A mathematician’s Divina Commedia - Websupport1websupport1.citytech.cuny.edu/faculty/hschoutens/PDF/SOC2010.pdf · A mathematician’s Divina Commedia his travels through the three

Size matters . . . and so does shape Symmetry and singularities Ultraworld

A mathematician’s Divina Commediahis travels through the three realms: the physical, the virtual,

and the platonic

H. Schoutens

Department of MathematicsNew York City College of Technology

April 19, 2010Scholar-on-Campus


Outline

1 Size matters

2 . . . and so does shape

3 The beauty and the beast: symmetry and singularities

4 A new utopia? Ultraworld

DisclaimerNone of what I say is potentially nearly as false as it soundsplausibly true.


Outline

1 Size matters






Outline

1 Size matters






Outline

1 Size matters






Outline

1 Size matters






Three scientific paradigms

First paradigmThe physical (inanimate) world can be modeled bymathematics.

Second paradigmMathematics is developed rigorously through logical reasoningand hence generates irrefutable truths.

Third paradigm

Computers can emulate this modeling, albeit only a finiteportion of it.

Sentimental ParadigmMathematics is most beautiful and most perfect.





Third paradigm







Third paradigm




Three scientific paradigms, and a sentimental one



Third paradigm




Outline

1 Size matters





Infinity∞

Definition (Galileo 1564–1642)A collection is infinite if it fits inside a smaller part of itself.


Infinity∞


Example (Hilbert’s Grand Hotel)Imagine a hotel with a room for each number.

Even if the hotelis booked full, a new guest can be accomodated by havingeveryone change to the next room.

1 2 3 4 5

. . .


Infinity∞


Example (Hilbert’s Grand Hotel)Imagine a hotel with a room for each number. Even if the hotelis booked full, a new guest can be accomodated by havingeveryone change to the next room.

1

,,

2

,,

3

,,

4

,,

5

** . . .


Infinity∞


Example (Hilbert’s Grand Hotel)Imagine a hotel with a room for each number. Even if the hotelis booked full, a new guest can be accomodated by havingeveryone change to the next room.

1

,,

2

,,

3

,,

4

,,

5

** . . .

CorollaryThere are infinitely many numbers.


Infinity∞


Example (Euclid’s line segment)Points on a line segment.

-

−1 −12 0 1

2 1s s

↘ ↓ ↙

-

−1 −12 0 1

2 1s s


Infinity∞


Example (Euclid’s line segment)Points on a line segment can be squeezed together.

-

−1 −12 0 1

2 1s s

↘ ↓ ↙

-

−1 −12 0 1

2 1s s


Finiteness: the Pigeon-Hole Principle

TheoremIf there are more pigeons than holes, at least two will have toshare a hole.

Proof.




Proof.




Proof.

))TTTTTTTTTTTTTTTTTTTTTTTTTTTTT

��

''NNNNNNNNNNNNNNNNNN

wwpppppppppppppppppp

wwpppppppppppppppppp




Proof.


Schools of thought

Philosophers and scientists about

Physical world

Plato: shadowsAristotle: apeiron = potential infinity.

Zoroaster: finite timeNewton: infinite Euclidean spaceEinstein: universe is bounded,

Planck: and discrete,Hawking: and expanding.

Turing: hardware = limited

Metaphysical world

ideasno actual infinityendless timemechanismrelativityuncertaintysingularitysoftware = unlimited


Schools of thought

(428 BC–347 BC)

Physical worldPlato: shadows

Aristotle: apeiron = potential infinity.Zoroaster: finite time

Newton: infinite Euclidean spaceEinstein: universe is bounded,



Metaphysical worldideas

no actual infinityendless timemechanismrelativityuncertaintysingularitysoftware = unlimited


Schools of thought

(384 BC–322 BC)


Aristotle: apeiron = potential infinity.

Zoroaster: finite timeNewton: infinite Euclidean spaceEinstein: universe is bounded,



Metaphysical worldideasno actual infinity

endless timemechanismrelativityuncertaintysingularitysoftware = unlimited


Schools of thought

(ca. 1500 BC)


Aristotle: apeiron = potential infinity.Zoroaster:!"#"$"%&!'()*++,&'!-+$+.&%!+*'/&01!'2*,&!'(+3+&223#&!'()&45&6!

