Why do mathematicians make things so complicated?

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Why do mathematicians make

things so complicated?

Zhiqin Lu, The Math Department

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Why do mathematicians make things so complicated? Introduction

What is Mathematics?

24,100,000

answers from

Goog

l

e.

Such a FAQ!

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What is Mathematics?

24,100,000

answers from

Goog

l

e.

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What is Mathematics?

24,100,000

answers from

Goog

l

e.

Such a FAQ!

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From Wikipedia

Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns,[2][3] formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.[4]

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An example

The Real World

vs.

The Math World

How to become a millionaire in a

month?

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An example

The Real World

vs.

The Math World

How to become a millionaire

in a

month?

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An example

The Real World

vs.

The Math World

How to become a millionaire in a

month?

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An example

Fact/Secret/Assumption: Most interest checking accounts generate interest at least one cent a

month.

Here is How:

1 Open 100,000,000 interest checking accounts

and deposit one cent to each account;

2 Wait for a month.

The profit?

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An example

month.

Here is How:

2 Wait for a month.

The profit?

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An example

month.

Here is How:

2 Wait for a month.

The profit?

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An example

month.

Here is How:

2 Wait for a month.

The profit?

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An example

month.

Here is How:

2 Wait for a month.

The profit?

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An example

...and that is not the end of the story...

Mathematicians like to say

Let n → ∞ (inf inity)

If we let the number of checking accounts go to infinity, what will happen?

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An example

...and that is not the end of the story... Mathematicians like to say

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An example

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An example

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An example

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An example

Since that is not possible, we get the following result by Reductio ad absurdum (proof by contradiction).

Theorem

No banks can afford a free $0.01 interest. (in the math world)

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An example

Theorem

No banks can afford a free $0.01 interest.

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An example

Theorem

No banks can afford a free $0.01 interest. (in the math world)

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Summary

I am going to talk about

1 Why everything has to be done in an indirect

way?

2 The power of symbols/abstractions.

3 How do we choose a problem/project to work

on?

4 Why do we care about other sciences? 5 Use of Computer.

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Summary

way?

on?

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Summary

way?

on?

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Summary

way?

on?

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Summary

way?

on?

4 Why do we care about other sciences?

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Summary

way?

on?

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Why do mathematicians make things so complicated? My field

Mathematics

Differential Geometry

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Mathematics

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Mathematics

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1 One of my projects is in the mathematical

aspects of Super String Theory.

2 It is related to the Mirror Symmetry.

3 Two Universes, quite different, but have the

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Figure: Brain Greene, The Elegant Universe, NY Times Best Selling Book.

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Why do mathematicians make things so complicated? Why everything has to be done in an ... indirect way?

A problem in Math 2E.

Triple integrals-A Problem in Math 2E

Compute

Z Z Z W

xdxdydz,

whereW is the region

bounded by the planes

x= 0, y = 0, and z = 2, and the surface

z =x2₊_y2 _{and lying in}

the quadrant

x≥0, y ≥0.

N II IV

~

N >< II II >< 0 N + ~ '<:

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Triple integrals-A Problem in Math 2E

Compute

Z Z Z W

xdxdydz,

the quadrant

x≥0, y ≥0.

N II IV

~

N >< II II >< 0 N + ~ '<:

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Triple integrals-A Problem in Math 2E

Compute

Z Z Z W

xdxdydz,

the quadrant

x≥0, y ≥0.

N II IV

~

N >< II II >< 0 N + ~ '<:

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How to compute integrations over an

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Figure: From the internet. It is the intersection of the quintic Calabi-Yau threefold to our three dimensional space.

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How to study high dimensional geometric object? Use Calculus;

PDE, functional analysis, complex analysis, etc

Use Linear Algebra; Lie algebra, commutative algebra, algebraic topology, etc

Use the results in all other mathematics fields. Euclidean Geometry methods usually do not apply.

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How to study high dimensional geometric object? Use Calculus; PDE, functional analysis, complex analysis, etc

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Use Linear Algebra;

Lie algebra, commutative algebra, algebraic topology, etc

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Use the results in all other mathematics fields.

Euclidean Geometry methods usually do not apply.

