Global in terms of mathematics - hokkyodai.ac.jp
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“Local” and “Global”in terms of mathematics
AOKI Masao
29 May 2019
AOKI Masao “Local” and “Global” 29 May 2019 1 / 19
Is mathematics useful?
Do you like mathematics?
Why do we study mathematics?
AOKI Masao “Local” and “Global” 29 May 2019 2 / 19
Is mathematics useful?
Do you like mathematics?
Why do we study mathematics?
AOKI Masao “Local” and “Global” 29 May 2019 2 / 19
Is mathematics useful?
Does mathematics help you solving (regional) problems?
No! (at least directly.)
But
Mathematics lies at the base of various studies.
Mathematics provides new viewpoints.
AOKI Masao “Local” and “Global” 29 May 2019 3 / 19
Is mathematics useful?
Does mathematics help you solving (regional) problems?
No! (at least directly.)
But
Mathematics lies at the base of various studies.
Mathematics provides new viewpoints.
AOKI Masao “Local” and “Global” 29 May 2019 3 / 19
Is mathematics useful?
Does mathematics help you solving (regional) problems?
No! (at least directly.)
But
Mathematics lies at the base of various studies.
Mathematics provides new viewpoints.
AOKI Masao “Local” and “Global” 29 May 2019 3 / 19
Mathematics lies at the base
statistics
physics
Natural Science
biologygeology
astronomychemistryeconomics
Social Science
jurisprudence
sociology
informatics
Engineering
architectureelectronics
mechanical engineering
music
art
literature psycology
education
Mathematics
AOKI Masao “Local” and “Global” 29 May 2019 4 / 19
Mathematics lies at the base
statistics
physics
Natural Science
biologygeology
astronomychemistryeconomics
Social Science
jurisprudence
sociology
informatics
Engineering
architectureelectronics
mechanical engineering
music
art
literature psycology
education
Mathematics
AOKI Masao “Local” and “Global” 29 May 2019 4 / 19
Mathematics lies at the base
statistics
physics
Natural Science
biologygeology
astronomychemistry
economics
Social Science
jurisprudence
sociology
informatics
Engineering
architectureelectronics
mechanical engineering
music
art
literature psycology
education
Mathematics
AOKI Masao “Local” and “Global” 29 May 2019 4 / 19
Mathematics lies at the base
statistics
physics
Natural Science
biologygeology
astronomychemistryeconomics
Social Science
jurisprudence
sociology
informatics
Engineering
architectureelectronics
mechanical engineering
music
art
literature psycology
education
Mathematics
AOKI Masao “Local” and “Global” 29 May 2019 4 / 19
Mathematics lies at the base
statistics
physics
Natural Science
biologygeology
astronomychemistryeconomics
Social Science
jurisprudence
sociology
informatics
Engineering
architectureelectronics
mechanical engineering
music
art
literature psycology
education
Mathematics
AOKI Masao “Local” and “Global” 29 May 2019 4 / 19
Mathematics lies at the base
statistics
physics
Natural Science
biologygeology
astronomychemistryeconomics
Social Science
jurisprudence
sociology
informatics
Engineering
architectureelectronics
mechanical engineering
music
art
literature psycology
education
Mathematics
AOKI Masao “Local” and “Global” 29 May 2019 4 / 19
Mathematics lies at the base
statistics
physics
Natural Science
biologygeology
astronomychemistryeconomics
Social Science
jurisprudence
sociology
informatics
Engineering
architectureelectronics
mechanical engineering
music
art
literature psycology
education
Mathematics
AOKI Masao “Local” and “Global” 29 May 2019 4 / 19
Mathematics lies at the base
statistics
physics
Natural Science
biologygeology
astronomychemistryeconomics
Social Science
jurisprudence
sociology
informatics
Engineering
architectureelectronics
mechanical engineering
music
art
literature psycology
education
Mathematics
AOKI Masao “Local” and “Global” 29 May 2019 4 / 19
Mathematics lies at the base
statistics
physics
Natural Science
biologygeology
astronomychemistryeconomics
Social Science
jurisprudence
sociology
informatics
Engineering
architectureelectronics
mechanical engineering
music
art
literature psycology
education
Mathematics
AOKI Masao “Local” and “Global” 29 May 2019 4 / 19
Mathematics lies at the base
statistics
physics
Natural Science
biologygeology
astronomychemistryeconomics
Social Science
jurisprudence
sociology
informatics
Engineering
architectureelectronics
mechanical engineering
music
art
literature psycology
educationMathematics
AOKI Masao “Local” and “Global” 29 May 2019 4 / 19
Mathematics provides new viewpoints
Population of Japan
AOKI Masao “Local” and “Global” 29 May 2019 5 / 19
Drawing graphs→ based on the idea of calculus
Mathematics provides new viewpoints
Population of Japan
AOKI Masao “Local” and “Global” 29 May 2019 5 / 19
Drawing graphs→ based on the idea of calculus
Mathematics provides new viewpoints
Population of Japan
AOKI Masao “Local” and “Global” 29 May 2019 5 / 19
Drawing graphs→ based on the idea of calculus
Mathematics to describe “region”
Key words
“Local” and “Global”
Local data and global data
Integration
Manifold (or Variety, Stack) – Local data and theirrelations determine global data.
