Thanks very much for the wonderful introduction.
So, I will be talking about the unreasonable effectiveness of physics in
mathematics, so
not quite as many branches of mathematics as was alluded to a minute ago.
And I wanna start by saying that my title is a bit of a twist on a famous
essay by the physicist, Eugene Wigner The Unreasonable Effectiveness of
Mathematics in the Natural Sciences.
And a little bit more familiar with.
From high school where we’re used to solving simple equations or
using basic concepts from geometry,
in order to solve problems of interest in physics or other natural sciences.
But today, I’d like to tell you about the field in which I work,
which really has new developments going in the opposite direction.
And so I'd like to argue that modern physics really has a lot to say
about mathematics, and
I believe that this is a really fruitful area of inter-disciplinary research.
And I think this period of this field is really well summarized by this quote by
Raoul Bott, a late mathematician at Harvard.
He says that although physicists and mathematicians,
we really have different sets of tools, we speak different languages sometimes.
There’s a push and pull nature of our relationship that really invigorates
Both of our fields, and it makes both fields stronger for the interaction.
And so, I'll be telling you one story from this interaction,
coming from my perspective as a theoretical physicist.
And in particular, the theory that I work on is called string theory.
So let me briefly give you the basic picture of what string theory is.
So as the name suggests,
String Theory is a theory of little vibrating one dimensional objects.
Strings and okay, the sound is a little bit innocuous.
Why would I care about a theory like this?
And it turns out that String Theory is what's called the Theory of
Quantum Gravity.
So what do I mean by that?
Well, there are two great pillars of modern physics,
at least two that I'm gonna talk about right now.
And one is general relativity, Einstein's theory of gravity.
And that's a theory that talks about large scale structures.
What is the shape of space time?
How do we understand galaxies, planets, large objects?
I mean, even just today,
we had a really exciting announcement that gravitational waves were discovered,
that's the theory of general relativity or one of it's predictions.
And on the other hand,
we have the theory of quantum mechanics which is a really beautiful and
extraordinarily effective theory for describing things that are very small.
Molecules, atoms, elementary particles, and so forth.
And the quantum theory of gravity really seeks to unite these two great pillars.
And this is an old dream called unification starting with Einstein.
That we have two great theories, but
why can't we combine them into one great theory that incorporates aspects of both?
And this was a notoriously hard problem.
And it wasn't until the 1970s when physicists were studying
actually something totally different than nuclear force in atoms that
they discovered this theory of strings.
And after looking at it a little bit more carefully, what they found was that
this very consistent quantum mechanical theory actually had gravity within it.
And they found that Einstein's equations emerge
from equations coming from this quantum mechanical theory of strengths.
And I don't have time to tell you too much more about string theory,
it might be the theory of quantum gravity that describes our universe.
But today, I'll be telling you about it's role as powerful and beautiful machine and
generating interesting questions in Mathematics.
And to that I'll only tell you one of the string theories prediction,
perhaps one of its more well-known ones.
And if you play with the equations of string theory,
they tell you something a little bit strange.
They say that there should ten spacetime dimensions.
What do I mean about that?
Well, in our daily life,
we use three spatial dimensions, forward back, left right, up and down.
And we’re also used to things moving in time or evolving in time.
Einstein’s Theory of Relativity has taught us that we should think of space and
time on the same footing.
So I say that we’re used to four large macroscopic space/time dimensions.
We’re used to four, but String Theory tells that maybe there’s ten.
And if there are ten, where are they?
And why don't we see them and what are their implications for physics?
And so, first let's pretend that string theory just predicted one additional
dimension that we don't see.
Well, how should I think about it?
So imagine you're standing very far away from a telephone wire.
If you're far enough away,
you only see one dimension that the wire has, its length.
But if you zoomed in closer to the wire or
if you were a little insect that could crawl around on the surface of the wire,
you'd become cognizant of the other dimension it has, the width.
The small circular extra dimension that you might not have been aware of if you
were standing very far away.
And so perhaps, if there was one extra space time dimension,
we'd be curled up very small in a little circle.
This is an idea dating back to the physicists, Kaluza and Klein.
And we don't see the circular dimension perhaps it's very small,
and perhaps someday experiments will tell us whether or not it's actually there.
But, we need experiments that can really zoom in close enough so, to speak.
But would elucidate the structure of space time for us.
But now, string theory says that there are ten space time dimensions and
if we believe it, they should be curled up into much more elaborate shapes or
geometries than circles.
And so, I've pictured here some of the candidate shapes that string theory says
are candidates for the extra six dimensions.
And these beautiful geometries have a very fancy name.
They're called Calabi-Yau manifolds, after the famous mathematicians Calabi and
Yau who really advanced their studies in mathematics.
These fancy, extra dimensions, rather horribly rendered here in two dimensions.
There were certainly of independent interests in mathematicians.
But also one should note that if these really are the extra
six dimensions of space time.
If we're interested in studying string theory and
figuring out what it's predictions are,
then the geometry of these manifolds should tell us something about physics.
