can we hear the shape of the universe?jmf/cv/seminars/pgcolloquium.pdf · can we hear the shape of...

Can we hear the shape of the universe?

Jose Figueroa-O’Farrill

School of Mathematics

PG Colloquium, 20 November 2003

1

The Big Questionsr

1

The Big Questionsr

• Is the (spatial) universe finite or infinite?

1

The Big Questionsr


• What is the geometry and topology of the universe?

1

The Big Questionsr



• What is the ultimate fate of the universe?

1

The Big Questionsr



• What is the ultimate fate of the universe?

And the answers are...

2

Space and time in Newton’s Principia

2


I do not define time, space, place and motion, since they are

well known to all.

2



well known to all.

Yet space and time are both absolute

2



well known to all.

Yet space and time are both absolute:

Absolute space, in its own nature, without relation to anything

external, remains always similar and immovable.

2



well known to all.




Absolute, true, and mathematical time, of itself, and from its

own nature, flows equably without relation to anything external.

2



well known to all.




Absolute, true, and mathematical time, of itself, and from its

own nature, flows equably without relation to anything external.

The Newtonian universe is thus R× R3.

3

Galilean relativity

3

Galilean relativity

• the motion of a point particle, say, is described by a path

3

Galilean relativity


x : [0, 1] → R3

3

Galilean relativity


x : [0, 1] → R3

subject to Newton’s equation

3

Galilean relativity


x : [0, 1] → R3


x(t) = F (x(t))

3

Galilean relativity


x : [0, 1] → R3


x(t) = F (x(t))

• if F = 0 particles move in straight lines

3

Galilean relativity


x : [0, 1] → R3


x(t) = F (x(t))

• if F = 0 particles move in straight lines, and Newton’s equation

4

is invariant under Galilean transformations

4


x 7→ x + tv

4


x 7→ x + tv t 7→ t + a

4


x 7→ x + tv t 7→ t + a

• velocities add

4


x 7→ x + tv t 7→ t + a

• velocities add:

x(t) 7→ x(t) + v

4


x 7→ x + tv t 7→ t + a

• velocities add:

x(t) 7→ x(t) + v

so there is no maximum velocity

4


x 7→ x + tv t 7→ t + a

• velocities add:

x(t) 7→ x(t) + v


• more generally, time intervals are invariant

4


x 7→ x + tv t 7→ t + a

• velocities add:

x(t) 7→ x(t) + v


• more generally, time intervals are invariant, and so are distances

4


x 7→ x + tv t 7→ t + a

• velocities add:

x(t) 7→ x(t) + v


• more generally, time intervals are invariant, and so are distances:

|x1 − tv − (x2 − tv)| = |x1 − x2|

5

Faraday’s field lines

5


• Michael Faraday envisioned novel objects moving in space

5


• Michael Faraday envisioned novel objects moving in space: space-

filling lines starting and ending on electric charges

6

Maxwell equations

6

Maxwell equations

James Clerk Maxwell formulated Faraday’s electrical theory

mathematically

6

Maxwell equations


mathematically:

curlE = −1c

∂B

∂t

curlB =1c

∂E

∂t

6

Maxwell equations


mathematically:

curlE = −1c

∂B

∂t

curlB =1c

∂E

∂t

div E = 0

div B = 0

6

Maxwell equations


mathematically:

curlE = −1c

∂B

∂t

curlB =1c

∂E

∂t

div E = 0

div B = 0

where E(x, t) and B(x, t) are the electric and magnetic fields,

respectively.

7

Maxwell’s equations give rise to wave equations:

∇2E =1c2

∂2E

∂t2

7


∇2E =1c2

∂2E

∂t2∇2B =

1c2

∂2B

∂t2

7


∇2E =1c2

∂2E

∂t2∇2B =

1c2

∂2B

∂t2

with speed c.

7


∇2E =1c2

∂2E

∂t2∇2B =

1c2

∂2B

∂t2

with speed c.

Its solutions are called electromagnetic waves

7


∇2E =1c2

∂2E

∂t2∇2B =

1c2

∂2B

∂t2

with speed c.

Its solutions are called electromagnetic waves, and they describe

the propagation of light.

7


∇2E =1c2

∂2E

∂t2∇2B =

1c2

∂2B

∂t2

with speed c.