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A simpler example.

An even simpler example

1 2π

I

circle

xdy −ydx

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A simpler example.

An even simpler example

1 2π

I

circle

xdy −ydx

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A simpler example.

Conclusion: Since Human Beings can’t

image or sense a high dimensional object, we have to study it indirectly. Mathematics is our seventh sense organ.

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Why do mathematicians make things so complicated? The power of symbols/abstractions.

An example

The mirror map (in the simplest case) is

(5ψ)−5exp

      5 ∞ X n=0 (5n)! (n!)5_(5ψ)5n

· ∞ X n=1 (5n)! (n!)5 ( _5n X j=n+1 1 j ) 1 (5ψ)5n

      ,

where |ψ| 1.

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An example

The mirror map (in the simplest case) is

(5ψ)−5exp

      5 ∞ X n=0 (5n)! (n!)5_(5ψ)5n

· ∞ X n=1 (5n)! (n!)5 ( _5n X j=n+1 1 j ) 1 (5ψ)5n

      ,

where |ψ| 1.

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An example

...and we denoted it as

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Another Example.

Newton’s Law of universal gravitation

F =Gm1m2 r2

The Coulomb’s Law

F =ke

q1q2

r2

In mathematics we study the function

y = C 1

r2

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Another Example.

F =Gm1m2 r2

The Coulomb’s Law

F =ke

q1q2

r2

y = C 1

r2

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Another Example.

F =Gm1m2 r2

The Coulomb’s Law

F =ke

q1q2

r2

y = C 1

r2

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Another Example.

The evolution of mathematics largely depends on the evolution of symbols.

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Why do mathematicians make things so complicated? How do we choose a problem/project to work on?

Mathematicians choose problems/projects in a counter-productive way.

1 Choose a problem that is unlikely to be solved. 2 Choose a problem whose outcome is

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1 Choose a problem that is unlikely to be solved.

2 Choose a problem whose outcome is

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1 Choose a problem that is unlikely to be solved. 2 Choose a problem whose outcome is

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1 Andrew Wiles proved the Fermat Last

Theorem, a conjecture that lasted 398 years.

2 Grigori Perelman solved Poincar´e Conjecture,

almost 100 years old, using the Ricci flow method.

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1 Andrew Wiles proved the Fermat Last

Theorem, a conjecture that lasted 398 years.

2 Grigori Perelman solved Poincar´e Conjecture,

almost 100 years old, using the Ricci flow method.

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Pros

1 Very creative and original;

2 Usually quite deep in the discovery of new

phenomena.

Cons

1 1-2 papers a year means very productive? 2 collaborative work becomes difficult.

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Why do mathematicians make things so complicated? Why do we care about other sciences?

The evolution of Mathematics.

The evolution of Mathematics

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1 generalization

(Differential Geometry=Calculus on curved space)

2 similar to bionical creativity engineering, get

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1 generalization (Differential Geometry=Calculus

on curved space)

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1 generalization (Differential Geometry=Calculus

on curved space)

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My results in the math aspect of super string theory.

There are some mathematical implications from Mirror Symmetry, one of which is the so-called BCOV Conjecture.

Bershadsky-Cecotti-Ooguri-Vafa Conjecture

1 _Let_F_A _{be an invariant obtained from symplectic}

geometry of one Calabi-Yau manifold;

2 _Let_F_B _{be an invariant obtained from complex geometry}

of the Mirror Calabi-Yau manifold.

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There are some mathematical implications from Mirror Symmetry, one of which is the so-called BCOV Conjecture. Bershadsky-Cecotti-Ooguri-Vafa Conjecture

1 _Let_F_A _{be an invariant obtained from symplectic}

geometry of one Calabi-Yau manifold;

2 _Let_F_B _{be an invariant obtained from complex geometry}

of the Mirror Calabi-Yau manifold.

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Setup of Conjecture (B)

LetX be a compact K¨ahler manifold.

Let∆ = ∆p,q be the Laplacian on(p, q) forms;

By compactness, the spectrum of∆ are eigenvalues:

0≤λ0 ≤λ1 ≤ · · · ≤λn→+∞.

