Why do mathematicians make things so complicated?zlu/talks/2010-Breakfast-Lecture/breakfast.pdf · Why do mathematicians make things so complicated? Why do mathematicians make things

Why do mathematicians make things so complicated?

Why do mathematicians makethings so complicated?

Zhiqin Lu, The Math Department

March 9, 2010


Introduction

What is Mathematics?

24,100,000 answers from Google.

Such a FAQ!


Introduction



Such a FAQ!


Introduction



Such a FAQ!


Introduction

From Wikipedia

Mathematics is the study of quantity, structure,space, and change. Mathematicians seek outpatterns,[2][3] formulate new conjectures, andestablish truth by rigorous deduction fromappropriately chosen axioms and definitions.[4]


Introduction

An example

The Real World vs. The Math World

How to become a millionaire in amonth?


Introduction

An example


How to become a millionaire

in amonth?


Introduction

An example


How to become a millionaire in amonth?


Introduction

An example

Fact/Secret/Assumption: Most interest checkingaccounts generate interest at least one cent amonth.

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?

100, 000, 000× $0.01 = $1,000,000!


Introduction

An example



and deposit one cent to each account;

2 Wait for a month.

The profit?

100, 000, 000× $0.01 = $1,000,000!


Introduction

An example




The profit?

100, 000, 000× $0.01 = $1,000,000!


Introduction

An example




The profit?

100, 000, 000× $0.01 = $1,000,000!


Introduction

An example




The profit?

100, 000, 000× $0.01 = $1,000,000!


Introduction

An example

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

Mathematicians like to say

Let n→∞ (infinity)

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

One can earn the whole Universe in a month!


Introduction

An example

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





Introduction

An example






Introduction

An example






Introduction

An example






Introduction

An example

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

Theorem

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


Introduction

An example


Theorem

No banks can afford a free $0.01 interest.

(in the math world)


Introduction

An example


Theorem

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


Introduction

Summary

I am going to talk about

1 Why everything has to be done in an indirectway?

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.


Introduction

Summary






Introduction

Summary



2 The power of symbols/abstractions.

3 How do we choose a problem/project to workon?

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


Introduction

Summary




on?

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


Introduction

Summary




on?4 Why do we care about other sciences?

5 Use of Computer.


Introduction

Summary






My field

Mathematics

Differential Geometry

Complex Geometry


My field

Mathematics


Complex Geometry


My field

Mathematics


Complex Geometry


My field

1 One of my projects is in the mathematicalaspects of Super String Theory.

2 It is related to the Mirror Symmetry.3 Two Universes, quite different, but have the

same Quantum Field Theory.


My field


2 It is related to the Mirror Symmetry.

3 Two Universes, quite different, but have thesame Quantum Field Theory.


My field


2 It is related to the Mirror Symmetry.3 Two Universes, quite different, but have the

same Quantum Field Theory.


My field

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


Why everything has to be done in an ... indirect way?

A problem in Math 2E.

Triple integrals-A Problem in Math 2E

Compute∫∫∫W

xdxdydz,

where W is the regionbounded by the planesx = 0, y = 0, and z = 2,and the surfacez = x2 + y2 and lying inthe quadrantx ≥ 0, y ≥ 0.

N II IV

~ N

><

II

II

>< 0

N

+ ~

'<:

'<: II





Compute∫∫∫W

xdxdydz,


N II IV

~ N

><

II

II

>< 0

N

+ ~

'<:

'<: II





Compute∫∫∫W

xdxdydz,


N II IV

~ N

><

II

II

>< 0

N

+ ~

'<:

'<: II




How to compute integrations over ann-dimensional object?




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




How to study high dimensional geometric object?

Use Calculus;

PDE, functional analysis,complex analysis, etc

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

Use the results in all other mathematics fields.

Euclidean Geometry methods usually do notapply.





Use Calculus; PDE, functional analysis,complex analysis, etc









Use Linear Algebra;

Lie algebra, commutativealgebra, algebraic topology, etc





























A simpler example.