E

7


∇2E =1c2

∂2E

∂t2∇2B =

1c2

∂2B

∂t2

with speed c.



E This is inconsistent with Galilean relativity.

7


∇2E =1c2

∂2E

∂t2∇2B =

1c2

∂2B

∂t2

with speed c.



E This is inconsistent with Galilean relativity.

Instead, Einstein found that Maxwell equations are consistent with

a “special relativity.”

8

Special Relativity

8

Special Relativity

In special relativity, the invariant quantity is the proper time

8

Special Relativity


τ = |x1 − x2|2 − c2(t1 − t2)2

8

Special Relativity


τ = |x1 − x2|2 − c2(t1 − t2)2

between two spacetime events (x1, t1) and (x2, t2)

8

Special Relativity


τ = |x1 − x2|2 − c2(t1 − t2)2

between two spacetime events (x1, t1) and (x2, t2).

Z

8

Special Relativity


τ = |x1 − x2|2 − c2(t1 − t2)2


Zτ = 0 if and only if two events can be joined by a light ray.

8

Special Relativity


τ = |x1 − x2|2 − c2(t1 − t2)2



Instead of Galilean transformations

8

Special Relativity


τ = |x1 − x2|2 − c2(t1 − t2)2



Instead of Galilean transformations, we have

9

Lorentz transformations

9

Lorentz transformations:

x 7→ x + (γ − 1)(v · x)v

v2− γtv

t 7→ γ(t− v · xc2

)

9


x 7→ x + (γ − 1)(v · x)v

v2− γtv

t 7→ γ(t− v · xc2

)where γ =

1√1− ‖v‖2

c2

9


x 7→ x + (γ − 1)(v · x)v

v2− γtv

t 7→ γ(t− v · xc2

)where γ =

1√1− ‖v‖2

c2

More generally, a Lorentz transformation is a linear map

Λ : R4 → R4 preserving the Minkowski inner product

9


x 7→ x + (γ − 1)(v · x)v

v2− γtv

t 7→ γ(t− v · xc2

)where γ =

1√1− ‖v‖2

c2


Λ : R4 → R4 preserving the Minkowski inner product:

〈(x1, t1), (x2, t2)〉 = x1 · x2 − c2t1t2

9


x 7→ x + (γ − 1)(v · x)v

v2− γtv

t 7→ γ(t− v · xc2

)where γ =

1√1− ‖v‖2

c2



〈(x1, t1), (x2, t2)〉 = x1 · x2 − c2t1t2

Z

9


x 7→ x + (γ − 1)(v · x)v

v2− γtv

t 7→ γ(t− v · xc2

)where γ =

1√1− ‖v‖2

c2



〈(x1, t1), (x2, t2)〉 = x1 · x2 − c2t1t2

Z 〈−,−〉 is indefinite

10

The electromagnetic universe is (R4, 〈−,−〉)

10

The electromagnetic universe is (R4, 〈−,−〉), which is called

Minkowski spacetime

10


Minkowski spacetime, and is written E1,3

10


Minkowski spacetime, and is written E1,3.

It has an indefinite metric

10



It has an indefinite metric:

η = −c2dt2 + dx21 + dx2

2 + dx23

10




η = −c2dt2 + dx21 + dx2

2 + dx23 ,

which is flat

10




η = −c2dt2 + dx21 + dx2

2 + dx23 ,

which is flat: its geodesics are straight lines.

11

General Relativity

11

General Relativity

• Galileo had discovered the equivalence principle

11

General Relativity

• Galileo had discovered the equivalence principle:

inertial mass = gravitational mass

11

General Relativity



and Einstein geometrised it

11

General Relativity




• GR mantra

11

General Relativity




• GR mantra:

Spacetime tells matter how to move

11

General Relativity




• GR mantra:

Spacetime tells matter how to move, and matter tells

spacetime how to curve.

12

• Mathematically

12

• Mathematically:

? Spacetime is a 4-dimensional lorentzian manifold (M, g)

12

• Mathematically:

? Spacetime is a 4-dimensional lorentzian manifold (M, g), where

g =3∑

µ,ν=0

gµν(x)dxµdxν

12

• Mathematically:


g =3∑

µ,ν=0

gµν(x)dxµdxν

? Gravitation is described by the Einstein equations

12

• Mathematically:


g =3∑

µ,ν=0

gµν(x)dxµdxν

? Gravitation is described by the Einstein equations:

Rµν − 12Rgµν = 8πGTµν

12

• Mathematically:


g =3∑

µ,ν=0

gµν(x)dxµdxν

? Gravitation is described by the Einstein equations:

Rµν − 12Rgµν = 8πGTµν

Which spacetime (M, g) describes our universe?

