２００９年１２月１０日 学術俯瞰講義 数学を創る

第10回 惑星の軌道を理解する

坪井 俊

Global Focus on Knowledgeg“Creating mathematics” lecture 12

GraduatedSchoolh l off Mathematicalhl Sciences

Takashi Tsuboi
The figures, photos and moving images with ‡marks attached belong to theirpy g holders. Reusing
g or reproducingpg them is p
prohibited unless p
permission is
copyright
obtained directly from such copyright holders.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

Planets

Reprinted from NASA Webpage

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi●ImpressionszI am impressed to see that even for simple polyhedrons we seein everyday life, the beautiful Euler polyhedron theorem always
holds.
holds！！I want you to be impressed！！
zIt was a surprise for me that Euler’s
Euler s polyhedron theorem isrelated to the orbits of planets.
zThat a theorem of p
polyhedronsyis applicablepp
to curved surfaces
is amazing.
！！That is why mathematics is interesting！！
zI couldn’t grasp the concept of a torus.
！！Umm！！
zTorus is found in video games (RPG map
map, DQ
DQ, etc)！！since mathematics is useful when creating video games,
many game developers are familiar with such mathematicalknowledge.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi●QuestionszI have heard that figures like tori are not natural in Euclidean
spaces, and I’m wondering what the true figure is like.
！！In
I fact,fsurfacef
off a torus is
i one off theh naturall forms.
f
Another natural form is a figure that is flat in two directions,and on such figures
figures, you would always return to the starting point
point.
zIt is not possible to fill the spaces with Archimedean polyhedrons,
whereas it is ppossible for diamond-shapedp ppolyhedrons.
yHow canwe demonstrate this?
！！You can show the former paying attention to the angle
b t
betweentwotfaces,fandd as tot theth latter,l tt play
l withith toy！！
t ！！
zI have heard the reason that planetary orbit do not change withtime is due to another conserved quantity
quantity. Is it true?
！！It is due to the inverse square law！！

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi●Questions
zDo
D concave polyhedronsl h dhhave any relations
l ti
tto planetslt or natural
t lphenomena?
！！Surfaces of genus more than 2 are, when they are in their most naturalform, concave at all points(though it cannot be realized in 3-dimensional spaces(Hilbert’s theorem)). These can also be space of states.！！
zAre there any applications of curved surface study?
！！There are few mathematical researches that are not related to curved
surfaces In addition
surfaces.
addition,…！！
！！
zAre there any applications of polyhedrons?
！！All the figures we observe are curved surfaces or polyhedrons
polyhedrons.
Designing automobiles, trains, and planes, and plastic products here andthere are made using metallic molds, and it is necessary to understand theproperties
ti off minimali i l surfacesfor th
the strengtht
th off curvedd resin.
i
Polyhedrons are also applied to studies of chemical reactions of interfacesof liquids and reactions of macromolecules.！！

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi●Questions
zAre there some more Platonic polyhedrons other than the fiveones?
！！We can demonstrate that with Euler’s polyhedrontheorem！！
zEuler characteristic of the Mobius’ ring is zero. Is the property ofMobius’ ring similar to that of a torus?
z ！！That！！That’ss true.
true On the surface of Mobius’
Mobius ring,
ring you can draw aflow that has no stationary points, and which is always
perpendicular to the boundary.！！

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

What are the problems that
Poincaré（1854－1912） was engaged
g g in?

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

z In Poincaré
Poincaré’ss era
era, there was a breakthrough in understanding
surfaces which are two‐dimensional manifolds (or one‐dimensional complex manifolds).
z Riemann’s mapping theorem became the present form.
z “Riemann’s mapping theorem”
a one‐dimensional
one dimensional, simply
simply‐connectedconnected complex manifold is,iscomplex analytically, the same as one of three objects, namely, aRiemann surface, a complex plane, or a unit open disk.
z Little was yet studied about higher dimensional manifolds.
z It was aroundd theth middle
iddl off th
the 20th centurytththatt theth ddebatesb t onthe definition of manifolds were settled.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

Poincaré did his mathematical researches in such an era:z Three-body problem is difficult.
z Even for three-body problem on flat surface,
the space of coordinates and velocity of threepoints are a (2+2)×3=12 dimensional space.
z If we take the coordinate which has its origin
at the center of mass of three points, thedimension of space of state becomes 8.
z Due
D tto conservationti off angularl momentumtand energy, dimension of the space of stateb
become66, bbutt thi
this iis ttoo hihighh a didimensioni ttounderstand.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

z For some time,
time we will study circular restricted
three-body problem.
z Assume that the sun and Jupiter is moving
circularly around their center of mass,
and we try to describe the motion of the thirdplanet.
National Astronomical Observatory of Japan‡