&#5&7&$8!+*'/&0!+*$92:&$&;!<&=!&)&'!-++)&1!>7;&'!-$2/&!-=!'?3(gaeTa)

finite timeNewton: infinite Euclidean spaceEinstein: universe is bounded,



Metaphysical worldideasno actual infinity

3/19/10 1:47AVESTA: KHORDA AVESTA: YASHTS (Hymns)

Page 1 of 324http://www.avesta.org/ka/yt_jamaspa.htm#yt10

Avesta -- Zoroastrian Archives Contents Prev yt_jamaspa Next Glossary

AVESTA: KHORDA AVESTA: YASHTS

(Hymns)

NOTE: You will need Avestan fonts in order to read this text.

1. Ohrmazd Yasht.

,eAdaX ,i ,dzvmrOh ,`,&dzaY ,i ,m&n ,ap) 0`,TAyAzaBa ,eharaX ,zrug ,InUzaBa ,i

,dNameharaX ,i ,dNamear ,i ,dzvmrOh ,rAdAd`,TAsar ,Vb ,mutVb ,i ,Oyniam ,&uyniam

`,mOn&mVSap ,titap ,hAnug ,Amah ,Zv ,tSvraWZud ,txUZud ,tamSud ,nItsiWarah ,Zv

,tsaj ,meaW ,drak ,meaW ,tfug ,meaW ,TInim ,IqVg ,ap ,nvm`,Tvtsv ,TUb ,nub ,meaW

,InSinuk ,InSvBag ,InSinam ,AhihAnug ,& ,Zv,&mVSap ,SxaBa ,exO ,In&uyniam ,IqVg ,In&wr ,Inat

,(.,mOh ,titap ,ap ,InSvBag ,vs ,ap

,ldzam ,eharuha ,arqoanCx `,SuVyniam ,eharMa ,itIdiOrat

.,mvmvtOCarvf ,AnsaW ,Tayh ,m&tSrawAyqiah,AcsaybiOtSrawh ,AcsaybiOtxUh ,AcsaybiOtamuh ,Eyutsarvf

,AcsaybiOBq&m,Eqiad ,Ayriagibia ,`,AcsaybiOwtSraW ,AcsaybiOBDvxaW

,AcAtxUh ,AcAtamuh ,ApsIW,AcAtxUZud ,AcAtamSud ,ApsIW ,Eqiad ,Ayciritiap

,`,AcAtSrawh.,AcAtSrawZud

(mainiiu)

endlesstimemechanismrelativityuncertaintysingularitysoftware = unlimited


Schools of thought

(ca. 1500 BC)






Metaphysical worldideasno actual infinityendless time

mechanismrelativityuncertaintysingularitysoftware = unlimited


Schools of thought

(1643–1727)



Newton: infinite Euclidean space

Einstein: universe is bounded,Planck: and discrete,

Hawking: and expanding.Turing: hardware = limited

Metaphysical worldideasno actual infinityendless timemechanism

relativityuncertaintysingularitysoftware = unlimited


Schools of thought

(1879–1955)






Metaphysical worldideasno actual infinityendless timemechanismrelativity

uncertaintysingularitysoftware = unlimited


Schools of thought

(1858–1947)




Planck: and discrete,

Hawking: and expanding.Turing: hardware = limited

Metaphysical worldideasno actual infinityendless timemechanismrelativityuncertainty

singularitysoftware = unlimited


Schools of thought

(1942–)






Metaphysical worldideasno actual infinityendless timemechanismrelativityuncertaintysingularity

software = unlimited


Schools of thought

(1912–1954)






Metaphysical worldideasno actual infinityendless timemechanismrelativityuncertaintysingularitysoftware = unlimited


Mind versus machine

Theorem (Godel ’31)Some mathematical facts can never be obtained by a computer.

Anti-theoremComputers can model facts that cannot be grasped by ourhuman minds.


Mind versus machine


Example (Matiyasevich ’70)No computer can check whether a random equation has asolution.