13

Empirical basis for FLRW cosmology

13


• Hubble (1920s)

13


• Hubble (1920s)

? uniform redshift of spectral lines in all directions

13


• Hubble (1920s)

? uniform redshift of spectral lines in all directions and growing

with distance

13


• Hubble (1920s)


with distance

? space between stars is expanding isotropically

13


• Hubble (1920s)


with distance


? “principle of mediocrity” =⇒ homogeneity

13


• Hubble (1920s)


with distance



• Penzias and Wilson (1965)

13


• Hubble (1920s)


with distance



• Penzias and Wilson (1965):

? discovered Cosmic Microwave Background

13


• Hubble (1920s)


with distance



• Penzias and Wilson (1965):

? discovered Cosmic Microwave Background

? confirms homogeneity and isotropy at large scales

14

At large scales, our universe is described by R× Σ

14

At large scales, our universe is described by R× Σ, with R the

cosmological time

14


cosmological time, and Σ a three-dimensional manifold of constant

curvature

14



curvature: our spatial universe.

14




The geometry is described by a

Friedmann–Lemaıtre–Robertson–Walker metric

14





Friedmann–Lemaıtre–Robertson–Walker metric:

−dt2 + a(t)2h

where h is the metric on Σ.

14






−dt2 + a(t)2h


a(t) is determined from the Einstein equations

14






−dt2 + a(t)2h


a(t) is determined from the Einstein equations, which depend on

the “matter” content of the universe

14






−dt2 + a(t)2h


a(t) is determined from the Einstein equations, which depend on

the “matter” content of the universe, currently

16

The geometry and topology of the spatial universe

16


FLRW cosmology

16


FLRW cosmology

=⇒ spatial universe Σ has constant sectional curvature

16


FLRW cosmology


=⇒ Σ is isometric to a space form

16


FLRW cosmology



=⇒ Σ is isometric to Σ/Γ

16


FLRW cosmology



=⇒ Σ is isometric to Σ/Γ, where Σ is a simply-connected three-

dimensional space form

16


FLRW cosmology



=⇒ Σ is isometric to Σ/Γ, where Σ is a simply-connected three-

dimensional space form, and Γ is a discrete subgroup of the

isometries of Σ

17

The metric h on Σ takes the form

17

The metric h on Σ takes the form:

h = dχ2 + f(χ)2(dθ2 + sin2 θ dϕ2

)

17



)where f depends on the sign of the curvature

17



)where f depends on the sign of the curvature:

• κ = 0

17




• κ = 0: f(χ) = χ

17




• κ = 0: f(χ) = χ

• κ = 1

17




• κ = 0: f(χ) = χ

• κ = 1: f(χ) = sin χ

17




• κ = 0: f(χ) = χ

• κ = 1: f(χ) = sin χ

• κ = −1

17




• κ = 0: f(χ) = χ

• κ = 1: f(χ) = sin χ

• κ = −1: f(χ) = sinhχ

18

• κ = 0

18

• κ = 0 (flat)

18

• κ = 0 (flat): Σ = E3

18

• κ = 0 (flat): Σ = E3:

E3 ⊂ E4

18

• κ = 0 (flat): Σ = E3:

E3 ⊂ E4 : x4 = 1

18

• κ = 0 (flat): Σ = E3:

E3 ⊂ E4 : x4 = 1

and Γ ⊂ O(3) n R3 ⊂ GL(4, R)

18

• κ = 0 (flat): Σ = E3:

E3 ⊂ E4 : x4 = 1

and Γ ⊂ O(3) n R3 ⊂ GL(4, R)

• κ > 0

18

• κ = 0 (flat): Σ = E3:

E3 ⊂ E4 : x4 = 1

and Γ ⊂ O(3) n R3 ⊂ GL(4, R)

• κ > 0 (spherical)

18

• κ = 0 (flat): Σ = E3:

E3 ⊂ E4 : x4 = 1

and Γ ⊂ O(3) n R3 ⊂ GL(4, R)

