Why do mathematicians make things so complicated?

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1 Why do mathematicians make things so complicated? Zhiqin Lu, The Math Department March 9, 2010

2 Introduction What is Mathematics?

4 Introduction What is Mathematics? 24,100,000 answers from Google. Such a FAQ!

5 Introduction 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]

6 Introduction An example The Real World vs. The Math World

7 Introduction An example The Real World vs. The Math World How to become a millionaire

8 Introduction An example The Real World vs. The Math World How to become a millionaire in a month?

9 Introduction An example Fact/Secret/Assumption: Most interest checking accounts generate interest at least one cent a month.

10 Introduction 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;

11 Introduction 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.

12 Introduction 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?

13 Introduction 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? 100, 000, 000 \$0.01 = \$1, 000, 000!

14 Introduction An example...and that is not the end of the story...

15 Introduction An example...and that is not the end of the story... Mathematicians like to say

16 Introduction An example...and that is not the end of the story... Mathematicians like to say Let n (infinity)

17 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 to infinity, what will happen?

18 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 to infinity, what will happen? One can earn the whole Universe in a month!

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

20 Introduction 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.

21 Introduction 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)

22 Introduction Summary I am going to talk about

23 Introduction Summary I am going to talk about 1 Why everything has to be done in an indirect way?

24 Introduction Summary I am going to talk about 1 Why everything has to be done in an indirect way? 2 The power of symbols/abstractions.

25 Introduction 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?

26 Introduction 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?

27 Introduction 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.

28 My field Mathematics

29 My field Mathematics Differential Geometry

30 My field Mathematics Differential Geometry Complex Geometry

31 My field 1 One of my projects is in the mathematical aspects of Super String Theory.

32 My field 1 One of my projects is in the mathematical aspects of Super String Theory. 2 It is related to the Mirror Symmetry.

33 My field 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 same Quantum Field Theory.

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

35 Why everything has to be done in an... indirect way? A problem in Math 2E. Triple integrals-a Problem in Math 2E

36 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 region bounded by the planes x = 0, y = 0, and z = 2, and the surface z = x 2 + y 2 and lying in the quadrant x 0, y 0.

37 II IV 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 W xdxdydz, where W is the region bounded by the planes x = 0, y = 0, and z = 2, and the surface z = x 2 + y 2 and lying in the quadrant x 0, y 0. N '<: II N >< II II >< 0 N + ~ ~ '<:

38 Why everything has to be done in an... indirect way? A problem in Math 2E. How to compute integrations over an n-dimensional object?

39 Why everything has to be done in an... indirect way? A problem in Math 2E. Figure: From the internet. It is the intersection of the quintic Calabi-Yau threefold to our three dimensional space.

40 Why everything has to be done in an... indirect way? A problem in Math 2E. How to study high dimensional geometric object? Use Calculus;

41 Why everything has to be done in an... indirect way? A problem in Math 2E. How to study high dimensional geometric object? Use Calculus; PDE, functional analysis, complex analysis, etc

42 Why everything has to be done in an... indirect way? A problem in Math 2E. How to study high dimensional geometric object? Use Calculus; PDE, functional analysis, complex analysis, etc Use Linear Algebra;

43 Why everything has to be done in an... indirect way? A problem in Math 2E. 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

44 Why everything has to be done in an... indirect way? A problem in Math 2E. 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.

45 Why everything has to be done in an... indirect way? A problem in Math 2E. 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.

46 Why everything has to be done in an... indirect way? A simpler example. An even simpler example

47 Why everything has to be done in an... indirect way? A simpler example. An even simpler example 1 xdy ydx = 1. 2π circle x 2 + y 2

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

49 The power of symbols/abstractions. An example The mirror map (in the simplest case) is (5ψ) 5 exp 5 (5n)! (n!) 5 (5ψ) 5n n=0 where ψ 1. { 5n } (5n)! 1 1 (n!) 5 j (5ψ) 5n j=n+1, n=1

50 The power of symbols/abstractions. An example The mirror map (in the simplest case) is (5ψ) 5 exp 5 (5n)! (n!) 5 (5ψ) 5n n=0 { 5n } (5n)! 1 1 (n!) 5 j (5ψ) 5n j=n+1, n=1 where ψ 1. Although complicated, it is very concrete.

51 The power of symbols/abstractions. An example...and we denoted it as q(ψ)

52 The power of symbols/abstractions. Another Example. Newton s Law of universal gravitation F = G m 1m 2 r 2

53 The power of symbols/abstractions. Another Example. Newton s Law of universal gravitation The Coulomb s Law F = G m 1m 2 r 2 F = k e q 1 q 2 r 2

54 The power of symbols/abstractions. Another Example. Newton s Law of universal gravitation The Coulomb s Law F = G m 1m 2 r 2 F = k e q 1 q 2 r 2 In mathematics we study the function y = C 1 r 2 which applies to both laws.

55 The power of symbols/abstractions. Another Example. The evolution of mathematics largely depends on the evolution of symbols.

