariants,
however, there are tons of applications
that fall completely outside
of this classical theory,
including my own work.
So, I've created a completely new method
to establish the relationships
between these invariants,
and it works in any classical cases,
as well as any non-classical case.
My methods runs in parallel
with representations.
I take a complicated mathematical system
and represent it as a simpler
one that's easier to solve.
I hope that my work in this area
can help us strengthen our understanding
of representations and invariants
that fall outside of
this established theory,
as well as further our knowledge
of any scenario in nature
where invariants and
abstract math can be found.
Who knows, maybe dark matter
is really just a mathematical invariant
waiting for us to find it.