An even simpler example

1

2π

∮circle

xdy − ydxx2 + y2

= 1.



A simpler example.

An even simpler example

1

2π

∮circle

xdy − ydxx2 + y2

= 1.



A simpler example.

Conclusion: Since Human Beings can’timage or sense a high dimensional object, wehave to study it indirectly. Mathematics isour seventh sense organ.


The power of symbols/abstractions.

An example

The mirror map (in the simplest case) is

(5ψ)−5 exp

5∞∑n=0

(5n)!

(n!)5(5ψ)5n

·∞∑n=1

(5n)!

(n!)5

5n∑

j=n+1

1

j

1

(5ψ)5n

,

where |ψ| 1.

Although complicated, it is very concrete.



An example

The mirror map (in the simplest case) is

(5ψ)−5 exp

5∞∑n=0

(5n)!

(n!)5(5ψ)5n

·∞∑n=1

(5n)!

(n!)5

5n∑

j=n+1

1

j

1

(5ψ)5n

,

where |ψ| 1.

Although complicated, it is very concrete.



An example

...and we denoted it as

q(ψ)



Another Example.

Newton’s Law of universal gravitation

F = Gm1m2

r2

The Coulomb’s Law

F = keq1q2r2

In mathematics we study the function

y = C1

r2

which applies to both laws.



Another Example.


F = Gm1m2

r2

The Coulomb’s Law

F = keq1q2r2


y = C1

r2




Another Example.


F = Gm1m2

r2

The Coulomb’s Law

F = keq1q2r2


y = C1

r2




Another Example.

The evolution of mathematics largelydepends on the evolution of symbols.


How do we choose a problem/project to work on?

Mathematicians choose problems/projects in acounter-productive way.

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

unexpected.




1 Choose a problem that is unlikely to be solved.

2 Choose a problem whose outcome isunexpected.




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

unexpected.



1 Andrew Wiles proved the Fermat LastTheorem, a conjecture that lasted 398 years.

2 Grigori Perelman solved Poincare Conjecture,almost 100 years old, using the Ricci flowmethod.



1 Andrew Wiles proved the Fermat LastTheorem, a conjecture that lasted 398 years.

2 Grigori Perelman solved Poincare Conjecture,almost 100 years old, using the Ricci flowmethod.



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.3 the work usually finishes in the last minute.


Why do we care about other sciences?

The evolution of Mathematics.

The evolution of MathematicsHow to push math forward?




1 generalization

(Differential Geometry=Calculuson curved space)

2 similar to bionical creativity engineering, gethints from other sciences




1 generalization (Differential Geometry=Calculuson curved space)





1 generalization (Differential Geometry=Calculuson curved space)




My results in the math aspect of super string theory.

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

Bershadsky-Cecotti-Ooguri-Vafa Conjecture

1 Let FA be an invariant obtained from symplecticgeometry of one Calabi-Yau manifold;

2 Let FB be an invariant obtained from complex geometryof the Mirror Calabi-Yau manifold.

Then FA = FB.




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

Bershadsky-Cecotti-Ooguri-Vafa Conjecture

1 Let FA be an invariant obtained from symplecticgeometry of one Calabi-Yau manifold;

2 Let FB be an invariant obtained from complex geometryof the Mirror Calabi-Yau manifold.

Then FA = FB.




Setup of Conjecture (B)Let X be a compact Kahler manifold.

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

By compactness, the spectrum of ∆ are eigenvalues:

0 ≤ λ0 ≤ λ1 ≤ · · · ≤ λn → +∞.

Definedet ∆ =

∏λi 6=0

λi.

ζ function regularization (for example: Riemannζ-function)







0 ≤ λ0 ≤ λ1 ≤ · · · ≤ λn → +∞.

Definedet ∆ =

∏λi 6=0

λi.








0 ≤ λ0 ≤ λ1 ≤ · · · ≤ λn → +∞.

Definedet ∆ =

∏λi 6=0

λi.








