0:00

I'm Dr. Carl, coming to you from the lands of

0:02

the Gadigal people of the Eora Nation.

0:05

I acknowledge Aboriginal and Torres Strait Islander

0:07

peoples as the first Australians and

0:09

traditional custodians of the lands where we

0:11

live, learn and work.

0:15

G'day, Dr. Carl here, Sheldon Lewis of Science, University

0:18

of Sydney, and today I'm very lucky to be joined

0:20

by Professor Lisa Piccirillo.

0:22

That's the right pronunciation, is it? Yes,

0:24

that's right. You've done something amazing

0:27

in knot theory.

0:28

Well, I have proven something

0:30

in knot theory. I don't know how amazing it is or isn't.

0:33

Professor Piccirillo, Dr. Lisa, proved

0:35

something in 10 days that

0:37

other people hadn't been able to prove in half a century.

0:40

Am I right on that?

0:41

Yes. You're awfully humble.

0:44

And then what

0:46

I'm trying to look at because of my medical training, I'm thinking

0:49

your work further down the line

0:51

will help change medicine

0:54

and pharmacology because of its reference

0:56

to shapes.

0:57

Yeah, so I don't really understand this because

0:59

I don't understand anything basically other than pure

1:02

math. But indeed, apparently

1:04

knot theory

1:05

is very useful in understanding DNA

1:07

and proteins. I studied kind of a four-dimensional

1:10

version of knot theory and I haven't heard that that

1:12

is in particular relevant to the

1:14

DNA folks yet.

1:15

Maybe if we wait long enough. Let's

1:17

tell people what a knot is. A knot is

1:19

something like a shoelace and the ends are free.

1:21

But in mathematics, the ends are joined

1:23

together. Do I get that bit right?

1:25

Yeah, that's right. I like to think about a knot

1:27

as being like if you go into your garage

1:29

and you get an extension cord and you plug the ends together,

1:32

it doesn't

1:33

matter what it looks like, it's a knot. So

1:35

I came across a word I'd never come across before, an unknot.

1:39

The unknot is what you would get if you're

1:41

very, very organized and your extension

1:44

cord was stored in a very nice

1:46

manner. So you could just lay the whole thing

1:47

flat on the ground and look like a circle. That's

1:50

the unknot. If you're a slob, it'll

1:52

be worse than that. And that's interesting

1:54

too. Right. So if it's got one

1:56

crossing, like a figure eight,

1:59

you can just folded and take it back to, I

2:02

love using this word I've never used before, unknot. And

2:05

if you've got two crossings you can bring it back to an unknot,

2:07

am I right so far? Yep, that's right. But

2:10

then with three, known as the three-four, you

2:12

can't, if you've got three crossings, no matter

2:14

what you do to it, you can't bring it back to an unknot. Well,

2:16

that's not true. You could

2:18

take an unknot and like do three

2:20

twists in it and it would have three crossings

2:22

and there would still be an unknot.

2:24

But then you fold it back again to an unknot?

2:26

That's right. So there are ways to draw the

2:28

unknot on a piece of paper where it has three crossings.

2:31

But I think what you're referencing is that there

2:33

is also a way to draw an unknot on a paper

2:35

that has three crossings, where that thing

2:37

you could never wiggle it around and turn

2:39

it into the unknot. Can you explain slice?

2:43

I have difficulty with this. Is used differently

2:45

in your field?

2:47

Yeah, so if used differently, let me go

2:49

back to unknot. Unknot is this

2:51

extension part that's unplugged into each other and

2:53

it could be all kinds of messy. But

2:55

if it's not messy and you could put it like on

2:57

the ground in your basement in one big circle, that's

3:00

the unknot. And there's a way I could characterize

3:03

that, which is that I could say

3:05

it bounds a two-dimensional disc, like

3:08

all the cement on the garage floor that is

3:10

kind of inside of it, that's a disc that

3:12

it bounds at a D2. And

3:14

that disc is existing in three-dimensional space

3:16

in your garage. All of that in summary

3:19

is just to say another way I could define the

3:21

unknot is I could say it's a knot which bounds a

3:23

disc

3:23

in three dimensions. It bounds a disc in

3:25

three dimensions. If I have

3:28

a cylinder, which is a three-dimensional

3:30

object, and I take a two-dimensional

3:33

slice through it, if I go through

3:35

at one angle, I'll end up with a

3:37

circle or different angles, I'll get ellipses. Is

3:40

that getting me closer to understanding what slice

3:42

means or am I trying to be too literal?

