imagine you are pac-man one day you go
for a walk you grab your two-dimensional
wallet leave your 2-dimensional house
and head off in some direction after a
bit of walking you find a wallet it’s
yours sure enough your pockets are empty
you put your wallet away and you
continue see pac-man is accustomed to
finding things he’s dropped without ever
turning around because when pac-man goes
up through the top of his screen he
comes in through the bottom when he goes
out through the right he comes in
through the left so what is this strange
two-dimensional surface the pac-man
lives on well – pac-man the top and
bottom of the screen are the same so we
may as well glue them together now we
have a cylinder the right and left of
the cylinder are also the same –
pac-man’s point of view
so let’s glue them together as you can
see here pac-man lives on the surface of
a doughnut delicious why a donut why not
some other two-dimensional surface
pac-man could very well be on a sphere
the gameplay would be a little bit
different but he could be on a sphere
looking for cherries and hiding from
ghosts or he could be on a two hole
donut same thing to hold surface of a to
hold donut it’s two-dimensional
pac-man’s two-dimensional this could
totally work or maybe the screen is set
up so that if pac-man goes up through
the top he comes in through the bottom
as usual but if he goes out through the
right at the top he comes in from the
left at the bottom upside down so what’s
going on here is that the right and left
are connected but in such a way that
there’s a twist when you glued them
together if you go through the same
process we did before
if you’ve seen such a object before
pac-man here would live on the surface
of a klein bottle I’ve given you a bunch
of examples of well of welds a pac-man
could live on this begs the question
what possible surfaces are there for him
to live on in order for us to make sense
of this question we first have to set
down some ground rules I’m going to
consider two surfaces to be the same if
I can stretch and squish one and
to the other for example a sphere is the
same as that thing because I can just
sort of warp it around it’s also the
same of as a cube I can just take my
sphere and squished sides in and make it
into a cube another example is that a
coffee cup I’m talking about the outside
surface of a coffee cup is the same as a
donut because I can take my coffee cup
put my hand through the handle and pull
out the inside and then I’ve got sort of
a handle with a blob and then squish
that blob around the handle and now the
hole through the donut is what used to
be a hole through the handle of the
coffee cup so in this world a coffee cup
and a donor are going to be considered
to be the same with this point of view
this becomes a question in a branch of
pure mathematics called topology the
question of which two-dimensional
surfaces can exist was answered about a
hundred years ago
it’s a beautiful theorem and we now have
a complete list of all the possible
surfaces what about a three-dimensional
background this is someone much like me
likes cherries hates ghosts but most
importantly lives in a three-dimensional
world but maybe the world I live in is
such that if I walk out that wall and
come in through there or if I look at
the back I’d see my own head or if I
look up I’m just thankful I’m not
wearing a kilt
or maybe that wall is glued to this wall
with a twist in it so if I look at the
back there I see myself standing on the
wall and if I look past myself I see
myself again standing on the ceiling or
maybe that walls glued to that wall with
a flip so if I raise my right hand and
look out that wall I’d see myself with
my left hand raised and then if I look
past myself I’d see myself with my right
hand raised or maybe I’m not in a cube
at all
maybe I’m in a dodecahedron or a
tetrahedron to some other
three-dimensional object and the faces
are glued together in some curious way
after you work in this on this problem
for a small amount of time it becomes
pretty clear that the three-dimensional
question of which three-dimensional
surfaces can exist
it’s far more complicated than the
2-dimensional one in fact we didn’t have
a solution or a classification of which
three-dimensional surfaces can exist
until 2003 and this problem was answered
this question was answered by Russian
mathematician Grigori Perelman it’s an
important and difficult enough problem
that it’s worth a million bucks and the
Fields Medal interestingly parallel Minh
declined both two dimensions three
dimensions were mathematicians what
about a four dimensional pac-man okay
let’s just talk about what we mean by
four dimensions first in order to
understand what I mean by four
dimensional surface or a four
dimensional world the pac-man can live
on let’s look at the two-dimensional
surface again this donut what makes it
two-dimensional right where does the two
come from the important point here is
that around any point on the donut if I
look at a little neighborhood of it
every point in that neighborhood is
determined by two numbers two
coordinates how far up and how far to
the left okay every point in that little
neighborhood and this happens everywhere
on the surface the three-dimensional
example I gave you of the cube where I
walk out there and coming through here
let’s look at a little neighborhood
around this point it’s like a little
sphere every point in that little sphere