So in particular, if we’re studying interactions of particles in the four
macroscopic dimensions, the predictions that we see in
four dimensions should be influenced by the geometry of these interesting and
beautiful structures that might be hiding curled up very small.
And so, the story I’d like to tell you about String Theory impacting mathematics
I'd like to call Making String Theory Count, quite literally.
Because the branch of math that I'd be telling you about is called Enumerative
geometry, which is counting numbers of solutions to certain geometric questions.
And this is an old and venerable subject in mathematics.
So to orient you let me tell you an example such question in Enumerative
geometry from around 200 BC.
And the question is, how many circles are tangent to three given circles in a plane?
And by tangent I mean it just touches each of the three circles at one point.
And here's one pictorial solution to the problem.
You can see that the black circles are affixed circles in the plane and
this purplish circle is tangent to them, it touches them once at each point.
And the answer to the enumerative geometry question as it turns out,
is that in general there are eight such solutions.
And you can even come up with that by spending a lot of time drawing circles
on paper.
You don't really need to much fancy intuition
to tell you the answer to this enumerative geometry question.
But, if I'm a modern mathematician, maybe drawing circles on paper isn't
Quite enough for me, maybe I'm interested in Calabi-Yau Manifolds,
these strange geometric structures.
And here I've just pictured one very beautiful and simple, perhaps the most
simple example of a Calabi-Yau Manifold, it's called the Quintic.
And if you're an equations person, you can see, I just called that,
because it's an equation in five variables and the raise to the fifth power.
So the number five shows up a lot.
And so, this because it's a Columbia manifold,
it's a candidate manifold on which strings might life.
But mathematicians don't necessarily care about string theory, at least not yet and
they were just asking the following question.
Can we count the number of lines in a quintic?
Or a line is an equation of degree one, I'll explain that in a minute.
And more generally,
what about the number of curves of degree D, I'll call this number N sub D.
So, here I've just pictured some example curves.
You see the line as curve of degree 1.
And if you don't like equation, you can just trust me
that equations have a number associated to them called the degree.
And each of these curves that are drawn have some equations describing them.
These are just some examples of such curves.
Actually, that's not a complete set of pictures by any means.
The number of different curves really proliferates as you increase the degree.
And if don't like equations just think of curves of higher D having more wiggles and
loops and more elaborate things going on, they're just more complicated looking.
And if you like equations, I'll tell you that,
the curves are described by polynomials and degree D, that's why they're called.
And so, it's a natural question for mathematicians,
how many such curves can live on this fancy manifold that you're interested in?
This is certainly not a question you can attack by drawing a lot of pictures on
a piece of paper.
Maybe not even if you had a six dimensional paper lying around.
And so, mathematicians worked very hard to try and answer this question.
They use the height of the geometric abstract tools that they had at the time.
They used all their grad students, every tool at their disposal, and
after all of these work, I think I got to maybe only n sub 3.
And I think n sub 3 even had a mistake in it, so didn't get very far.
It was very hard.
I mean it's extremely difficult problem, but on the other end of the campus or
a few buildings away, physicist were studying string theory and string
theory had these manifolds, hiding very quietly in the structure of spacetime.
And the physicists were asking perhaps more normal physics questions,
how do particles interact?
How strongly do they interact?
What's the nature of the particles in the fore space time dimensions?
As you can imagine the strings are moving around, they're probing the geometry
it's based on, they're traveling around on these Calabi–yau manifolds.
And in fact strings as little one dimensional objects are wiggling around,
and they look like curves crawling all over the surface of these manifolds,
well these geometries if you will.
And so, the physicists had some answers to some physical questions and they found
some numbers hidden in their answers, some large interesting looking integers.
So they went to their mathematician friends.
Well, is the physics telling us anything interesting about the geometry?
And the mathematicians said well, we've been computing these curve counts and
we've got to n sub 3, and it's pretty hard.
We think we're right.
And the physicist said, I see.
Well, I think we can do a little better with string theory.
And in fact, it turned out that string theory can count these curves naturally.
Based on the strings moving around in these spaces.
String theory generated an infinite number of predictions.
Mathematicians later worked hard and were able to prove this.
And these numbers I've shown you I just computed for the but this is true for
more general geometries that might string theory.
However, not only is this conjunction approved, but new ingredients and
ideas from string theory, following up on this observation,
lengthen the development of a whole new subject in geometry.
It's called mearsymmetry.
Unfortunately, not as simple as its name suggest.
It's very active area of mathematical research today going on even now.
And what I really wanna stress is that this is just one success story
of string theory influencing mathematics.
In fact, we're still only scratching the surface of this incredible edifice
about string theory.
An incredible and beautiful theoretical structure.
My research as I've mentioned briefly tries to bring together a lot of
areas of mathematics and new interesting unifying ways.
And I think we're making many discoveries such that
physics is really helping mathematics develop, and
I think mathematics is of course still teaching us a great deal about physics.
And I thank you for your attention.
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