3:45

So I think that's too literal.

3:47

It's really going to be kind of an analogy to

3:49

this definition of the unknot that I was just giving,

3:51

which is the unknot, it bounds a disc

3:53

in three-dimensional space. And

3:55

we're going to say that the knot is slice if it

3:57

bounds a disc in four-dimensional

3:59

space.

4:00

What that means is if you took like a frisbee and

4:02

it was really wobbly and

4:05

you wanted to try to get the rim of the frisbee

4:07

to look like some complicated knot,

4:10

you could try and you maybe get it close, but at some point

4:12

you would need to shove the frisbee through

4:15

the frisbee, which is not a thing

4:17

in three dimensions. But if you had a little

4:19

more ambient space, then maybe you could shove

4:22

the frisbee through the frisbee. Those

4:24

knots, knots that would bound a frisbee

4:26

that could intersect itself, that's kind of

4:28

how you should be thinking about flice. I'm

4:31

sort of feeling it bubble to the surface

4:33

in my brain. Let me give

4:35

a definition where everything

4:37

is a dimension down of fliceness. And here

4:39

it'll make sense because we'll be able to kind of picture

4:41

everything. You know, I'm not is a

4:43

one-dimensional object, right? That's our extension cord. And

4:46

I want to drop that down. I want to think about a zero-dimensional

4:48

object. Zero dimensions is just that point. And

4:52

so if I have like four points

4:55

sitting on the edge of a piece

4:57

of paper, you know, color one red,

4:59

then one blue, then one red, one blue. So right

5:01

at piece of paper, I got four points right on the edge of it, red,

5:04

blue, red, blue. Then I could

5:06

ask myself, okay, I want to make this zero-dimensional

5:09

thing. I want to make it bound to intervals.

5:13

And I want like the red ones to be the end

5:16

of one interval and the blue ones to be the end of

5:18

one interval. If I only

5:20

allow those intervals to live in the boundary

5:23

of a piece of paper, like in that edge, then

5:26

I can't do it, right? Because if I try to put

5:28

the interval like kind of down on the

5:30

edge of the paper, like let's say I try to put the red one

5:32

down, while the blue is in the way,

5:34

it's in the middle. So I

5:36

have this issue. But if I allow

5:39

myself to put the interval in the entire piece of paper,

5:41

not just on the very edge, then

5:44

I can kind of make the red interval go down

5:46

into the page, then it doesn't have to

5:48

intersect this blue dot in the middle.

5:51

So what is all of that about it? It's saying if

5:53

you have objects and you want to make them be the

5:55

boundary of other objects, sometimes

5:57

that's not possible in one

5:59

space.

5:59

But if you allow another dimension of freedom,

6:02

then it is possible. And that's what license

6:04

is about. And we're going to go into higher dimensions, aren't we?

6:06

Depends on where you want to go. Because you quite

6:08

cheerfully go into, in your working

6:10

four dimensions, and we're not talking of the fourth

6:13

dimension being time, as in

6:15

the regular universe we deal in, but

6:17

being another space dimension

6:20

that's at right angle to the other three dimensions

6:22

which are backwards, forwards, left,

6:24

right, and up, down. And you're adding in

6:27

a fourth dimension in space, which

6:29

is at right angles to all of those who have got that right.

6:32

That's

6:32

one way to think about it. Yeah.

6:34

Is there another way?