Mind versus machine




Mind versus machine



Example (Knuth ’76)Recursion can create unfathomable numbers.

a ↑n b := a ↑ . . . ↑ b︸︷︷︸n


Mind versus machine



Example (Knuth ’76)Recursion can create unfathomable numbers. For instance, thisis a self-referential definition of Knuth’s arrow

a ↑n b := a ↑n−1 a ↑n−1 . . . ↑n−1 a︸︷︷︸b


Mind versus machine




3 ↑ 3 = 33 = 27

3 ↑↑ 3 = 333= 7, 625, 597, 484, 987

3 ↑↑↑ 3 is an immense number;9 ↑9 9 is unfathomable, and to our mind looks like infinity.


Mind versus machine




3 ↑ 3 = 33 = 273 ↑↑ 3 = 333

= 7, 625, 597, 484, 987

3 ↑↑↑ 3 is an immense number;9 ↑9 9 is unfathomable, and to our mind looks like infinity.


Mind versus machine




3 ↑ 3 = 33 = 273 ↑↑ 3 = 333

= 7, 625, 597, 484, 9873 ↑↑↑ 3 is an immense number;

9 ↑9 9 is unfathomable, and to our mind looks like infinity.


Mind versus machine




3 ↑ 3 = 33 = 273 ↑↑ 3 = 333

= 7, 625, 597, 484, 9873 ↑↑↑ 3 is an immense number;9 ↑9 9 is unfathomable, and to our mind looks like infinity.


Dealing with big numbers

Adjustmentprecision

Example (engineering)Number of seconds in a(Julian) year:3600× 24× 365.25 =31557600



Adjustmentprecision approximation

Example (engineering)Number of seconds in a(Julian) year:3600× 24× 365.25 =30000000 ≈ 3× 107



Adjustmentprecision approximationunrestricted

Example (cosmology)

Collision speed of two lightparticles:vcoll = v1 + v2 = c + c



Adjustmentprecision approximationunrestricted restricted

Example (cosmology)

Collision speed of two lightparticles:vcoll = v1 + v2 = c + c = c



Adjustmentprecision approximationunrestricted restrictedaccuracy

Example (computing)In a 1K-computer, we have:881×883 = 777923

≡ 923mod 1000



Adjustmentprecision approximationunrestricted restrictedaccuracy modularity

Example (computing)In a 1K-computer, we have:881×883 = 777923 ≡ 923mod 1000



Adjustmentprecision approximationunrestricted restrictedaccuracy modularityentropy

Example (cryptography)Multiplying :881×883 99K

= 777923

≡ 923mod 1000



Adjustmentprecision approximationunrestricted restrictedaccuracy modularityentropy algorithm

Example (cryptography)Multiplying versus factoring:881×883 L99

= 777923

≡ 923mod 1000


Outline

1 Size matters





Dimension of a line

Definition (Dimension)Dimension is the number of independent directions in which wecan move.

We can go left

or right

Going left is the same as going right backwards, so, as adirection, it is dependent:

� � line is one-dimensional

. . . |0

. . .


Dimension of a line


We can go left

or rightGoing left is the same as going right backwards, so, as adirection, it is dependent:


�� |0

. . .


Dimension of a line


We can go left or right

Going left is the same as going right backwards, so, as adirection, it is dependent:


�� |0

� ��


Dimension of a line


We can go left or rightGoing left is the same as going right backwards, so, as adirection, it is dependent:


�� |0

� ��←−


Dimension of a line


We can go left or rightGoing left is the same as going right backwards, so, as adirection, it is dependent:


. . . |0

. . .


Dimension and curvature

Descartes’ plane is 2-dimensional

Newton’s universe is 3-dimensionalEarth’s surface is 2-dimensionalEinstein’s universe is 4-dimensionalString theory:

10-dimensionaluniverse(but about 10500 possible ones)

Witten’s solution: make universe11-dimensional

OO

��

oo //|0

Flat geometry

particles are little strings

they can wrap around aCalabi-Yau manifold



Descartes’ plane is 2-dimensionalNewton’s universe is 3-dimensional

Earth’s surface is 2-dimensionalEinstein’s universe is 4-dimensionalString theory:



OO

��

oo //|0

��

AA

Flat geometry





Descartes’ plane is 2-dimensionalNewton’s universe is 3-dimensionalEarth’s surface is 2-dimensional

Einstein’s universe is 4-dimensionalString theory:



Elliptic geometry





Descartes’ plane is 2-dimensionalNewton’s universe is 3-dimensionalEarth’s surface is 2-dimensionalEinstein’s universe is 4-dimensional