AOKI Masao “Local” and “Global” 29 May 2019 6 / 19
Mathematics to describe “region”
Key words
“Local” and “Global”
Local data and global data
Integration
Manifold (or Variety, Stack) – Local data and theirrelations determine global data.
AOKI Masao “Local” and “Global” 29 May 2019 6 / 19
Mathematics to describe “region”
Key words
“Local” and “Global”
Local data and global data
Integration
Manifold (or Variety, Stack) – Local data and theirrelations determine global data.
AOKI Masao “Local” and “Global” 29 May 2019 6 / 19
Local and global
Global – Observe the whole object.
Local – only the neighborhood of a certain point.
rrr
P
AOKI Masao “Local” and “Global” 29 May 2019 7 / 19
Local and global
Global – Observe the whole object.
Local – only the neighborhood of a certain point.
rrr
P
AOKI Masao “Local” and “Global” 29 May 2019 7 / 19
Local and global
Global – Observe the whole object.
Local – only the neighborhood of a certain point.
r
rr
P
AOKI Masao “Local” and “Global” 29 May 2019 7 / 19
Local and global
Global – Observe the whole object.
Local – only the neighborhood of a certain point.
r
r
r
P
AOKI Masao “Local” and “Global” 29 May 2019 7 / 19
Local and global
Global – Observe the whole object.
Local – only the neighborhood of a certain point.
rr
r
P
AOKI Masao “Local” and “Global” 29 May 2019 7 / 19
Local and global
Global – Observe the whole object.
Local – only the neighborhood of a certain point.
rrr
P
AOKI Masao “Local” and “Global” 29 May 2019 7 / 19
Example : local and global
Closed curve on a plane.
AOKI Masao “Local” and “Global” 29 May 2019 8 / 19
Example : local and global
Length, area — global data.
Area S
Length L
AOKI Masao “Local” and “Global” 29 May 2019 8 / 19
Example : local and global
Tangent lines, curveture — local data.
AOKI Masao “Local” and “Global” 29 May 2019 8 / 19
Example : local and global
Tangent lines, curveture — local data.
Neighborhood of this point.
AOKI Masao “Local” and “Global” 29 May 2019 8 / 19
Example : local and global
Tangent lines, curveture — local data.
Neighborhood of this point.
AOKI Masao “Local” and “Global” 29 May 2019 8 / 19
Example : local and global
Tangent lines, curveture — local data.
Neighborhood of this point.
AOKI Masao “Local” and “Global” 29 May 2019 8 / 19
Integration
a b
S
y = f (x)
Value of a function — local data.
Area – global data.
IntegrationGet global data from the collection of local data.
AOKI Masao “Local” and “Global” 29 May 2019 9 / 19
Integration
a b
Value of a function — local data.
Area – global data.
IntegrationGet global data from the collection of local data.
AOKI Masao “Local” and “Global” 29 May 2019 9 / 19
Integration
a b
Value of a function — local data.
Area – global data.
IntegrationGet global data from the collection of local data.
AOKI Masao “Local” and “Global” 29 May 2019 9 / 19
Integration
a b
Value of a function — local data.
Area – global data.
IntegrationGet global data from the collection of local data.
AOKI Masao “Local” and “Global” 29 May 2019 9 / 19
Integration
a b
S =
∫b
a
f (x) dx
Value of a function — local data.
Area – global data.
IntegrationGet global data from the collection of local data.
AOKI Masao “Local” and “Global” 29 May 2019 9 / 19
Integration
a b
S =
∫b
a
f (x) dx
Value of a function — local data.
Area – global data.
IntegrationGet global data from the collection of local data.
AOKI Masao “Local” and “Global” 29 May 2019 9 / 19
Length and area
AOKI Masao “Local” and “Global” 29 May 2019 10 / 19
Length and area
L =
∫b
a
√x ′(t)2 + y ′(t)2 dt
S =
∫b
a
{x(t)y ′(t) − x ′(t)y(t)} dt
AOKI Masao “Local” and “Global” 29 May 2019 10 / 19
Manifold
Hard to grasp the whole.
Cannot draw picture.
Cannot measure or calculate.
⇒ Easy to study locally.
