Its a Mathematical World Cristian Rios University of Calgary PIMS - - PowerPoint PPT Presentation

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It’s a Mathematical World

Cristian Rios

University of Calgary

PIMS Lunchbox Lecture Series April 16 2015

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Mathematics everywhere

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Celestial mechanics

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General relativity

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Particle scatter

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Wave interference

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Quantum interference

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Quantum scattering

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Crossing a street

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The streets of Buenos Aires

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Social Mathematics

Every decision taken, or action performed, as a result of the use of our frontal cortex has a mathematical base. Basic Principles:

1

Minimize risk.

1

Accident prevention or avoidance.

2

Minimize losses.

3

Self preservation.

2

Maximize profit.

1

Economical advancement.

2

Social advancement.

3

Professional advancement.

4

Betterment of society.

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Social Mathematics

Every decision taken, or action performed, as a result of the use of our frontal cortex has a mathematical base. Basic Principles:

1

Minimize risk.

1

Accident prevention or avoidance.

2

Minimize losses.

3

Self preservation.

2

Maximize profit.

1

Economical advancement.

2

Social advancement.

3

Professional advancement.

4

Betterment of society.

Deterministic: Optimization problems. Probabilistic: Computing the odds, choosing the path with best expected outcome.

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Social mathematics

Example of a social mathematics problem: What is the optimal time to call for an election?

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Mathematics of sports

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Money ball math

"Sabermetrics is the empirical analysis of baseball, especially baseball statistics that measure in-game activity. The term is derived from the acronym SABR, which stands for the Society for American Baseball

• Research. It was coined by Bill James, who is one of its pioneers and is
• ften considered its most prominent advocate and public face." (Lewis,

Michael M. (2003). Moneyball: "The Art of Winning an Unfair Game")

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That sounds like math

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The miracle of feeding the multitude

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The Banach-Tarski Paradox - Paradoxical sets

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The Banach-Tarski Paradox (paradoxical sets)

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Foundations of mathematics

Principia Mathematica (1910-1913), by Alfred North Whitehead and Bertrand Russell. Intended to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could be proven.

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THE ROOTS (DNA) - Zermelo-Frankel axioms - Axiomatic set theory. Axiom of extensionality ∀A∀B (∀X (X ∈ A ⇐ ⇒ X ∈ B)) = ⇒ A = B Axiom of empty set ∃X∀Y¬ (Y ∈ X) Axiom of pairing ∀A∀B∃C∀D [D ∈ C ⇐ ⇒ (D = A ∨ D = B)] Axiom of union ∀A∃B∀c (c ∈ B ⇐ ⇒ ∃D (c ∈ D ∧ D ∈ A)) Axiom of infinity ∃I (∅ ∈ I ∧ ∀x ∈ I ((x {x}) ∈ I)) Axiom of replacement "the image of a set is a set" Axiom of power set ∀A∃P∀B [B ∈ P ⇐ ⇒ B ⊂ A] Axiom of regularity ∀x (x = ∅ = ⇒ ∃y ∈ x (y x = ∅)) Axiom schema of specification "a subclass of a set is a set" Axiom of choice ∀X

• /

∅ / ∈ X = ⇒ ∃f : X → X ∀A ∈ X (f (A) ∈ A)

• Cristian Rios

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One odd family

Equivalence classes Z = {integers} = {0, 1, −1, 2, −2, 3, −3, . . . } . 2 Z = {even integers} = {0, 2, −2, 4, −4, 6, −6, . . . } . Define the equivalence p ≈ q (p is "related" to q) if p − q ∈ 2 Z. 2 equivalence classes (families): even integers, and odd integers. Representatives of the classes M = {0, 1} .

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One odd family

Equivalence classes Z = {integers} = {0, 1, −1, 2, −2, 3, −3, . . . } . 2 Z = {even integers} = {0, 2, −2, 4, −4, 6, −6, . . . } . Define the equivalence p ≈ q (p is "related" to q) if p − q ∈ 2 Z. 2 equivalence classes (families): even integers, and odd integers. Representatives of the classes M = {0, 1} . Each "family" is affinely equivalent to the whole set Z. q ∈ 2Z (even family) define T0 : q → q/2. Then T0 (2Z) = Z (biyective) p ∈ 2Z + 1 (odd family) define T1 : p → (p − 1) /2. Then T1 (2Z + 1) = Z (biyective)

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5 Families

Z = {integers} = {0, 1, −1, 2, −2, 3, −3, . . . } . 5 Z = {multiples of five } = {0, 5, −5, 10, −10, 15, −15, . . . } . Define the equivalence p ≈ q (p is "related" to q) if p − q ∈ 5 Z. 5 equivalence classes (families): 5 Z, 5 Z + 1, 5 Z + 2, 5 Z + 3, 5 Z + 4. Representatives of the classes M = {0, 1, 2, 3, 4} .

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Group of rotations in the circle

Q = {rationals} = p

q, p, q ∈ Z, q = 0

• T = {group of rotations} = [0, 2π).

R = {group of rational rotations} = 2πQmod 2π. Define the equivalence. p, q ∈ T, p ≈ q if p − q ∈ R. ∞ equivalence classes (families). Axiom of choice: We can build a set M of representatives of the classes. Given a rotation p, its family is p + R. T = disjoint union of all families =

·

• p∈M {p + R} =

·

• p∈RpM.

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Rationals are countable

Q = {rationals} = p

q, p, q ∈ Z, q = 0

• = {p1, p2, p3, p4,...}

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A paradoxical decomposition

We enumerate the rationals Q = {p1, p2, p3, p4, . . . } Then, we let Mi = piM (ith rotation of the representatives) T = (circle) =

·

∞

i=1Mi =

·

• i evenMi
• ·
• ·
• i oddMi
• .

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A paradoxical decomposition

We enumerate the rationals Q = {p1, p2, p3, p4, . . . } Then, we let Mi = piM (ith rotation of the representatives) T = (circle) =

·

∞

i=1Mi =

·

• i evenMi
• ·
• ·
• i oddMi
• .

The paradox T =

·

• i evenp i

2 p−1

i

Mi and T =

·

• i oddp i+1

2 p−1

i

Mi.

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Banach-Tarski Paradox for the unit ball in 3-space

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Mathematics evolution

Axioms are the "atoms" of mathematics. Basic accepted truths. These atoms "interact" and combine, according to the rules of logic. These interactions lead to new "molecules" (theorems). Molecules combine and group into "proteins", (theories, areas).

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The tree of known mathematics

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Mathematical subjects

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Math Zen

By connecting mathematical objects apparently dissociated, one glimpses into the notion the whole contained in the parts, of the mysteries of the universe, the intricate connections between all that exists. 1 12 + 1 22 + 1 32 + 1 42 + 1 52 + · · · = π2 6 . eiπ + 1 = 0.

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