Define

det ∆ = Y

λi6=0

λi.

ζ function regularization (for example: Riemann

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Setup of Conjecture (B)

0≤λ0 ≤λ1 ≤ · · · ≤λn→+∞.

Define

det ∆ = Y

λi6=0

λi.

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Setup of Conjecture (B)

0≤λ0 ≤λ1 ≤ · · · ≤λn→+∞.

Define

det ∆ = Y

λi6=0

λi.

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Setup of Conjecture (B)

0≤λ0 ≤λ1 ≤ · · · ≤λn→+∞.

Define

det ∆ = Y

λi6=0

λi.

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Setup of Conjecture (B)

0≤λ0 ≤λ1 ≤ · · · ≤λn→+∞.

Define

det ∆ = Y

λi6=0

λi.

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Setup of Conjecture B

Bershadsky-Ceccotti-Ooguri-Vafa defined

T def= Y

p,q

(det ∆p,q)(−1)

p+q_pq

.

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Setup of Conjecture B

Bershadsky-Ceccotti-Ooguri-Vafa defined

T def= Y

p,q

(det ∆p,q)(−1)

p+q_pq

.

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Conjecture

(B) Letk · k be the Hermitian metric on the line bundle

(π∗KW/CP1)

⊗62_⊗_(T_(CP1₎₎⊗3_|

CP1\D

induced from theL2_{-metric on} _π

∗KW/CP1 and from the

Weil-Petersson metric onT(CP1). Then the following identity holds:

τBCOV(Wψ) = Const. 1 F_1,Btop(ψ)3

Ωψ

y0(ψ) 62 ⊗ q d dq 3 2 3 ,

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Conjecture (B) was proved by Fang-L-Yoshikawa.

Fang-L-Yoshikawa

Asymptotic behavior of the BCOV torsion of Calabi-Yau moduli

ArXiv: 0601411 JDG (80), 2008, 175-259,

Aleksey Zinger proved Conjecture (A). Combining the two results, we proved the BCOV Conjecture, which is an evidence that Super String Theory may be true.

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Conjecture (B) was proved by Fang-L-Yoshikawa.

Fang-L-Yoshikawa

Asymptotic behavior of the BCOV torsion of Calabi-Yau moduli

ArXiv: 0601411 JDG (80), 2008, 175-259,

Aleksey Zinger proved Conjecture (A). Combining the two results, we proved the BCOV Conjecture, which is an evidence that Super String Theory may be true.

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String theorists believe that there are parallel

universes to our Universe. Ashok-Douglas developed a method to count the number of those parallel universes.

Joint with Michael R. Douglas, we proved that, if string theory is true, the the number of parallel universes is finite.

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String theorists believe that there are parallel

universes to our Universe. Ashok-Douglas developed a method to count the number of those parallel universes.

Joint with Michael R. Douglas, we proved that, if string theory is true, the the number of parallel universes is finite.

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Why do mathematicians make things so complicated? The use of computer

Computer usage is absolutely important in pure math.

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Two kinds of math theorems

Theorem

π2 > 9.8

Theorem

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Two kinds of math theorems

Theorem

π2 > 9.8

Theorem

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From Wikipedia

A computer-assisted proof is a mathematical proof that has been at least partially generated by

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The Antunes-Freitas Conjecture.

Antunes-Freitas Conjecture

A triangle drum with its longest side equal to 1. Let

λ1, λ2 be the two lowest frequencies. Then

λ2 −λ1 ≥

64π2

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The conjecture was recently solved by Betcke-L-Rowlett, with an extensive use of computer.

It is a computer assisted

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The key part is, although there are infinitely many different triangles, we proved that by checking the conjecture for finitely many of them (In fact, we checked 10,000 triangles), the conjecture must be true for any triangles.

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We proved that

1 for triangles with hight < 0.04, the conjecture

is true;

2 for triangles closed enough to the equilateral

triangle, the conjecture is true;

3 If for any triangle the gap is more than 64π2/9,

there is a neighborhood such that for any triangle in that neighborhood, the

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We proved that

is true;

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We proved that

is true;

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Why do mathematicians make things so complicated? Thank you!