is determined by three numbers how far
up how far that way and how far this way
there are three coordinates the three is
what makes this a three-dimensional
surface so a four-dimensional surface is
just some space where around each point
if you look at a neighborhood it looks
like four dimensional space that is
every point is determined by four
numbers four coordinates me and mortals
like us have a really hard time thinking
about what this looks like but
mathematics doesn’t care that we grew up
in a three dimensional world in our
brains and feeble and can’t do this the
mathematics can deal with this no
problem is just a fourth number right
it’s a fourth coordinate so which four
dimensional surfaces can exist for a
four dimensional pac-man to live on we
don’t know this is one of the most
important
open problems in topology today it’s a
question that motivates some of the
stuff that I work on so let me tell you
a little bit about that one way to build
four dimensional surfaces is to take two
two-dimensional surfaces and smoosh them
together in such a way that the
coordinate system so there’s four
numbers that determine my four
dimensional surface the first two of
them belong to one of my two-dimensional
surfaces and the second two of them
belong to another one the take-home
message here is that because two plus
two equals four I can take two
two-dimensional surfaces stitch them
together in such a way and get a four
dimensional surface out yeah this is the
key point here if you want to understand
these four dimensional surfaces it turns
out that you can gain some understanding
about them by looking at symmetries of
the underlying two-dimensional surfaces
or at these building blocks that made
this four dimensional surface so what’s
a symmetry symmetry is tougher you hear
a lot about in mathematics and what’s
the symmetry so symmetry is something I
can do to an object while you’re looking
away having an app not paying attention
that you can’t tell I did when you wake
up again for example if I had a glass of
wine here and I turned it and you were
looking away while I did that you
wouldn’t be able to tell that I did that
that’s a symmetry of the glass of wine
if I drank some you could tell because
the level would have gone down I don’t
study glasses of wine unfortunately but
I knew study symmetries of
two-dimensional surfaces so here’s an
example of a symmetry if I take a
two-hole doughnut I could flip it if you
were looking away while I did that you
couldn’t tell it happened here’s an
example of a three hole donor an example
of a symmetry sorry of a three hole
doughnut if I rotate by 120 degrees
again if you were looking away you
wouldn’t be able to tell that I did that
if you take a surface a two-dimensional
surface and look at the entire group of
symmetries so there’s a whole bunch of
symmetries for any one surface that
entire group of symmetries has structure
on it
in the same way that the whole numbers
have addition you can take two whole
numbers and get another one out you can
take two symmetries and get another
symmetry out if you can understand this
structure it turns out that you can use
that understanding to understand a four
dimensional surface that you built so
that’s that’s the area I work in is the
symmetries of two-dimensional surfaces
but at this point I want to address just
to finish off I want to address a
question that a lot of you might have
right now which is a perfectly
reasonable question who cares
I get this a lot it’s much more polite
usually but this is what people are
thinking when I talk to them about what
I do right who cares why should you care
about something abstract like pure
mathematics or topology I’m gonna give
you two reasons and if you take nothing
else away from this talk I want you to
take these two reasons away first
topology is part of the gigantic machine
that starts at pure mathematics and ends
at new technology and understanding
mathema the universe really does seem to
be written in the language of
mathematics philosopher Eugene Wigner
calls this the unreasonable
effectiveness of mathematics in the
Natural Sciences time and time again
seemingly abstract nonsense pieces of
mathematics have found applications
important ones well after they were
discovered for example complex numbers
were initially discovered simply to
solve equations that couldn’t be solved
using real numbers
now they form the framework in which to
understand electricity this is the
natural framework in which to understand
electricity minkowski’s geometry for
curved space was discovered well before
Einstein used it to formulate his
general theory of relativity without
which today we wouldn’t have GPS there
have been cases where fundamental
particles of the universe have been
predicted to exist with certain
properties simply because an abstract
mathematical object exists the
topological foundations laid down 100
years ago are being used today in
quantum computing and even featured in
last year’s Nobel
chemistry pure mathematics is useful and
indispensable to society that’s the
first reason it’s a pretty good reason I
reckon here’s the second reason pure
mathematics is beautiful and awesome and
amazing like music and art it has this
incredible aesthetic quality pure
mathematics and especially topology is
one of the most beautiful and compelling
things anyone can experience this is
more than enough of a reason for me to
do what I do
thank you
[Applause]
you