String theory:



Hyperbolic geometry





Descartes’ plane is 2-dimensionalNewton’s universe is 3-dimensionalEarth’s surface is 2-dimensionalEinstein’s universe is 4-dimensionalString theory:

10-dimensionaluniverse

(but about 10500 possible ones)


•••••

•••••

•••••

•••••

•••••

•••••

•••••

x//

yOO









•••••

•••••

•••••

•••••

•}

•••

•••••

•••••

x//

yOO









•••••

•••••

•••••

•••••

•}

•••

•••••

•••••

x//

yOO

particles are little stringsthey can wrap around aCalabi-Yau manifold



Descartes’ plane is 2-dimensionalNewton’s universe is 3-dimensionalEarth’s surface is 2-dimensionalEinstein’s universe is 4-dimensionalString theory: 10-dimensionaluniverse

(but about 10500 possible ones)Witten’s solution: make universe11-dimensional

}x//

yOO





Descartes’ plane is 2-dimensionalNewton’s universe is 3-dimensionalEarth’s surface is 2-dimensionalEinstein’s universe is 4-dimensionalString theory: 10-dimensionaluniverse(but about 10500 possible ones)


}x//

yOO





Descartes’ plane is 2-dimensionalNewton’s universe is 3-dimensionalEarth’s surface is 2-dimensionalEinstein’s universe is 4-dimensionalString theory: 10-dimensionaluniverse(but about 10500 possible ones)Witten’s solution: make universe11-dimensional





Descartes’ plane is 2-dimensionalNewton’s universe is 3-dimensionalEarth’s surface is 2-dimensionalEinstein’s universe is 4-dimensionalString theory: 10-dimensionaluniverse(but about 10500 possible ones)Witten’s solution: make universe11-dimensional



DefinitionA manifold looks in each point like one of these worlds.


Mirror symmetry and Enumerative Geometry

TheoremOn a Calabi-Yau manifold, the number of parametric curves ofdegree 5 is n5 = 229305888887625 ≈ 2× 1014.




Preliminaries.Mirror images are the same (up to chirality).




Preliminaries.To cheat at Blind Man’s Bluff poker, look into a mirror.




Preliminaries.With a 3-dimensional mirror, we could see inside a tree anddetermine its age by counting its rings.




String theoretic proof.

String theory: our universe has a mirror image

in this mirror image, we can read off the value of n5 of itscorresponding Calabi-Yau manifoldmathematical truth is universe-independent, so n5 is thecorrect number in our as well as any alternate universe!





String theory: our universe has a mirror imagein this mirror image, we can read off the value of n5 of itscorresponding Calabi-Yau manifold

mathematical truth is universe-independent, so n5 is thecorrect number in our as well as any alternate universe!





String theory: our universe has a mirror imagein this mirror image, we can read off the value of n5 of itscorresponding Calabi-Yau manifoldmathematical truth is universe-independent, so n5 is thecorrect number in our as well as any alternate universe!


Inside a virtual world

Virtual worlds have finite length, often a prime number p.

Example (World of length p = 7)

1

7 PPPPPP6 nnnnnn

5��

4999999

3

2��

&&&&&&

“clock-wise” addition:to add 3 to 5,

move 3 points up3 + 5 = 8 ≡ 1 mod 7.

Virtual worlds can have arbitrary high dimension d . Such amulti-dimensional virtual world has therefore size pd .





1

0 PPPPPP6 nnnnnn

5��

4999999

3

2��

&&&&&&








1

0 PPPPPP6 nnnnnn

5←��

4999999

3

2��

&&&&&&








1

0 PPPPPP6↖

nnnnnn

5��

4999999

3

2��

&&&&&&

“clock-wise” addition:to add 3 to 5,move 3 points up

3 + 5 = 8 ≡ 1 mod 7.






1

0

↑

PPPPPP6 nnnnnn

5��

4999999

3

2��

&&&&&&


3 + 5 = 8 ≡ 1 mod 7.






1↗

0 PPPPPP6 nnnnnn

5��

4999999

3

2��

&&&&&&

“clock-wise” addition:to add 3 to 5,move 3 points up, so3 + 5 = 8 ≡ 1 mod 7.