6:35

Well, yeah. You can have

6:37

a four dimensional space any time you have four,

6:40

let's say, real parameters that are independent

6:43

from each other. So let's say, like,

6:45

I want the temperature in Sydney, the temperature

6:47

in Boston, the temperature in Texas, and the temperature on the

6:49

Moon. If I measure all

6:51

of those and say, what is the total space

6:54

of all of the temperatures that could be going on

6:56

there, never mind some stuff about temperatures

6:58

that can't go below a certain thing, then

7:00

I have a four dimensional space because, well,

7:03

that's four values and they're all unrelated,

7:06

more or less. Oh my God, there's a thought of

7:08

that. You know, this is something, I guess, that is actually

7:10

used in physics, but people sometimes say time

7:12

is the fourth dimension. It sounds very, like,

7:15

mystical. And, you know, you don't really

7:18

understand that because it really makes sense and you can't picture

7:20

it and you just think, like, well, that's some physics wizardry

7:22

I'll never understand. But in fact, it's just the same

7:25

thing as this, like, temperature in four places

7:27

thing. It's just another data point. How

7:30

did you stumble across this idea of fix this problem

7:32

that

7:33

eluded people for such a long time? In

7:35

the article, in Quadri, you say, very

7:37

humbly, I didn't realize it was a big deal and you

7:40

didn't think of it as real mess. You only did it in your own spare

7:42

time.

7:44

There was some sort of twist that you did.

7:46

I didn't know about this problem.

7:48

I really study kind of four dimensional space

7:50

most of the time. I don't study knots.

7:53

They come up and I know a little bit about them. But

7:56

I was at a conference and I was going to talk about knots

7:58

and placeness and somebody mentioned, like... We

8:01

don't even know if this 11 crossing knot is slice, isn't

8:03

that ridiculous? And I

8:05

thought like, yeah, that's ridiculous. There's this

8:07

way of thinking about whether or not

8:09

they're slice, which is in terms of some

8:11

other four-dimensional stuff. And that other four-dimensional

8:14

stuff is what I really study most of the time. I

8:16

already knew about this connection. I just thought

8:18

like, I'll think about that 11 crossing knot,

8:21

surely just like nobody cares, but maybe I could do it, and I could

8:23

just like write an email to this woman and she

8:25

would be happy. So that's why I thought

8:28

about it, and that's why I thought I could think about it, is

8:30

because I knew this connection to four-dimensional

8:32

spaces, and I knew something about those.

8:36

I was thinking about shapes from

8:38

my medical training. You know in pharmacology

8:40

that everything works on a kind of shape,

8:43

lock and key basis, have you run across that concept? Nope.

8:46

So you've got 37 trillion cells. Each

8:49

cell has got hundreds of

8:52

receptors on the outside, which

8:55

will then activate something inside that cell

8:57

if a drug or chemical goes into it.

9:01

For insulin to work, the chemical

9:03

insulin, which is made by pancreas, has

9:05

to land on insulin

9:07

receptors on all sorts of cells all

9:10

over your body, and then things happen

9:12

and the glucose goes down and you're a happy person.

9:15

But we've found a family

9:17

of circus athletes in

9:20

Pakistan who have an abnormality

9:23

in one of their sodium receptors,

9:26

and they don't feel pain. And

9:28

they normally die in their teens

9:30

or 20s from having a broken leg sticking

9:32

through their skin and they don't feel any pain. So

9:35

this whole lock and key thing is incredibly

9:38

important to medical people trying to get shapes

9:40

and understand shapes, and it's just busting

9:42

their heads. Because if you knew the exact

9:45

shape of the receptor and a chemical,

9:48

you could say,

9:50

oh, that cancer cell is working

9:52

on that receptor, I'll just block it by making this

9:54

shape. And so you're working with shapes,

9:57

with topology?

