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

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?

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How to become a millionaire

in amonth?

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How to become a millionaire in amonth?

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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!

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and deposit one cent to each account;

2 Wait for a month.

The profit?

100, 000, 000× $0.01 = $1,000,000!

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The profit?

100, 000, 000× $0.01 = $1,000,000!

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The profit?

100, 000, 000× $0.01 = $1,000,000!

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The profit?

100, 000, 000× $0.01 = $1,000,000!

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...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!

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

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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)

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Theorem

No banks can afford a free $0.01 interest.

(in the math world)

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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

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.

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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

'<: II

Compute∫∫∫W

xdxdydz,

N II IV

'<: II

Compute∫∫∫W

xdxdydz,

N II IV

'<: 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

∮circle

xdy − ydxx2 + y2

A simpler example.

An even simpler example

∮circle

xdy − ydxx2 + y2

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

(n!)5(5ψ)5n

·∞∑n=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

(n!)5(5ψ)5n

·∞∑n=1

(5ψ)5n

where |ψ| 1.

Although complicated, it is very concrete.

An example

...and we denoted it as

Another Example.

Newton’s Law of universal gravitation

F = Gm1m2

The Coulomb’s Law

F = keq1q2r2

In mathematics we study the function

y = C1

which applies to both laws.

Another Example.

F = Gm1m2

The Coulomb’s Law

F = keq1q2r2

y = C1

Another Example.

F = Gm1m2

The Coulomb’s Law

F = keq1q2r2

y = C1

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.

1 Very creative and original;2 Usually quite deep in the discovery of new

phenomena.

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)

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: