Global Focus on Knowledge ～Creating Mathematics～
L tLecture
ffourCo‐evolution
Cl ti off Numbers
N b anddGeometry
Creating words, Originating worlds
2009.10.29
The figures,g,pphotos and movingg images
g with ‡marks attached belongg totheir copyright holders. Reusing or reproducing them is prohibited unlesspermission is obtained directly from such copyright holders.

Establishment of arithmetic
geometry(late 20th century)• Research on the figures defined byequations with integer coefficient,coefficient
like Xn + Yn = Znor• Research on the properties ofintegers using such figures

The relation between number
and figure‐coordinatesy• Numbers and figures areth twothet sides
id offthe same thing
Descartes 1637
『Discourse on the method』

(7,5)
5x7

Descartes (1596.3.31 ‐ 1650.2.11)
Reprinted fromhttp://ja.wikipedia.org/wiki/デカルト(2010/09/03)

High dimensional spaces
High‐dimensional• Also in the case of n>3,
A set of n numbers(x1,x2,…,xn)
indicates
A point on a n‐dimensional
space(a,b,c)cba

High dimensional space =”theworld of linear algebra”• invisible• Based on simultaneous linear equation
reduced to linear algebra
mathematically well known

Paradigm shift in geometry
researchFigures
gsubjectin planes or spaces

SpacepitselfRiemann 1854
on the hypothesis which lies in the“onfundamental of geometry”

Elliptic curves and Fermat’s last
theorem• In the case of n = 3,, 4
Demonstrated by studying the rational number
solution of the equationsqwhich determine certain
elliptic curves
ｙ2 = x3 ‐ x((Fermat))ｙ3 = x3 ‐ 1(Euler)• In the case that n is a prime number and larger thanor equal to 5
Demonstrated by showing that the equation that
determines an elliptic curve ｙ2 = x (x‐an) (x‐cn)
does not exist

Riemann (1826.9.17 ‐ 1866.7.20)
Reprinted fromhttp://ja.wikipedia.org/wiki/ファイ
ル:Georg_Friedrich_Bernhard_Riemann.jpeg(2010/09/03)

Paradigm shift in physics
Quantum mechanics

Reprinted fromhttp://www1.iwate‐ed.jp/tantou/joho/material/tajuurokou/index.htm(2010/09/03)

elementary particles

Super-K
Kamio
okandde‡‡Reprinted fromhttp://www‐sk.icrr.u‐tokyo.ac.jp/sk/gallery/index.html(2010/09/03)

Paradigm shift in biology
molecular biology

Reprinted fromhttp://commons.wikimedia.org/wiki/File:Hauskatze_in_Abendsonne.jpg(2010/09/03)

genes

Paradigm shift in geometry
modern mathematics

Figures
gSpacein planes or spaces

itselfmanifolds

Where the paradigm shift ingeometry come from?
• Riemann surfaces• Gauss’ theory on curved surfaces• Non‐Euclidean
N E lidgeometriesi• Physics
Ph i............

Multi‐valued function(somewhat contradictory name)• function ： the rule that relates a point((=variable)) to one value• The square root of z
if z is a real number
you can take the positive one
if z is a complex number
impossible to specify
which one you should take

Riemann surfaces• You can consider the complex plane
as the superposition of two sheetson the one sheeton the other ‐• The two sheets are
actually connected
Reprinted fromhttp://en.wikipedia.org/wiki/File:Riemann_sqrt.jpg(2010/09/03)

Establishment of the concept offunction• Consider the “double” covering ofcomplex plane:“multi‐valued” function(somewhat(contradictory name) now consistent• “double”“d bl ” covering
i is
i sticking
i ki out offcomplex plane = Riemann surface

Gauss’ theory on curved surfaces
Gauss
he discovered a geometricproperty of curved surfaces,
surfacesand the property is unrelated toexternal 3‐dimensional spacespStudying on the space?
• The description of the object
figures in unknown spaces
how to make up?

• Method for the study
what does it mean to understand spaces?

Set as a term in mathematics• By regarding a set as a point or an element ofsome other set,,
we can create a new objectThe emergence of “modernmodern mathematics”
mathematics=A “math” based on the concept of “set andtopology”
p gyMathematics as a language innatural sciences• the universe ...is written in the language ofmathematics
Galileo Galilei “the Assayer”• Modern mathematics is written by the wordsof set theory

Paradigm shift
The common sense：changes with times
How times
Htichange：hwhen a new generation appears,appears newways of thinking become popular“Modernization”
Modernization in math education(1971‐79)
education(1971 79)‡Reprinted fromhttp://www.dainippon‐tosho.co.jp/math_history/history/age03_el/age03_el_03.html#main(2010/09/03)
By courtesy of Dainippontosho Co Ltd,.

Dedekind cut (1858.11.24)(1858 11 24)• A method that treats real number regarding
rational numbers as elements of set• A real number x is a set of rational numbers｛r | r is rational and r < x }• Hence,
Hence a real number equation x = y meansr<xr<yfor any given rational number r

manifolds = geometrical objects
patchwork of already known spaces

create a new spacemanifolds = geometrical objects• A space that cannot be embedded in
an n‐dimensionaln dimensional space Ｒｎ on whichtthee coo
coordinates
d ates aaree aalreadyeady dedefineded• The Einstein’s universe：
fundamentals of general relativity(1916)

Riemann sphere
i

Q

1Ｚ

1i

P

invariant = methods in geometry
dimension, for example,
independent of the construction or theexpression,

They are the same value for thesame spacedistinguish spaces using invariantsEuler characteristic for polyhedra• (number of vertices)‐(number ofg ) (of faces))edges)+(number• 4 – 6 + 4 = 2 9 – 16 + 9 = 2

Elliptic curves
x,yreall

Reprinted fromhttp://upload.wikimedia.org/wikipedia/commons/5/5b/ECexamples01.png(2010/09/03)

Elliptic curves as Riemann surfaces• ｙ2=ｘ3‐ｘ

x , y complex number

Elliptic curves=
opposite sides
are identified

Euler characteristic for elliptic curves

Euler characteristic for elliptic curves

2
1 ‐ 2 + 1 = 0

Euler characteristic for Riemann surfaces

2‐2g

g：genusg=2

Elliptic curves=Riemann surface
with genus 1

Riemann surface with genus 0 =Riemann sphereHomology• Riemann surface X with genus g2g –dimensional
dimensional space H1(X)• Higher‐dimensional manifolds Ｘ
space Hq(Ｘ)•1900

Poincaré“Analysis Situs”

Homology of elliptic curves
Ｅ

β

=α

opposite sides are identified

H1(Ｅ)= ｛ aα+bβ|α+bβ| a, b real｝
l｝

Reprinted fromhttp://ja.wikipedia.org/wiki/ファイル:Henri_Poincare.jpg(2010/09/03)

Poincaré (1854.4.29 ‐ 1912.7.17)

Poincaré conjecture• “a space that has the same
homology as 3‐sphere
3 sphere”
Is it identical to 3‐spherep
as a spacep?
construction and detailed descriptionof a counterexample(P i(Poincaréé (1904))

Poincaré conjecture• A space has the samef dfundamentalt l group as 3‐sphere3 h
it is identical to 33‐spheresphereas a space((Perelman(2003))())

hierarchical structure of nature
Matter of various sizes‡

The size of matterphysics deals withranges from 10‐30
cm to 1030cm

By courtesy of Yasushi Suto, school of science, the university of Tokyo

Hierarchical structure inmathematics• Individual number and figure• Systems and spaces of numbers .... sets
• Systems of numbers and relations between
pcategories
gspaces....
• mappings that associate mathematical objects
with other kinds of objects ....functors

Hierarchical structure inmathematics crude objects

numbers
0eπfigures

Hierarchical structure inmathematics Sets

number systems

spacesRational
number2‐adic2
adic number
3‐adic numberRealnumber....

....