9:58

I don't know, though, if... topology is

10:00

more appropriate here or geometry. So

10:02

geometry, people know maybe what that is studied

10:04

in school, it's a study of triangles

10:07

and squares and things like this. And in

10:09

particular in geometry you measure things, you

10:11

care about how big things are. I

10:13

wonder if in the biology you do care

10:15

about the size of these things, if maybe one

10:17

of these receptors is different, if it's twice

10:20

the size, even if it otherwise

10:22

looks the same.

10:23

So that would be like geometric properties.

10:26

And in topology we don't worry

10:28

about geometric, we don't worry about distances.

10:30

We study the same kind of things but

10:33

I'm not going to make a distinction between a little circle

10:35

and a big circle. I am going to make a distinction

10:37

though between one circle and two circles, those are

10:39

like really different objects but just a big one and

10:41

a little one they're only different in size I don't care. That's

10:44

kind of the difference between topology and geometry.

10:46

So one of them is probably relevant

10:49

in these medical questions but I don't know which.

10:51

I'm kind of hoping that it will because

10:54

if you look at your phone there's

10:56

five different technologies each of which was

10:58

invented by scientists doing government

11:00

work on purely abstract stuff.

11:04

In fact it was the search for black holes

11:07

that gave us Wi-Fi and

11:09

if you look at medical research

11:12

papers they're totally abstract

11:15

within 10 years something like 50% of them

11:17

are quoted as some sort of future

11:19

patent. So I'm kind of

11:21

thinking that in some way further down

11:23

the line like with those dimensions that your work

11:26

will become useful. I'm not saying it's

11:28

not useful but I'm saying it will become relevant

11:31

to the human condition maybe in 10 years maybe in a

11:33

century maybe in two centuries. I don't

11:35

worry

11:36

about that too much because I think you know the way I think

11:38

about basic research especially in abstract things

11:40

is you know you don't know what's going to be useful and

11:43

any individual thing maybe isn't going

11:45

to be huge but some of it will. But

11:47

I don't worry too much about whether mine will

11:50

or won't be and I don't think there's any particular expectation

11:52

that it will be. It's probably something.

11:56

11th-place

12:00

knot, you shall be mine. How do you do it?

12:03

The process of doing math

12:06

is it's very

12:08

careful and in many ways it can be

12:10

a lot more like a trade than

12:12

like a kind of just sitting around and waiting

12:14

for the eureka to come to you. In

12:17

math we decide we're going to study some particular

12:20

object, so we make very careful definitions of exactly

12:22

what we're going to study. Things like circles or the

12:24

real numbers or things like this. And

12:26

then we have some questions about them and we want to

12:28

try to answer it. One makes an argument that says like

12:30

this is absolutely statement is true

12:33

or false or

12:33

something like this. The way you actually

12:35

go about doing math is very incremental. You

12:38

take what you know and

12:39

you know how the objects you understand might interact

12:41

with other ones and if you want to know something

12:43

about object A but you know it's similar

12:45

to object B and you know this fact about object

12:47

B then maybe you can try to think about whether

12:49

that fact from object B is also true

12:51

for object A. It's these kind of things. So

12:54

really the progress you do on

12:56

a day-to-day basis is just these

12:57

like little tiny connections and

13:00

very slowly this process

13:02

builds up to understanding. But

13:05

no, you don't wake up in the morning and just know

13:07

I wish you did. That'd be great. Are there

13:10

many blind endings where

13:12

you look back and say I wasted five days

13:14

there? Every day. Yeah

13:17

just like you should expect you know

13:20

probably somebody's math will be useful but it's

13:22

probably not going to be yours on any given day.

13:24

The stuff you think about probably not going to be useful

13:27

but like five days a year maybe it'll

13:29

actually go somewhere. What was the process

13:31

you did over that 10-day period to solve this

13:33

Conway-Not thing? The

13:36

process was really like I just needed to carry

13:38

out a long computation ultimately.