0 ≤ λ0 ≤ λ1 ≤ · · · ≤ λn → +∞.

Definedet ∆ =

∏λi 6=0

λi.








0 ≤ λ0 ≤ λ1 ≤ · · · ≤ λn → +∞.

Definedet ∆ =

∏λi 6=0

λi.





Setup of Conjecture B

Bershadsky-Ceccotti-Ooguri-Vafa defined

Tdef=∏p,q

(det ∆p,q)(−1)p+qpq.

Why define such a strange quantity?




Setup of Conjecture B

Bershadsky-Ceccotti-Ooguri-Vafa defined

Tdef=∏p,q

(det ∆p,q)(−1)p+qpq.

Why define such a strange quantity?




Conjecture

(B) Let ‖ · ‖ be the Hermitian metric on the line bundle

(π∗KW/CP 1)⊗62 ⊗ (T (CP 1))⊗3|CP 1\D

induced from the L2-metric on π∗KW/CP 1 and from the

Weil-Petersson metric on T (CP 1). Then the following identityholds:

τBCOV(Wψ) = Const.

∥∥∥∥∥ 1

F top1,B (ψ)3

(Ωψ

y0(ψ)

)62

⊗(qd

dq

)3∥∥∥∥∥

23

,

where Ω is the local holomorphic section of the (3, 0) forms.




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

Fang-L-Yoshikawa

Asymptotic behavior of the BCOV torsion of Calabi-Yaumoduli

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

Aleksey Zinger proved Conjecture (A). Combining the tworesults, we proved the BCOV Conjecture, which is an evidencethat Super String Theory may be true.




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

Fang-L-Yoshikawa

Asymptotic behavior of the BCOV torsion of Calabi-Yaumoduli

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

Aleksey Zinger proved Conjecture (A). Combining the tworesults, we proved the BCOV Conjecture, which is an evidencethat Super String Theory may be true.




String theorists believe that there are paralleluniverses to our Universe. Ashok-Douglas developeda method to count the number of those paralleluniverses.

Joint with Michael R. Douglas, we proved that, ifstring theory is true, the the number of paralleluniverses is finite.




String theorists believe that there are paralleluniverses to our Universe. Ashok-Douglas developeda method to count the number of those paralleluniverses.Joint with Michael R. Douglas, we proved that, ifstring theory is true, the the number of paralleluniverses is finite.


The use of computer

Computer usage is absolutely important in puremath.


The use of computer

Two kinds of math theorems

Theorem

π2 > 9.8

Theorem

x2 + y2 ≥ 2xy


The use of computer

Two kinds of math theorems

Theorem

π2 > 9.8

Theorem

x2 + y2 ≥ 2xy


The use of computer

From Wikipedia

A computer-assisted proof is a mathematical proofthat has been at least partially generated bycomputer.


The use of computer

The Antunes-Freitas Conjecture.

Antunes-Freitas Conjecture

A triangle drum with its longest side equal to 1. Letλ1, λ2 be the two lowest frequencies. Then

λ2 − λ1 ≥64π2

9


The use of computer


The conjecture wasrecently solved byBetcke-L-Rowlett, withan extensive use ofcomputer.It is a computer assistedproof !


The use of computer


The key part is, although there are infinitely manydifferent triangles, we proved that by checking theconjecture for finitely many of them (In fact, wechecked 10,000 triangles), the conjecture must betrue for any triangles.


The use of computer


We proved that

1 for triangles with hight < 0.04, the conjectureis true;

2 for triangles closed enough to the equilateraltriangle, the conjecture is true;

3 If for any triangle the gap is more than 64π2/9,there is a neighborhood such that for anytriangle in that neighborhood, theAntunes-Freitas Conjecture is true.


The use of computer


We proved that





The use of computer


We proved that





Thank you!

Documents

Why do mathematicians make things so complicated?zlu/talks/2010-Breakfast-Lecture/breakfast.pdf · Why do mathematicians make things so complicated? Why do mathematicians make things