Hierarchical structure inmathematics category
rational number
and the category of its
extensions

equiv‐ The absoluteanddareadded to the world ofrational number
is added to theworld of rational number

The world of rational
numberalentGalois groupof rational
number field

GaloistheoryGQ

Extension with equations• ｘ2 + 1 = 0a+b(a,(a b rational number)• ｘ2 ‐ 2 = 0
a+b(a, b rational number)Reprinted fromhttp //ja wikipedia org/wiki/ファイル E Galois Portrait No 1 jpg(2010/09/03)
http://ja.wikipedia.org/wiki/ファイル:E._Galois_Portrait_No.1.jpg(2010/09/03)

Galois (1811.10.25‐1832.5.31)

The number systemy(the first half of the 20th century)
rational number andits extensions

2‐adicnumber andits
extensions

3‐adicnumberand its
extensions5‐adicnumber
and its
extensions

7‐adicnumber
and its
extensions........

real
number andcomplex number• The fundamental framework of modern number
theory, that enables one to reach even for thesettlement of Fermat’s last theorem,, was originated.
glocal global principle
local‐global
The absolute
Thb lGalois
G l i group off
rational number field GQ
The absolute
Galoisl group off
2‐adic number
field GQ2
The absolute
Galois group of
3‐adic number
field GQ3

The absolute Galois group ofreal number field
GR=｛1, complex conjugate｝
The absolute
Galois group of
5‐adic number
field GQ5

・・

....

….

local‐global principle(in the case that xn‐1)• ｎ , a 1: natural number, coprime
there are infinite number of prime
numbers that leave a remainder of awhen divided by n :theorem on arithmetic
progressions (1837)

remainder when divided by 4:
2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 37, 41,43, 47, 53, 59, 61, 67, 71, 73, 79, 83, ....

Dirichlet (1805.2.13–1859.5.5)
Reprinted fromhttp://ja.wikipedia.org/wiki/ファイル:Peter_Gustav_Lejeune_Dirichlet.jpg(2010/09/03)

local global principle
local‐global
The absolute
Thb lGaloisG l i group off
rational number field GQ
The absolute
Galoisl group off
2‐adic number
field GQ2
The absolute
Galois group of
3‐adic number
field GQ3

The absolute Galois group ofreal number field
GR=｛1, complex conjugate｝
The absolute
Galois group of
5‐adic number
field GQ5・・

....

….

The Hierarchical structure ofmathematics Functor
The category ofmanifolds

The categoryof linear space

RnHomology
H1(E)Hq((X))

The world of arithmetic geometry
The categoryof ringTh category offTheGalois
representations

The world ofnumber and formula
space definedby equations

the categoryof scheme

etale cohomology

The world of arithmetic geometryTh category offTheThe categoryGalois
linearof ringalgebra representations
algebraggeometryy

local and global

The categoryof scheme

Elliptic curves and Galois representation
elliptic curves

ＥGalois representation
pH1((Ｅ,, Qp)= ｛ aα+bβ | a, b are p-adic number ｝

Elliptic curves and Galois representation
The “same” Galois representation H1(Ｅ, Qp)

the “same”
same elliptic curve Ｅ(Faltingsg 1983)establishment of arithmetic geometry

similarities between number andformula• Division with remainders32 ÷

5= 6 remainder 2ｘ5 ÷ (x2 + 1) = x3 ‐ x remainder xsimilarities between numberand formula• Prime number decomposition
15624 = 23 ・ 32 ・ 7 ・ 31
x6 ‐ 1 = (x ‐ 1)(x + 1)(x2 ‐ x + 1)
(x2 + x + 1)

Similarities between number andpoint
a function on the number line
= 1 + x + x2 + ....

T l expansion
Taylorithe behavior around the point x = 0

Similarities between number andpoint= 1 + p + p2 + ....
p‐adicpadic expansion
Spec Z2・・ 3 5 ・ ・ ・prime number …point・ ・Integer … function

number and point duality
function =determines the value at a point
point = determines the value of function= 1 + p + p2 + ....
Spec Z2・・ 3 5 ・ ・ ・・ ・

prime number …point
Integer … function

The proof of Fermat
Fermat’ss last theorem
1①

自明でない(a, b, c)(a,b,c)
Nontrivial解
solution

楕円曲線
E: ｙ2 = x (x‐an) (x‐cn)
Elliptic
p curve
2②Corresponding
Cdi 保型形式
automorphic
hi formsf
対応する3③

矛盾
Contradiction

The settlement of Taniyama‐
Shimura conjecture
Elliptic curves

Automorphicforms

Galois representation

The world of arithmetic geometryTh category off
TheThe category
Galoislinearof ringalgebra representation
algebraggeometryy

local and global

The categoryof scheme

The settlement of Taniyama‐
Shimura conjecture
Elliptic curves

Automorphicforms

Galois representation