13:41

People study whether or not they're slice and they have

13:43

ways to show that knots aren't slice

13:46

by computing something called an invariant for

13:48

them. This is an algorithm and

13:50

we know that if you take this knot and you perform the

13:52

algorithm you'll get a number and if the number is

13:54

not zero then the knot is not slice. That's

13:57

something people already know. The

13:59

trouble is that when you do that

13:59

that for the Conway knot you always do get zero,

14:02

so you can't prove it's not slice. That's why

14:04

it was like hard. So I knew this

14:06

other fact about

14:07

slicing, which is that if you have a knot

14:09

and it shares the same four dimensional

14:12

space in some particular way as

14:14

some other knot, so let's say one knot's called

14:17

K and one knot's called J. If K and J have the

14:19

same four dimensional space, whatever that means, then

14:21

either they're both slice or they're both not slice.

14:25

That was kind of something people who study

14:27

these spaces knew about. And

14:29

so

14:31

when I wanted to understand the Conway knot, I

14:33

thought, well,

14:34

apparently it's hard to do it for the Conway

14:36

knot. You run the algorithm, you don't learn anything. I'll

14:39

build this other knot J and

14:41

then I'll run the algorithm for J. So in

14:43

fact, the thing that was happening for those 10 days

14:45

was me trying to write down J and that's

14:48

like itself a big long algorithm. That's

14:50

something I studied, so I did that. That took a long time. And

14:52

then I ran the other algorithm to compute the invariant

14:54

for J and then it wasn't zero,

14:56

so not slice. Do you do this

14:58

by hand or on a computer?

15:02

The first part, writing down the southern knot J was

15:04

by hand and the second parts

15:07

of the algorithm were computer-assisted

15:09

and there's like a very big combinatorial process

15:11

that has to happen, the computer does that part. And what

15:14

were your emotions like when you came up

15:16

with this thing? Did you think, oh, this is a big deal,

15:18

I'll finish that, I'll go and have a couple of clean house.

15:21

Yeah, much more the latter. You

15:24

know, I didn't know anybody cared. I just thought

15:26

like, well, it's not slice,

15:28

it's slice.

15:29

You've got a question for me to stimulate

15:31

your mind. You're saying, who can be successful

15:35

in mathematics? That's

15:38

really deep actually. Can you take me through

15:40

your thinking on that?

15:41

I think that the process of doing

15:44

math and also the objects that mathematicians

15:46

study, those are very different

15:49

than the kind of math, the objects

15:51

you meet in math class in school and the

15:54

types of things you do in math class in school. In

15:56

fact, what you do in math class in school is mostly

15:58

run algorithms. Maybe they're complicated. like

16:00

you need to factor a polynomial.

16:03

That's like a process that you know the teacher

16:05

hands you a polynomial and then you're like I

16:07

eat polynomial I spit factored polynomial.

16:10

Like that's what math looks like in school. But

16:13

that's not what it's like for real. What

16:15

it's really like is you have these objects

16:17

and you're trying to make arguments about them and

16:19

the objects are probably not polynomials

16:22

or integers or anything like that. I never studied

16:25

numbers. I'm terrible at multiplication

16:28

and you definitely don't want to ask me about calculus.

16:31

But I'm a mathematician and so

16:33

I think that people have this idea that

16:35

if they didn't like that stuff in school where they were fed

16:38

polynomials and we're supposed to factor

16:40

them and then that means they wouldn't like math

16:42

or they wouldn't be good at math and then it turns out

16:44

that it's not even irrelevant because

16:46

it's not what math is like. That's a

16:48

deep insight for people in year 10.

16:51

Yeah I just think math is a lot more like

16:54

learning a foreign language than it is like being

16:57

smart. And so you know oftentimes

16:59

maybe somebody in a math class more advanced

17:01

than you will say something they'll be like a

17:04

squared plus b squared equals c squared. But they say

17:07

something you don't understand it and you think like I

17:09

got I'm not clever I don't understand math.

