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