BASIC codes for the threebody restricted problem

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

z For some time,
time we should study circular
restricted three-body problem.
z Assume that the sun and Jupiter is moving
circularly around their center of massmass, we try todescribe the motion of the third planet.
Reprinted from NASA Webpage‡

Basic codes for the circular
restricted three-body problem

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

z In circular restricted three
three‐bodybody problem,
problem
we can take the coordinate system inwhich circularly moving Sun and Jupiterare fixed.
z In this coordinate system, there are
centrifugal
t if l andd Coriolis
C i li fforce other
th th
than
gravitational force.
z the space of state that represents thethird planet is 2+2=4 dimensional space.
z This motion has a constant quantity called
Jacobi Integral that corresponds to energy,
th dimension
thedii off theth space off statet t iis 3.
3z By studying flows on three‐dimensional
manifolds, we can understand circular
restricted three‐body problem.

Basic codes for
restricted three-bodyproblems in rotating
frame.

Poincaré（1854－1912）

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

As to three‐dimensional manifolds:z Poincaré‐Hopf theorem holds.
In three dimensional problems, there are
more kind of stationary points that must
be considered.
considered If we should give ± sign(integers in general ) to each points andg the Euler
sum them all,, we getcharacteristic of that manifold.
z Euler characteristic of a compact threeddimensionall manifolds
f ld is alwayslzero.
z Are there any other invariants for three‐dimensional manifolds?
z The number that Betti（1823－1892）
defined is, sure to be an invariant.

Betti（1823 1892）
Betti（1823－1892）

Poincaré（1854－1912）

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi TsuboizPoincaré advanced Betti’s theory and constructed the
Homology theory.

z

He conjecturedjthat “anyy three manifolds whose
homology group are isomorphic to the three‐dimensional sphere are homeomorphic to three‐dimensional spheresphere” (1900)

z

Then Poincaré recognized that the proposition is false,and he constructed a counter‐example(1904).
counter example(1904)z

Poincaré demonstrated that this is in fact a counter‐examplel byb defining
d fi i “f“fundamentaldl groups”.
”z

And he proposed the problem “Is every simply‐
connected (trivial fundamental group), compact, three‐dimensional manifold homeomorphic to three‐dimensional sphere?
p”
Poincaré（1854－1912）

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

the counter‐examplethtl thatth t Poincaré
P ié constructedt t d isi called
ll d Poncare‐Phomology sphere.
Each polytope of the regular 120‐polytopes, one of 4‐dimensional
polyhedrons, is fundamental region of Poincaré‐homology sphere.

Poincaré‐homology spheres are now recognized to beimportant in the classification theory of three‐dimensional
manifolds.
‡

DIMENSIONS
http://www.dimensions-math.org/

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

Many mathematician,hwhoh are contemporary withh
Poincaré, studied about the figures of spaces.
Cantor(1845－1908) studiedcomparison of sets
and what topology is.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

Cantor discovered that there are manyy kinds of“infinities”.
He “demonstrated”
demonstrated that “rationalrational numbers are
isolated within real numbers”
The problem “what is number?” also came up.
Peano’s definition ofnatural numbers, anddefinition of real
numbers by
Dedekind’s cut.
Peano（1858－1932）

Dedekind（1831－1916）

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

Teijij Takagi
g went to Berlin and Gottingeng to studyy around
1900.
Japanese mathematical teaching and research isbased on the tradition ofHigh level mathematics (Wa(Wa-san)san)Teiji Takagi teached and studied
as the third professor ofmathematics
at e at cs at the
t e University
U e s tyof Tokyo.
Teiji Takagi’s research on classfield theory is highly appreciatedand his theory led to the proof of
Teiji Takagi（1875－1960）
http://www.ms.u-tokyo.ac.jp/summary/gallery.html
Fermat’s last theorem.
©copyright 2010- Graduate School of Mathematical Sciences, The University of Tokyo