for a walk you grab your two-dimensional
wallet leave your 2-dimensional house
and head off in some direction after a
bit of walking you find a wallet it’s
yours sure enough your pockets are empty
you put your wallet away and you
continue see pac-man is accustomed to
finding things he’s dropped without ever
turning around because when pac-man goes
up through the top of his screen he
comes in through the bottom when he goes
out through the right he comes in
through the left so what is this strange
two-dimensional surface the pac-man
lives on well – pac-man the top and
bottom of the screen are the same so we
may as well glue them together now we
have a cylinder the right and left of
the cylinder are also the same –
pac-man’s point of view
so let’s glue them together as you can
see here pac-man lives on the surface of
a doughnut delicious why a donut why not
some other two-dimensional surface
pac-man could very well be on a sphere
the gameplay would be a little bit
different but he could be on a sphere
looking for cherries and hiding from
ghosts or he could be on a two hole
donut same thing to hold surface of a to
hold donut it’s two-dimensional
pac-man’s two-dimensional this could
totally work or maybe the screen is set
up so that if pac-man goes up through
the top he comes in through the bottom
as usual but if he goes out through the
right at the top he comes in from the
left at the bottom upside down so what’s
going on here is that the right and left
are connected but in such a way that
there’s a twist when you glued them
together if you go through the same
process we did before
if you’ve seen such a object before
pac-man here would live on the surface
of a klein bottle I’ve given you a bunch
of examples of well of welds a pac-man
could live on this begs the question
what possible surfaces are there for him
to live on in order for us to make sense
of this question we first have to set
down some ground rules I’m going to
consider two surfaces to be the same if
I can stretch and squish one and
to the other for example a sphere is the
same as that thing because I can just
sort of warp it around it’s also the
same of as a cube I can just take my
sphere and squished sides in and make it
into a cube another example is that a
coffee cup I’m talking about the outside
surface of a coffee cup is the same as a
donut because I can take my coffee cup
put my hand through the handle and pull
out the inside and then I’ve got sort of
a handle with a blob and then squish
that blob around the handle and now the
hole through the donut is what used to
be a hole through the handle of the
coffee cup so in this world a coffee cup
and a donor are going to be considered
to be the same with this point of view
this becomes a question in a branch of
pure mathematics called topology the
question of which two-dimensional
surfaces can exist was answered about a
hundred years ago
it’s a beautiful theorem and we now have
a complete list of all the possible
surfaces what about a three-dimensional
background this is someone much like me
likes cherries hates ghosts but most
importantly lives in a three-dimensional
world but maybe the world I live in is
such that if I walk out that wall and
come in through there or if I look at
the back I’d see my own head or if I
look up I’m just thankful I’m not
wearing a kilt
or maybe that wall is glued to this wall
with a twist in it so if I look at the
back there I see myself standing on the
wall and if I look past myself I see
myself again standing on the ceiling or
maybe that walls glued to that wall with
a flip so if I raise my right hand and
look out that wall I’d see myself with
my left hand raised and then if I look
past myself I’d see myself with my right
hand raised or maybe I’m not in a cube
at all
maybe I’m in a dodecahedron or a
tetrahedron to some other
three-dimensional object and the faces
are glued together in some curious way
after you work in this on this problem
for a small amount of time it becomes
pretty clear that the three-dimensional
question of which three-dimensional
surfaces can exist
it’s far more complicated than the
2-dimensional one in fact we didn’t have
a solution or a classification of which
three-dimensional surfaces can exist
until 2003 and this problem was answered
this question was answered by Russian
mathematician Grigori Perelman it’s an
important and difficult enough problem
that it’s worth a million bucks and the
Fields Medal interestingly parallel Minh
declined both two dimensions three
dimensions were mathematicians what
about a four dimensional pac-man okay
let’s just talk about what we mean by
four dimensions first in order to
understand what I mean by four
dimensional surface or a four
dimensional world the pac-man can live
on let’s look at the two-dimensional
surface again this donut what makes it
two-dimensional right where does the two
come from the important point here is
that around any point on the donut if I
look at a little neighborhood of it
every point in that neighborhood is
determined by two numbers two
coordinates how far up and how far to
the left okay every point in that little
neighborhood and this happens everywhere
on the surface the three-dimensional
example I gave you of the cube where I
walk out there and coming through here
let’s look at a little neighborhood
around this point it’s like a little
sphere every point in that little sphere
is determined by three numbers how far