17:11

If you instead heard

17:12

two people talking in the hallway and they were just speaking in

17:14

Russian you don't speak Russian you wouldn't think like

17:17

I'm not very clever. You would just think

17:19

I don't speak Russian. Math is much

17:22

more like the language. When people

17:24

are saying these sentences you don't understand it's because

17:26

like the word sliceness. In fact they just

17:28

like

17:29

they really are speaking another language. They're

17:31

talking about objects that you don't know the definition

17:33

of. Most people when I

17:35

meet them and I tell them I'm a mathematician they say like oh

17:37

god I'm terrible at math. I'm like that's

17:39

not true you just don't speak Russian. But you're not

17:41

terrible at Russian. I tend to surprisingly

17:45

say the same thing about the explanation

17:47

of why the sky is blue. I

17:49

cannot explain this to you in the words

17:52

of English or Spanish

17:54

or Mandarin. The only language I can explain

17:56

it to you in is a language called mathematics which

17:58

is just another language. I feel very justified

18:01

hearing you saying that. Thank you. Now, obviously

18:04

you're awfully busy, Professor at both

18:06

University of Texas at Austin and MIT.

18:10

By the way, I would recommend to people to look up

18:12

the Quanta article about you, which

18:14

is just nice things about you, and also your Wikipedia

18:16

entry, as well as the Wikipedia entries on

18:19

Knots and also on Conway. But

18:22

suppose somebody was inspired

18:24

by you and wanted to be your student or get

18:26

inspiration from you. Do you have students?

18:29

I have students. I have PhD students both at MIT.

18:32

And I actually don't have any at UT yet, but I'm

18:34

taking PhD students at UT. I

18:36

try to give lectures and

18:39

do podcasts and be available

18:41

to people. I can't just talk to everybody

18:43

who's interested in Knots. It doesn't

18:46

work out. There's only one of me in my inbox. It's

18:48

just like, you don't want to know. You don't

18:50

want to see it. So

18:53

I think if you're interested in Knots, then

18:56

probably the best thing to do is not to email me about

18:58

it. But instead, to read something

19:01

about it or harass your math teacher at school

19:03

about learning about it, there's this really wonderful

19:06

book by Colin Adams called The Knot Book.

19:08

How many of you remember The Knot Book, Colin Adams?

19:11

And that's awesome. That's a great place to

19:13

start learning about Knots. There's another book

19:15

called Flatland by Edwin Abbott

19:17

Abbott. That's also pretty cool. He

19:20

was a monk, maybe a little more philosophical than

19:22

mathematical. But I'll still start you thinking about some

19:25

of these things that I was talking about,

19:26

where you can't make the two dots

19:28

be connected on the edge of the page, but you could make

19:30

them be connected if you use the whole page. That

19:32

kind of stuff, you want this Flatland. I

19:35

remember reading that as a kid and being inspired.

19:38

And then if they're really good, and they've done all of that,

19:40

and they got their master's degree or they're heading

19:42

for a university, then maybe they could

19:44

apply to a university where you

19:46

happen to be and maybe intersect

19:49

you one day, possibly further down the

19:51

line.

19:52

Absolutely. Get an undergrad degree in math,

19:54

and then be a PhD student at UT

19:56

or MIT, and we could talk. But

19:59

I should also.

19:59

say that there's lots of not theorists

20:02

doing really interesting work that aren't me.

20:04

So if you're interested in us and you want to stay in Australia,

20:07

you have options too. You can get your undergrad degree in

20:09

math and then become the PhD student of a number of like

20:11

really incredible Australian not-teachers.

20:13

It's not impossible. Thank you very much Professor

20:15

Lisa Pigarello for giving us

20:16

your time this morning. Thank you. It's

20:19

getting hotter. Our population's

20:22

aging. We're glued to our screens

20:25

and AI? Well, it's

20:28

changing the world as we speak. We're

20:30

facing big challenges. We

20:32

need big solutions. I'm

20:36

Mark Scott, the Vice-Chancellor at the University

20:39

of Sydney and I've got a backstage

20:41

pass to the people making

20:43

change happen. The Solutionists.

20:46

Look for it in your favourite podcast app.