• κ > 0 (spherical): Σ = S3

18

• κ = 0 (flat): Σ = E3:

E3 ⊂ E4 : x4 = 1

and Γ ⊂ O(3) n R3 ⊂ GL(4, R)

• κ > 0 (spherical): Σ = S3:

S3 ⊂ E4

18

• κ = 0 (flat): Σ = E3:

E3 ⊂ E4 : x4 = 1

and Γ ⊂ O(3) n R3 ⊂ GL(4, R)


S3 ⊂ E4 : x21 + x2

2 + x23 + x2

4 =1κ2

18

• κ = 0 (flat): Σ = E3:

E3 ⊂ E4 : x4 = 1

and Γ ⊂ O(3) n R3 ⊂ GL(4, R)


S3 ⊂ E4 : x21 + x2

2 + x23 + x2

4 =1κ2

and Γ ⊂ O(4)

19

• κ < 0

19

• κ < 0 (hyperbolic)

19

• κ < 0 (hyperbolic): Σ = H3

19

• κ < 0 (hyperbolic): Σ = H3:

H3 ⊂ E1,3

19


H3 ⊂ E1,3 : −x20 + x2

1 + x22 + x2

3 =−1κ2

19


H3 ⊂ E1,3 : −x20 + x2

1 + x22 + x2

3 =−1κ2

x0 > 0

19


H3 ⊂ E1,3 : −x20 + x2

1 + x22 + x2

3 =−1κ2

x0 > 0

and Γ ⊂ O(1, 3)

19


H3 ⊂ E1,3 : −x20 + x2

1 + x22 + x2

3 =−1κ2

x0 > 0

and Γ ⊂ O(1, 3)

Spherical and hyperbolic spaceforms are rigid

19


H3 ⊂ E1,3 : −x20 + x2

1 + x22 + x2

3 =−1κ2

x0 > 0

and Γ ⊂ O(1, 3)

Spherical and hyperbolic spaceforms are rigid: their topology

determines their geometry!

19


H3 ⊂ E1,3 : −x20 + x2

1 + x22 + x2

3 =−1κ2

x0 > 0

and Γ ⊂ O(1, 3)


determines their geometry! (e.g., Mostow rigidity)

19


H3 ⊂ E1,3 : −x20 + x2

1 + x22 + x2

3 =−1κ2

x0 > 0

and Γ ⊂ O(1, 3)



Big question

19


H3 ⊂ E1,3 : −x20 + x2

1 + x22 + x2

3 =−1κ2

x0 > 0

and Γ ⊂ O(1, 3)



Big question:

Which topology do we inhabit?

20

Large scale anisotropy in the CMB

20

Large scale anisotropy in the CMB

The COsmic Background Explorer showed large scale anisotropy

21

The Wilkinson Microwave Anisotropy Probe refined the data

21

The Wilkinson Microwave Anisotropy Probe refined the data:

22

Physics of the CMB

22

Physics of the CMB

• CMB results from photons produced in the early universe

22

Physics of the CMB


• WMAP data is a snapshot of the “surface of last scatter”

22

Physics of the CMB



• CMB is measured in temperature

22

Physics of the CMB



• CMB is measured in temperature: / 3◦K

22

Physics of the CMB




• anisotropy is measured as “temperature fluctuations”

22

Physics of the CMB




• anisotropy is measured as “temperature fluctuations”, and is due

to density fluctuations in the early universe

23

We introduce a “scalar perturbation” to the FLRW metric

23

We introduce a “scalar perturbation” to the FLRW metric:

−(1 + 2Φ)dt2 + a(t)2(1− 2Φ)h

23


−(1 + 2Φ)dt2 + a(t)2(1− 2Φ)h

in the form of a “gravitational potential”

23


−(1 + 2Φ)dt2 + a(t)2(1− 2Φ)h

in the form of a “gravitational potential” Φ : R× Σ → R.

23


−(1 + 2Φ)dt2 + a(t)2(1− 2Φ)h


The time evolution of Φ is fixed by the Einstein equations

23


−(1 + 2Φ)dt2 + a(t)2(1− 2Φ)h


The time evolution of Φ is fixed by the Einstein equations, whence

it is determined by its “initial value” Φ0 : Σ → R.

23


−(1 + 2Φ)dt2 + a(t)2(1− 2Φ)h


The time evolution of Φ is fixed by the Einstein equations, whence

it is determined by its “initial value” Φ0 : Σ → R.