Set coordinates.
Algebra and analysis work.
AOKI Masao “Local” and “Global” 29 May 2019 11 / 19
Manifold
Hard to grasp the whole.
Cannot draw picture.
Cannot measure or calculate.
⇒ Easy to study locally.
Set coordinates.
Algebra and analysis work.
AOKI Masao “Local” and “Global” 29 May 2019 11 / 19
The Earth is ...
a sphere, locally a plane.
AOKI Masao “Local” and “Global” 29 May 2019 12 / 19
The Earth is ...
Example : the EarthThe Earth (Globe) is a sphere (globe).⇒ You may think earth is flat if you look locally.
Locally approximate a sphere by a plane.
Draw maps.
Measure length and area on a plane.
Use coordinates to point locations.
AOKI Masao “Local” and “Global” 29 May 2019 13 / 19
The Earth is ...
Example : the EarthThe Earth (Globe) is a sphere (globe).⇒ You may think earth is flat if you look locally.
Locally approximate a sphere by a plane.
Draw maps.
Measure length and area on a plane.
Use coordinates to point locations.
AOKI Masao “Local” and “Global” 29 May 2019 13 / 19
Manifold
Locally approximate by Euclidian spaces.
“twist” in intersections.
AOKI Masao “Local” and “Global” 29 May 2019 14 / 19
Manifold
Locally approximate by Euclidian spaces.
“twist” in intersections.
AOKI Masao “Local” and “Global” 29 May 2019 14 / 19
Manifold
Locally approximate by Euclidian spaces.
“twist” in intersections.
AOKI Masao “Local” and “Global” 29 May 2019 14 / 19
Manifold
Locally approximate by Euclidian spaces.
“twist” in intersections.
X
Ua
Ub
AOKI Masao “Local” and “Global” 29 May 2019 14 / 19
Manifold
Locally approximate by Euclidian spaces.
“twist” in intersections.
X
Ua
Ub
AOKI Masao “Local” and “Global” 29 May 2019 14 / 19
Patching
Patching two maps requires twist.
AOKI Masao “Local” and “Global” 29 May 2019 15 / 19
Local approximation and patching
X
Ua Ub
Rab
manifold
local approximation
patching
AOKI Masao “Local” and “Global” 29 May 2019 16 / 19
Local approximation and patching
X
Ua Ub
Rab
manifold
local coordinate
coordinate transformation
AOKI Masao “Local” and “Global” 29 May 2019 16 / 19
Local approximation and patching
X
Ua Ub Uc Ud · · ·
Rab
manifold
local coordinate system
— Cover the whole.
coordinate transformation
AOKI Masao “Local” and “Global” 29 May 2019 16 / 19
Local approximation and patching
X
Ua Ub Uc Ud · · ·
Rab Rbc
Rac Rcd · · ·
manifold
local coordinate system
— Cover the whole.
coordinate transformations
— For all intersections.
AOKI Masao “Local” and “Global” 29 May 2019 16 / 19
Local approximation and patching
X
U
R
manifold
local coordinate system
coordinate transformations
AOKI Masao “Local” and “Global” 29 May 2019 16 / 19
Viewpoint of modern geometry
R U X
PrincipleLocal coordinate system and data of coordinatetransformations detrmine the whole manifold.
AOKI Masao “Local” and “Global” 29 May 2019 17 / 19
Viewpoint of modern geometry
Thus we can use algebraic or analytic method to studymanifolds, even if it no longer lies in a Euclidian space.
If we introduce more general concepts of “covering” and“patching”, we can treat much incredible objects such asdifferential orbifolds or algebraic stacks. We cannot drawpictures of them, but they are new “spaces” wheremathematics works.
For example, “Hom stacks” in my thesis.
AOKI Masao “Local” and “Global” 29 May 2019 18 / 19
Viewpoint of modern geometry
Thus we can use algebraic or analytic method to studymanifolds, even if it no longer lies in a Euclidian space.
If we introduce more general concepts of “covering” and“patching”, we can treat much incredible objects such asdifferential orbifolds or algebraic stacks. We cannot drawpictures of them, but they are new “spaces” wheremathematics works.
For example, “Hom stacks” in my thesis.
AOKI Masao “Local” and “Global” 29 May 2019 18 / 19
Sources
Sources of the images
the Earth (the Blue Marble) : NASA / Apollo 17 crewPublic Domain, copied from Wikimedia Commons.
maps : c⃝OpenStreetMap contributorsCC BY-SA, https://www.openstreetmap.org/
Typeset with LATEX and Beamer, Tikz packages.
AOKI Masaoaoki.masao@h.hokkyodai.ac.jp
AOKI Masao “Local” and “Global” 29 May 2019 19 / 19
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