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

In “Kaiseki Gairon”
Gairon”, there
are theories on real numberthat began to be recognizedworldwide.
This year is Teiji Takagi’s 50
year anniversary,anniversary so there will
be a citizen lecture on February
20th at the large lecture room atthe department of mathematical
study.
‡
http://mathsoc.jp/meeting/takagi50/index.html
By courtesy of Mathematical Society of Japan‡

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi‡

Whitney（1907－1989）

de Rham（1903 －1990）

http://www.math.ias.edu/people/past-faculty‡‡

Hopf（1894 － 1971）

Theory of manifolds
are paved by manymathematicians.
Thom（1923 － 2002）

Copyright(c)MFO
http://owpdb.mfo.de/detail?photo_id=4170

Milner（1931－ ）Copyright(c)MFO
http://owpdb.mfo.de/detail?photoID=9831

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

z Poincaré conjecturez Firstly, the same problem with 5 or higher
dimension was settled by Smale (1960).
z After that , by Freedman, 4‐dimensional
4 dimensional
Poincaré conjecture is proved(1982).
‡

Copyright(c) George M. Bergman, Berkeley
http://owpdb.mfo.de/detail?photo_id=5121

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

One of the reasons that the higher-dimensional
higher dimensional
manifolds are easy to handle is that theintersection of projection of two-dimensional
two dimensional disks
and the disk itself cancel.
study
d off manifolds
if ld bbegan bby considering
id i ffunctions
ion them, which are called Morse functions. Thenmathematiciansthti ittriedi d tto simplify
i lif ththeir
i gradientdi t flflows,or the maximal gradient lines when Morse functions
are seen as heighth i ht ffunctions.
ti
In other words
words, they studied manifolds by seeing
their shadows (functions) and thinking of thepatterns(flows) (solving ordinary differential
equations) on them.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

For three-dimensional
three dimensional manifolds
manifolds, it is shown that there aremany 3-dimensional manifolds with the same fundamentalgroupgp but not homeomorphic,
p , which led some ppeoplep tosearch for a counter-example of Poincaré conjecture.

Though there are not so many kinds of manifolds as above,
more invariantsii t were neededd d ffor ffurther
th classification.
lifi ti
It wasexpected that with such invariants mathematicians would be
able to distinguish counter-examples to Poincaré conjecture
conjecture.
On the other hand, it was proved that, all compact threedimensional manifolds can be made by connecting finite
number of pieces along spheres or two
two-dimensional
dimensional tori
tori.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

Thurston proposed thatg ggeometricalone of eight
structure can be in eachof such pieces
pieces, which iscalled geometrization
conjecture (early 1980s)
1980s).
If this is true, Poincaré
conjecturej t
iis ttrue.

‡

Copyright(c) George M. Bergman, Berkeley
http://owpdb.mfo.de/detail?photo_id=6119

ThurstonDecember 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

Thurston formulated the condition that a metricwith constant curvature (hyperbolic metric) can
exist in the piece. The proof is so massive that it
is called “Monster” theorem.
Hamilton came up with an idea‡
making use of Ricci flow, thesolution of partial differential
equations obtained fromdeformations of Riemann metrics
on manifolds,, to deform the metricon manifolds with positive
curvature into one with constantbyy courtesyy of international mathmaticians unioncurvature.

January 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View

G. Perelman has discovered certain
relations between Ricci flow and thedecomposed parts of certain class of
manifolds, and therefore showed thatfor any such decomposed parts, themetric on them always converges into
one of eight types
types.

Takashi Tsuboi‡

As a corollary, the 3-dimensional
Poincaré conjecture was verified.
G.Perelman
After this Perelman’s work, such atransformation of metric has become
pop lar as a ne
popularnew mathematical method
method.

byy courtesyy of Matthias Webe

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

For a coordinate near a point on the three‐dimensional manifold,
The metric can be represented as
While the Ricci curvature can be represented as
the following equation describes the variation of metric, the solutions ofwhich are called Ricci flow:(This is an equation for 3 x 3 symmetric matricesddepending
di on position
iti andd time)ti )December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

As such, the three-dimensional Poincaré conjecture wasdemonstrated hearing sounds, observing waves, and measuring how
heat is transferred (to solve partial differential equations) on
manifolds.
Since manifolds are developed to do calculations on them, methods tosolve
l partial
ti l diffdifferential
ti l equations
ti
on manifolds
if ld are studied
t di d from
f
their
th i
early history.
Theories constructed by Atiyah and Singer , Yau, and Donaldson aredidiscovered
d and
d the
th world
ld off manifolds
if ld are ddeepened.
dPerelman’s proof is also based on these fundamentals.
‡‡