The “Sachs–Wolfe effect” relates the temperature fluctuations in

the CMB to Φ0.

24

Strategy

24

Strategy: Try to work out Φ0 from WMAP data

24

Strategy: Try to work out Φ0 from WMAP data, and hence the

geometry of Σ.

24


geometry of Σ.

• WMAP data defines Ψ : S2 → R

24


geometry of Σ.

• WMAP data defines Ψ : S2 → R, which we expand in spherical

harmonics

24


geometry of Σ.


harmonics

• gravitational potential Φ0 : Σ → R

24


geometry of Σ.


harmonics

• gravitational potential Φ0 : Σ → R, which we can also expand in

harmonics on Σ

24


geometry of Σ.


harmonics


harmonics on Σ

• lowest lying modes in expansion of Φ0 carry information about the

topology of Σ

24


geometry of Σ.


harmonics


harmonics on Σ


topology of Σ, and feed into the lowest lying modes in expansion

of Ψ

24


geometry of Σ.


harmonics


harmonics on Σ


topology of Σ, and feed into the lowest lying modes in expansion

of Ψ

There is a related mathematical problem...

25


25


We can determine the length of a string by hearing it vibrate

25


We can determine the length of a string by hearing it vibrate.

Mark Kac (1966)

25



Mark Kac (1966): Can we hear the shape of a drum?

25




Mathematically, given a riemannian manifold (M, g), consider the

Laplacian operator ∆ acting on functions

25





Laplacian operator ∆ acting on functions:

∆f(x) =∑i,j

gij(x)∇i∇jf(x)

25





Laplacian operator ∆ acting on functions:

∆f(x) =∑i,j

gij(x)∇i∇jf(x)

with ∇ the riemannian connection.

26

The eigenvalues of ∆ define the spectrum of (M, g)

26

The eigenvalues of ∆ define the spectrum of (M, g).

Kac’s question is

26


Kac’s question is:

Are isospectral 2-manifolds isometric?

26




The answer is known to be false in general

26




The answer is known to be false in general:

Milnor (1964)

26





Milnor (1964): R16/Λ1 and R16/Λ2 are isospectral but not

isometric.

26






isometric.

More recent counterexamples

26






isometric.

More recent counterexamples: planar domains.

26






isometric.

More recent counterexamples: planar domains.

But it is true for three-dimensional space-forms!

27

Spherical harmonics

27

Spherical harmonics

• F : R3 → R: a homogeneous polynomial of degree `

27

Spherical harmonics


• F = r`f

27

Spherical harmonics


• F = r`f , where f : S2 → R

27

Spherical harmonics


• F = r`f , where f : S2 → R

• F is harmonic ⇐⇒ ∆f = −`(` + 1)f on S2

27

Spherical harmonics


• F = r`f , where f : S2 → R


• all eigenfunctions of ∆ are obtained in this way

27

Spherical harmonics


• F = r`f , where f : S2 → R



• the eigenspace V` with eigenvalue −`(`+1) has dimension 2`+1

27

Spherical harmonics


• F = r`f , where f : S2 → R



• the eigenspace V` with eigenvalue −`(`+1) has dimension 2`+1

• V` is an irreducible representation of SO(3)

28

• spherical harmonics Y` m

28

• spherical harmonics Y` m, m = −`, ..., `

28

• spherical harmonics Y` m, m = −`, ..., `, span V`

28


• we expand

Ψ =∑`≥0

∑m=−`

a` mY`,m

28


• we expand

Ψ =∑`≥0

∑m=−`

a` mY`,m

• power spectrum: {C`}

28


• we expand

Ψ =∑`≥0

∑m=−`

a` mY`,m

• power spectrum: {C`}, where

C` =∑

m=−`

|a` m|2

28


• we expand

Ψ =∑`≥0

∑m=−`

a` mY`,m

• power spectrum: {C`}, where

C` =∑

m=−`

|a` m|2

• WMAP data gives {C`} up to ` ∼ 100

29

A spherical universe

29


Consider a spherical universe Σ = S3/Γ

29


Consider a spherical universe Σ = S3/Γ.