up how far that way and how far this way
there are three coordinates the three is
what makes this a three-dimensional
surface so a four-dimensional surface is
just some space where around each point
if you look at a neighborhood it looks
like four dimensional space that is
every point is determined by four
numbers four coordinates me and mortals
like us have a really hard time thinking
about what this looks like but
mathematics doesn’t care that we grew up
in a three dimensional world in our
brains and feeble and can’t do this the
mathematics can deal with this no
problem is just a fourth number right
it’s a fourth coordinate so which four
dimensional surfaces can exist for a
four dimensional pac-man to live on we
don’t know this is one of the most
important
open problems in topology today it’s a
question that motivates some of the
stuff that I work on so let me tell you
a little bit about that one way to build
four dimensional surfaces is to take two
two-dimensional surfaces and smoosh them
together in such a way that the
coordinate system so there’s four
numbers that determine my four
dimensional surface the first two of
them belong to one of my two-dimensional
surfaces and the second two of them
belong to another one the take-home
message here is that because two plus
two equals four I can take two
two-dimensional surfaces stitch them
together in such a way and get a four
dimensional surface out yeah this is the
key point here if you want to understand
these four dimensional surfaces it turns
out that you can gain some understanding
about them by looking at symmetries of
the underlying two-dimensional surfaces
or at these building blocks that made
this four dimensional surface so what’s
a symmetry symmetry is tougher you hear
a lot about in mathematics and what’s
the symmetry so symmetry is something I
can do to an object while you’re looking
away having an app not paying attention
that you can’t tell I did when you wake
up again for example if I had a glass of
wine here and I turned it and you were
looking away while I did that you
wouldn’t be able to tell that I did that
that’s a symmetry of the glass of wine
if I drank some you could tell because
the level would have gone down I don’t
study glasses of wine unfortunately but
I knew study symmetries of
two-dimensional surfaces so here’s an
example of a symmetry if I take a
two-hole doughnut I could flip it if you
were looking away while I did that you
couldn’t tell it happened here’s an
example of a three hole donor an example
of a symmetry sorry of a three hole
doughnut if I rotate by 120 degrees
again if you were looking away you
wouldn’t be able to tell that I did that
if you take a surface a two-dimensional
surface and look at the entire group of
symmetries so there’s a whole bunch of
symmetries for any one surface that
entire group of symmetries has structure
on it
in the same way that the whole numbers
have addition you can take two whole
numbers and get another one out you can
take two symmetries and get another
symmetry out if you can understand this
structure it turns out that you can use
that understanding to understand a four
dimensional surface that you built so
that’s that’s the area I work in is the
symmetries of two-dimensional surfaces
but at this point I want to address just
to finish off I want to address a
question that a lot of you might have
right now which is a perfectly
reasonable question who cares
I get this a lot it’s much more polite
usually but this is what people are
thinking when I talk to them about what
I do right who cares why should you care
about something abstract like pure
mathematics or topology I’m gonna give
you two reasons and if you take nothing
else away from this talk I want you to
take these two reasons away first
topology is part of the gigantic machine
that starts at pure mathematics and ends
at new technology and understanding
mathema the universe really does seem to
be written in the language of
mathematics philosopher Eugene Wigner
calls this the unreasonable
effectiveness of mathematics in the
Natural Sciences time and time again
seemingly abstract nonsense pieces of
mathematics have found applications
important ones well after they were
discovered for example complex numbers
were initially discovered simply to
solve equations that couldn’t be solved
using real numbers
now they form the framework in which to
understand electricity this is the
natural framework in which to understand
electricity minkowski’s geometry for
curved space was discovered well before
Einstein used it to formulate his
general theory of relativity without
which today we wouldn’t have GPS there
have been cases where fundamental
particles of the universe have been
predicted to exist with certain
properties simply because an abstract
mathematical object exists the
topological foundations laid down 100
years ago are being used today in
quantum computing and even featured in
last year’s Nobel
chemistry pure mathematics is useful and
indispensable to society that’s the
first reason it’s a pretty good reason I
reckon here’s the second reason pure
mathematics is beautiful and awesome and
amazing like music and art it has this
incredible aesthetic quality pure
mathematics and especially topology is
one of the most beautiful and compelling
things anyone can experience this is
more than enough of a reason for me to
do what I do
thank you
[Applause]
you
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