Copyright: MFO

by courtesy of
Dr. Simon Donaldson

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

Summary up to herez In the study of mechanics (differential equations), theconcept of manifolds which is a generalization of
curved surfaces are formulated.
z To understand the figure of manifolds, Homology
groupsg
oups anda d fundamentalu da e a ggroupsoups aaree dedefined.
edz Poincaré proposed a problem that “Are any compact,
three-dimensional, simply-connected manifolds
homeomorphic to a three-dimensional sphere?”(Poincaré conjecture)z Thanks to developments of the theory of manifolds,y g differential equationsq
on manifolds,, Poincaré
analyzingconjecture was verified.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

Since Poincaré conjecturejof each dimension is verified,,attention is shifted from manifolds themselves to thestructure on manifolds.
Though studies on manifolds themselves are based on the
research on structures on manifolds, but now more kinds ofstructures are being studiedstudied.
Also for curved surfaces, studies on their complex structure
are now going onon.
On the other hand, there are a lot of recent studies on areal
forms on curved surfaces.
For three-dimensional manifolds, studies on the tangent
structure on them are now going on
on.
For four-dimensional manifolds, studies on complex
structures and symplectic structure are now going onon.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

Studies of areal forms on curved surfaces are
related to circular restricted three-body problem.
Solutions to the circular restricted three
three-bodybody problem
are known to be flows on three-dimensional space ofstate,, and the volume is conserved.

For such a flow, if curved surfaces are crossed,
the orbit from the surface returns to the surface.
This is known to bea recursive map.
Foro a recursive
ecu s e map,
ap, tthe
ea
area
ea is
s co
conserved.
se ed
Poincaré watched this phenomena.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi‡

It is visualized
as right figure.

ArnoldSmall denominators andproblems of stability of
motion in classical andcelestial mechanics

Figure 6 , Page 93"Small denominators and problems of stability of motion in classical and celestial mechanics"
Vladimir I Arnol'd, 1963, Russian Mathematical Surveys,18, pp85-191
By courtesy of London Mathematical Sosiety

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

PoincaréPiéh
hadd a novell id
idea on theh circulari lrestricted three-body problem.
Poincaré came to thinking about observing thedifference of recursive maps of flows, varying theparameterpa
a ete oof ccircular
cu a restricted
est cted tthree-bodyee body
problem, namely, the ratio between the mass of
Jupiterp
and the Sun,, from zero.
Though I can’t explain the details, but if you see
the fraction of them, it is almost the same as thefollowing figures.
BASIC source codeDecember 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

When you see this recursive maps, you can observe that there aremanyy circles with the same central p
points, and that there are signs
g ofchaos at areas filled with many colors.

In fact, there are clear explanation with the Kolmogorov-ArnoldMoser theory, and Aubry-Mather theory.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

many contemporary mathematical studies arefocused on the structure on manifolds.
As an example of structures that can easily be visualized,
there are foliated structures, and I am also studying them.
There are animated figuresfavailable:
http://faculty.ms.u-tokyo.ac.jp/users/showroom/

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboiz Now,
Now let us return to the beginning of the today’s
today slecture.
z Today I talked about the Poincaré conjecture
conjecture, but
in the beginning the motivation is that we wanted
to investigate on the spaces of statesstates. With theknowledge about them, we can restrict the way
flows flow and we can get the information
information.
z Whether curved surfaces are the same or not isis,you may think, is distinguished if we carefully
observe themthem.
z But how can we distinguish whether spaces or
structures on spaces are the same or not ?

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

the samez To determine whether manifolds are the same or
not is not easy.
z In principle, the definition of “the same” is needed.
z If two manifolds are homeomorphic(the same
figure) that there exists a continuous map from one
manifold to another that has the inverse map that iscontinuous.
continuous
z To demonstrate two manifolds are the same,, youyshould either make up such a map, or show thatmaking such a map is possible.
z To demonstrate they are not the same, you should
show making such a map is not possible
possible.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

The difference: how to distinguish shapesz To do this, you can use a reduction to absurdity.
z First y
you can find out some number that is thesame if two manifolds are homeomorphic.
z When you find this number above are different
for two manifolds, you can conclude these are
not theh same manifolds.
if ld
z Within my talk so far
far, Euler characteristic andfundamental groups are such numbers.
z Numbers like these are called invariants.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

definition of spatial invariants
A certain number is defined for each structure on
the space, and if you calculate them and get thesame quantity for the same space, irrespective of
the structure on them
them, the number is called spatial
invarinats.