1

0 PPPPPP6 nnnnnn

5��

4999999

3

2��

&&&&&&


3 + 5 = 8 ≡ 1 mod 7.



Outline

1 Size matters





Symmetry

(group theory) symmetry = transformation preserving themanifold

(crystallography) many physical objects have nice symmetries(tessalation) we can ‘tile’ the manifold to reveal its symmetry(geometry) tile + symmetry manifold


Symmetry


(crystallography) many physical objects have nice symmetries

(tessalation) we can ‘tile’ the manifold to reveal its symmetry(geometry) tile + symmetry manifold


Symmetry


(crystallography) many physical objects have nice symmetries(tessalation) we can ‘tile’ the manifold to reveal its symmetry

(geometry) tile + symmetry manifold


Symmetry


(crystallography) many physical objects have nice symmetries(tessalation) we can ‘tile’ the manifold to reveal its symmetry(geometry) tile + symmetry manifold


The tile space of a symmetry

DefinitionA tile is the smallest part whose copies under the symmetriescover the manifold.

DefinitionEdges of the tile get identified by the symmetry.

Glueing theseedges gives the tile space (or, quotient space).

. . .

. . . .

. . .

. . . .






. . .

. . . .

.

}}}}}}} .

}}}}}}} .

. . . .






. . .

. . v /o/o/o/o/o/o/o . .

.

u }}}}}}}

}}}}}}} v /o/o/o/o/o/o/o .

u }}}}}}}

}}}}}}} .

. . . .




DefinitionEdges of the tile get identified by the symmetry. Glueing theseedges gives the tile space (or, quotient space).

�Oj*m-o//oq1t4

�OO�*j -m /oo/ 1q 4t

O�

*j -m /oo/ 1q 4tO��

��

��

uv

v




DefinitionEdges of the tile get identified by the symmetry. Glueing theseedges gives the tile space (or, quotient space).


Tile spaces are more general than manifolds

ExampleConsider the rotational symmetries of a triangle.

Each slice is atile. To build the tile space, we glue together its sides. Theresult is a cone, with a sharp vertex.

DefinitionAny point where the space does no longer look like a manifold,is called a singularity.

.

� .

.

�T__j��T __ j

��

��

))))))))))))•



ExampleConsider the rotational symmetries of a triangle. Each slice is atile.

To build the tile space, we glue together its sides. Theresult is a cone, with a sharp vertex.


.

•

.

.

ppppppppppppppp

�T__j��T __ j

��

��

))))))))))))•



ExampleConsider the rotational symmetries of a triangle. Each slice is atile. To build the tile space, we glue together its sides.

Theresult is a cone, with a sharp vertex.


.

•

.

.

pppppppp

�T__j��T __ j

��

��

))))))))))))•



ExampleConsider the rotational symmetries of a triangle. Each slice is atile. To build the tile space, we glue together its sides. Theresult is a cone, with a sharp vertex.


.

•

.

.

pppppppp �T__j��T __ j

��

��

))))))))))))•vertex





.

•

.

.


��

��

))))))))))))•singularity


Singularities come in many shapes and sizes. . .

(curve) self-intersection = node

(surface) fold line = pinch(catastrophe theory) sharp edge = cusp(quantum gravity) black hole = sink(cosmology) Big Bang?

Example



(curve) self-intersection = node(surface) fold line = pinch

(catastrophe theory) sharp edge = cusp(quantum gravity) black hole = sink(cosmology) Big Bang?

Example




(catastrophe theory) sharp edge = cusp

(quantum gravity) black hole = sink(cosmology) Big Bang?

Example




(catastrophe theory) sharp edge = cusp(quantum gravity) black hole = sink

(cosmology) Big Bang?

Example




(catastrophe theory) sharp edge = cusp(quantum gravity) black hole = sink(cosmology) Big Bang?

Example


Outline

1 Size matters





Frobenius symmetry of virtual worlds

Double product formula (student version)

(x + y)2 = x2 + y2

Theorem (Frobenius (1849–1917))In a finite world of length p, the student’s version of the binomialtheorem does hold true: (x + y)p = xp + yp

CorollaryThe p-th power map is a symmetry of any virtual world of lengthp, called the Frobenius symmetry.