56 How do we choose a problem/project to work on? Mathematicians choose problems/projects in a counter-productive way.

57 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.

58 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 unexpected.

59 How do we choose a problem/project to work on? 1 Andrew Wiles proved the Fermat Last Theorem, a conjecture that lasted 398 years.

60 How do we choose a problem/project to work on? 1 Andrew Wiles proved the Fermat Last Theorem, a conjecture that lasted 398 years. 2 Grigori Perelman solved Poincaré Conjecture, almost 100 years old, using the Ricci flow method.

61 How do we choose a problem/project to work on? Pros 1 Very creative and original; 2 Usually quite deep in the discovery of new phenomena. Cons papers a year means very productive? 2 collaborative work becomes difficult. 3 the work usually finishes in the last minute.

62 Why do we care about other sciences? The evolution of Mathematics. The evolution of Mathematics How to push math forward?

63 Why do we care about other sciences? The evolution of Mathematics. 1 generalization

64 Why do we care about other sciences? The evolution of Mathematics. 1 generalization (Differential Geometry=Calculus on curved space)

65 Why do we care about other sciences? The evolution of Mathematics. 1 generalization (Differential Geometry=Calculus on curved space) 2 similar to bionical creativity engineering, get hints from other sciences

66 Why do we care about other sciences? 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.

67 Why do we care about other sciences? 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. Then F A = F B.

68 Why do we care about other sciences? My results in the math aspect of super string theory. Setup of Conjecture (B) Let X be a compact Kähler manifold.

69 Why do we care about other sciences? My results in the math aspect of super string theory. Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms;

70 Why do we care about other sciences? My results in the math aspect of super string theory. Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +.

71 Why do we care about other sciences? My results in the math aspect of super string theory. Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +. Define det = λ i 0 λ i.

72 Why do we care about other sciences? My results in the math aspect of super string theory. Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +. Define det = λ i 0 λ i. ζ function regularization (for example: Riemann ζ-function)

73 Why do we care about other sciences? My results in the math aspect of super string theory. Setup of Conjecture B Bershadsky-Ceccotti-Ooguri-Vafa defined T def = (det p,q ) ( 1)p+qpq. p,q

74 Why do we care about other sciences? My results in the math aspect of super string theory. Setup of Conjecture B Bershadsky-Ceccotti-Ooguri-Vafa defined T def = (det p,q ) ( 1)p+qpq. p,q Why define such a strange quantity?

75 Why do we care about other sciences? My results in the math aspect of super string theory. Conjecture (B) Let be the Hermitian metric on the line bundle (π K W/CP 1) 62 (T (CP 1 )) 3 CP 1 \D induced from the L 2 -metric on π K W/CP 1 and from the Weil-Petersson metric on T (CP 1 ). Then the following identity holds: ( ) 62 ( 1 Ωψ τ BCOV (W ψ ) = Const. F top q d ) 2 3 3, 1,B (ψ)3 y 0 (ψ) dq where Ω is the local holomorphic section of the (3, 0) forms.

76 Why do we care about other sciences? My results in the math aspect of super string theory. Conjecture (B) was proved by Fang-L-Yoshikawa. Fang-L-Yoshikawa Asymptotic behavior of the BCOV torsion of Calabi-Yau moduli ArXiv: JDG (80), 2008, ,

77 Why do we care about other sciences? My results in the math aspect of super string theory. Conjecture (B) was proved by Fang-L-Yoshikawa. Fang-L-Yoshikawa Asymptotic behavior of the BCOV torsion of Calabi-Yau moduli ArXiv: JDG (80), 2008, , 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.

78 Why do we care about other sciences? My results in the math aspect of super string theory. String theorists believe that there are parallel universes to our Universe. Ashok-Douglas developed a method to count the number of those parallel universes.

79 Why do we care about other sciences? My results in the math aspect of super string theory. 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.

80 The use of computer Computer usage is absolutely important in pure math.

81 The use of computer Two kinds of math theorems Theorem π 2 > 9.8

82 The use of computer Two kinds of math theorems Theorem π 2 > 9.8 Theorem x 2 + y 2 2xy

83 The use of computer From Wikipedia A computer-assisted proof is a mathematical proof that has been at least partially generated by computer.

84 The use of computer The Antunes-Freitas Conjecture. Antunes-Freitas Conjecture A triangle drum with its longest side equal to 1. Let λ 1, λ 2 be the two lowest frequencies. Then λ 2 λ 1 64π2 9

85 The use of computer The Antunes-Freitas Conjecture. The conjecture was recently solved by Betcke-L-Rowlett, with an extensive use of computer. It is a computer assisted proof!

86 The use of computer The Antunes-Freitas Conjecture. 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.

87 The use of computer The Antunes-Freitas Conjecture. We proved that 1 for triangles with hight < 0.04, the conjecture is true;

88 The use of computer The Antunes-Freitas Conjecture. 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;

89 The use of computer The Antunes-Freitas Conjecture. 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 64π 2 /9, there is a neighborhood such that for any triangle in that neighborhood, the Antunes-Freitas Conjecture is true.

90 Thank you!

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