Definition of invariants of spatial structure
Suppose one is given a space with certain structure
structure,(then one defines some more structure), and if you
calculate certain qquantityy defined for this structure
and always get the same result for spaces with thesame structure (and that value is independent offi
finer
structure)t t ) we callll thi
this quantity
tit ththe iinvariants
i t off
spaces with structure.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

Elegant theorem 1．
Invariants defined using different ways ofdefinition take the same value.
“essential theorem”
Elegant theorem 2.
Two spaces (or structures) with the sameinvariant is true.
“classification theorem”

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

Example :linear algebra

Let us call that ｎ×ｎ matrix Ａ，Ｂ
Ａ Ｂ are “equal”
equal when there exist
n-dimensional proper matrices P,Q such that PAQ = B.
If there are nonzero component among the （ｎＣｋ）２
determinants of submatrices of order k of a matrix and the（ｎＣｋ＋1）２ submatricesb ti
off order
d k+1 are allll zero, th
the rank
k offthat matrix is defined to be k.

“classification theorem”
That A and B are “the same” and that the rank of
A and the rank of B is equal is equal
equal.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

For curved surfaces,
surfaces dividing them with triangles
triangles, the value
Number of vertices – number of edges + number of faces
is independent of separation.
This value is called Euler number.

“Morse’s theorem”
For functions on curved surfaces
surfaces, consider critical points composed ofminimum points, saddle points, and maximum points and the value
Number of minimum points – number of saddle points + number ofmaximumi
pointsi t
is equal to Euler characteristic (independent of the function
considered)“Poincaré Hopf theorem”
For flows on curved surfaces and if their stationary points
are source, saddle, and sink, the value
number of sources – number of saddles + number of sinks
is equal to Euler characteristic(independent of the flow considered)December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

Theorem aboutTh
b t
Curved surfaces“Gauss-Bonnet theorem”
The integral of curvature of a closed curved
surface in three dimensional Euclidean space isequal to 2π times its Euler characteristic.
characteristic“classification
classification theorem”
theorem
ClosedCl
d surfacesfiin ththree didimensional
il
Euclidean space are, if their Euler
characteristics are equal, homeomorphic.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

Th
Theoremoff threethdimensionaldiil manifolds
if ld“Poincaré
Poincaré-Hopf
Hopf theorem
theorem”
If a flow on three
three-dimensional
dimensional compact
manifolds has finite stationary points, the sumof indices of stationary points is zero
zero.
Poincaré conjecture
conjecture”“Poincaré
If the fundamental group of a threedimensional compact manifold is trivial, themanifold is homeomorphic to a threedimensional sphere.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

How tto distinguishHdi ti
i h diff
differentt shapes:hDefine the invariants and calculate them.
This is not necessarily so difficult.
We simply define some quantity that we
think would be useful.
Hard point is that the quantity so definedcorrectly gives “thethe same value for the same
figures”.
It is not until the time when there are a
sufficient ways to distinguish figures that we
can tackle the classification problem.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

things I sometimes experience as a mathematical researcher

The most is that the invariant I defined with awith pains is, actually, always zero.
This is not a waste.
waste This is a theorem that statessome kind of objects have certain nature.

The second kind is that the value I havedefined is, actually, already known.
This is a theorem
theorem.
But I rarely find the definition of a new andnontrivial invatiant.

December 14th, 2010 Global Focus on Knowledge Creating Mathematics
Lecture 12 How to Distinguish Different Shapes, and the Mathematical Point of View Takashi Tsuboi

References
Teiji Takagi
http://mathsoc.jp/meeting/takagi50/index.html
Milnor：Poincaré conjecture
http://faculty.ms.u-tokyo.ac.jp/users/kokaikoz/milnor-j.pdf
DIMENSIONS
http://www.dimensions-math.org/
The entropy formula for the Ricci flow and its geometric applications, 2002
http://arxiv.org/abs/math.DG/0211159
Ricci flow with surgery on three-manifolds, 2003
http://arxiv.org/abs/math.DG/0303109
Finite extinction time for the solutions to the Ricci flow on certain threemanifolds, 2003
http://arxiv.org/abs/math.DG/0307245
pg