Double product formula

(x + y)2 = x2 + 2xy + y2





Newton’s (1643–1727) Binomial Formula

(x + y)n = xn +(n1)xn−1y +

(n2)xn−2y2 + · · ·+ yn





Khayyam’s (1048–1131) Binomial Formula

(x + y)n = xn +(n1)xn−1y +

(n2)xn−2y2 + · · ·+ yn






(x + y)n = xn +(n1)xn−1y +

(n2)xn−2y2 + · · ·+ yn






(x + y)n = xn +(n1)xn−1y +

(n2)xn−2y2 + · · ·+ yn




Ultraworlds

DefinitionAn ultraworld is obtained by averaging infinitely many worlds.

Theorem (Łos ’55)A computation in an ultraworld is true, if it is true in a randomlychosen world.

Proof.We construct the corresponding parallel quantumultra-computer, with each processor working in somerandom world.If, on average, something is true in each world, then ourultra-computer will detect this with probability one.


Ultraworlds





Ultraworlds





Ultraworlds





Ultraworlds





Some ultraworld travelers

Skolem ’38: precursor to ultraworld: the non-standard numbersŁos ’55: ultraworld as a concept from logic

Keisler–Tarski ’64: use ultraworlds of infinity to get even largerinfinities

Robinson ’66: use the ultraworld of the real world to definenon-standard analysis (=foundation for Leibnitz’infinitesimal calculus)

Ax–Kochen ’68: use ultraworlds to study average properties offinite worlds

Schmidt–van den Dries ’84: use ultraworlds to obtaincomputability

S. ’00: use ultraworlds as a gateway between finite andinfinite worlds


Some ultraworld travelers

Skolem ’38: precursor to ultraworld: the non-standard numbersŁos ’55: ultraworld as a concept from logic

Keisler–Tarski ’64: use ultraworlds of infinity to get even largerinfinities

Robinson ’66: use the ultraworld of the real world to definenon-standard analysis (=foundation for Leibnitz’infinitesimal calculus)

Ax–Kochen ’68: use ultraworlds to study average properties offinite worlds

Schmidt–van den Dries ’84: use ultraworlds to obtaincomputability

S. ’00: use ultraworlds as a gateway between finite andinfinite worlds


Lefschetz world

DefinitionLefschetz world is the ultraworld of all finite worlds.

CorollaryLefschetz world has ultra-Frobenius symmetry.

Proof.Finite worlds have Frobenius symmetry (given by taking p-thpowers).

Averaging this out yields ultra-Frobeniussymmetry.

Theorem (S. ’03, Aschenbrenner-S. ’07)Any infinite world embeds into Lefschetz world.


Lefschetz world







Lefschetz world







Lefschetz world



Proof.Finite worlds have Frobenius symmetry (given by taking p-thpowers). Averaging this out yields ultra-Frobeniussymmetry.



Lefschetz world



Proof.Finite worlds have Frobenius symmetry (given by taking p-thpowers). Averaging this out yields ultra-Frobeniussymmetry.



Rationality of tile spaces

Theorem (Smith ’06)Tile spaces inside finite worlds have nice singularities.

Main Theorem (S. ’08)Any tile space has nice singularities.

Proof.embed tile space in Lefschetz world,

redo the above argument with ultra-Frobenius symmetry.





.

•

.

.


��

��

))))))))))))•vertex




Proof.This follows from two conflicting properties of Frobeniussymmetry:

1 it contracts subspaces,2 yet preserves cohomology.

This can only happen if cohomology is already zero, which iswhat is meant with nice singularities.








1 it contracts subspaces,

2 yet preserves cohomology.This can only happen if cohomology is already zero, which iswhat is meant with nice singularities.






































Proof.embed tile space in Lefschetz world,redo the above argument with ultra-Frobenius symmetry.


Epilogue

CorollaryMathematics is heaven!

Proof.Did you not just pay attention!

!"#$%"&'(("

)*"'+,%&-"


Epilogue



!"#$%"&'(("

)*"'+,%&-"


Epilogue



!"#$%"&'(("

)*"'+,%&-"


Thank you!

Documents

A mathematician’s Divina Commedia - Websupport1websupport1.citytech.cuny.edu/faculty/hschoutens/PDF/SOC2010.pdf · A mathematician’s Divina Commedia his travels through the three