• functions on S3/Γ are Γ-invariant functions on S3

29




• S3 and S3/Γ are locally isometric

29





=⇒ laplacians on S3 and on S3/Γ agree

29





=⇒ laplacians on S3 and on S3/Γ agree

• eigenfunctions on S3/Γ are Γ-invariant spherical harmonics

30

Spherical harmonics on S3

30


• F : R4 → R: a homogeneous polynomial of degree k

30



• F = rkf

30



• F = rkf , where f : S3 → R

30




• F is harmonic ⇐⇒ ∆f = −k(k + 2)f on S3

30






30






• the eigenspace Vk with eigenvalue −k(k + 2) has dimension

(k + 1)2

30






• the eigenspace Vk with eigenvalue −k(k + 2) has dimension

(k + 1)2

• Vk is an irreducible representation of SO(4)

31

Subgroups Γ ⊂ SO(4)

31


S3 = {q ∈ H | qq = 1}

31


S3 = {q ∈ H | qq = 1}

Left- and right-multiplication by unit quaternions

31


S3 = {q ∈ H | qq = 1}

Left- and right-multiplication by unit quaternions:

q 7→ q1qq2

31


S3 = {q ∈ H | qq = 1}


q 7→ q1qq2

define an action of SU(2)× SU(2).

31


S3 = {q ∈ H | qq = 1}


q 7→ q1qq2


The element (−1,−1) acts trivially

31


S3 = {q ∈ H | qq = 1}


q 7→ q1qq2


The element (−1,−1) acts trivially, so we have an action of

SO(4) = (SU(2)× SU(2)) /Z2

32

Let Γ ⊂ SO(4) be a finite subgroup

32

Let Γ ⊂ SO(4) be a finite subgroup.

Three types

32


Three types:

• left-acting: q 7→ γq, γ ∈ Γ

32


Three types:


• right-acting: q 7→ qγ, γ ∈ Γ

32


Three types:



• doubly-acting

32


Three types:



• doubly-acting

For the first two types

32


Three types:



• doubly-acting

For the first two types: Γ ⊂ SU(2).

33

Finite subgroups Γ ⊂ SU(2)

33


McKay correspondence

33


McKay correspondence:

Γ Dynkin diagram Name

33




A` e e e e e· · · binary cyclic

33





D` e e e e e· · ·

ebinary dihedral

33





D` e e e e e· · ·

ebinary dihedral

E6 e e e e ee

binary tetrahedral

33





D` e e e e e· · ·

ebinary dihedral

E6 e e e e ee

binary tetrahedral

E7 e e e e e ee

binary octahedral

33





D` e e e e e· · ·

ebinary dihedral

E6 e e e e ee

binary tetrahedral

E7 e e e e e ee

binary octahedral

E8 e e e e e e ee

binary dodecahedral

34

And the answer is...

34


Luminet, Weeks, Riazuelo, Lehoucq and Uzan (2003)

34


Luminet, Weeks, Riazuelo, Lehoucq and Uzan (2003) found that

the first year WMAP data suggests

34



the first year WMAP data suggests:

Σ ∼=

34




Σ ∼= S3/

34




Σ ∼= S3/E8 (!)

34




Σ ∼= S3/E8 (!)

• the universe is finite!

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Σ ∼= S3/E8 (!)


• π1(Σ) ∼= E8

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Σ ∼= S3/E8 (!)


• π1(Σ) ∼= E8 =⇒ the universe is not simply-connected!

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• [E8, E8] = E8

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• [E8, E8] = E8 =⇒ H1(Σ) = π1/[π1, π1] = 0

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• [E8, E8] = E8 =⇒ H1(Σ) = π1/[π1, π1] = 0=⇒ Σ is a homology 3-sphere

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• [E8, E8] = E8 =⇒ H1(Σ) = π1/[π1, π1] = 0=⇒ Σ is a homology 3-sphere

• Σ is counterexample to the “wrong” Poincare conjecture!

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Caveat emptor

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Caveat emptor

• few data points

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Caveat emptor

• few data points: only C` for ` = 2, 3, 4 were matched

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Caveat emptor


• statistical significance in doubt

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Caveat emptor



• circles in the sky?

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Caveat emptor




• a little too exceptional?

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Caveat emptor




• a little too exceptional?

• awaiting more data: WMAP, Planck surveyor,...

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Watch this space.

can we hear the shape of the universe?jmf/cv/seminars/pgcolloquium.pdf · can we hear the shape of...

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