옥스퍼드대 수학과 김민형 교수와 함께 우리가 잘 모르고 느끼지 못했던 수학의 매력과 깊이 속으로 빠져들어 보자.

1. Global Mathematics and Local Culture
Kim Minhyong
August, 2018
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2. Coin Tossing
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3. Coin Tossing
Tossing 40 coins. Write H for head and T for tail.
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4. Coin Tossing
Tossing 40 coins. Write H for head and T for tail.
Suppose the result were
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5. Coin Tossing
Tossing 40 coins. Write H for head and T for tail.
Suppose the result were
HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
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6. Coin Tossing
Tossing 40 coins. Write H for head and T for tail.
Suppose the result were
HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
HHHHHHHHHHHHHHHHHHHHTTTTTTTTTTTTTTTTTTTT
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7. Coin Tossing
Tossing 40 coins. Write H for head and T for tail.
Suppose the result were
HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
HHHHHHHHHHHHHHHHHHHHTTTTTTTTTTTTTTTTTTTT
HTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHT
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8. Coin Tossing
Tossing 40 coins. Write H for head and T for tail.
Suppose the result were
HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
HHHHHHHHHHHHHHHHHHHHTTTTTTTTTTTTTTTTTTTT
HTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHT
HHTTHTTHTTTTHHTHHTHHHHTHHHTHHTTTTHTHTTHH
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9. Coin Tossing
How many possible outcomes?
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10. Coin Tossing
How many possible outcomes?
One toss:
H, T
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11. Coin Tossing
How many possible outcomes?
One toss:
H, T
Two tosses:
HH, HT, TH, TT.
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12. Coin Tossing
How many possible outcomes?
One toss:
H, T
Two tosses:
HH, HT, TH, TT.
Three tosses:
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.
Kim Minhyong Global Mathematics and Local Culture August, 2018 3 / 25

13. Coin Tossing
How many possible outcomes?
One toss:
H, T
Two tosses:
HH, HT, TH, TT.
Three tosses:
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.
For 40 tosses, have
240
possible outcomes. This is much larger than the number of atoms in our
body.
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14. Coin Tossing
Any given sequence of tosses has probability 1
240 of arising. So why are we
more surprised by
HTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHT
than
HHTTHTTHTTTTHHTHHTHHHHTHHHTHHTTTTHTHTTHH
?
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15. Information Theory
Figure: Andrei Kolmogorov (1903-1987)
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16. Information Theory
We are surprised by
HTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHT
because it is easy to describe.
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17. Information Theory
We are surprised by
HTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHT
because it is easy to describe.
The sequence
HHTTHTTHTTTTHHTHHTHHHHTHHHTHHTTTTHTHTTHH
seems to admit no easier description than it itself.
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18. Kolmogorov Complexity
The Kolmogorov complexity of a sequence is the length of the shortest
computer programme that will print the sequence.
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19. Kolmogorov Complexity
The Kolmogorov complexity of a sequence is the length of the shortest
computer programme that will print the sequence.
In our example, the surprising sequences are the ones of low Kolmogorov
complexity.
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20. Kolmogorov Complexity
The Kolmogorov complexity of a sequence is the length of the shortest
computer programme that will print the sequence.
In our example, the surprising sequences are the ones of low Kolmogorov
complexity.
Kolmogorov proved that for long sequences, most sequences are
incompressible.
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21. Kolmogorov Complexity
The sequences
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
obviously have low complexity. How about
5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1
?
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22. Gelfand
Figure: Israel M. Gelfand (1913-2009)
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23. Kontsevich
Figure: Maxim Kontsevich (1964- )
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24. Arnold
Figure: Vladimir I. Arnold (1937-2010 )
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25. Arnold’s Puzzle:
lim
x→0
sin(tan(x)) − tan(sin(x))
arcsin(arctan(x)) − arctan(arcsin(x))
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26. Bourbaki
Figure: Bourbaki, Circa 1935
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27. Weil
Figure: Andre Weil (1906-1998)
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28. Grothendieck
Figure: Alexander Grothendieck (1928-2014)
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29. Grothendieck
Figure: Alexander Grothendieck (1928-2014)
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30. Grothendieck
Figure: Alexander Grothendieck (1928-2014)
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31. Grothendieck
Figure: Teiji Takagi (1875-1960)
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32. Probability: A Few Puzzles
A small hospital had 10 births this week. A large hospital had 100 births
this week. Which event is more probable: 6 boys or more in the small
hospital, 60 boys or more in the large hospital?
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33. Probability: A Few Puzzles
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34. Probability: A Few Puzzles
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35. Probability: A Few Puzzles
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36. Probability: A Few Puzzles
When there are n babies, the probability that at least 60 percent are boys is
approximately
1
√
2π
∞
√
n/2
e−x2/2
dx.
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37. Probability: A Few Puzzles
When there are n babies, the probability that at least 60 percent are boys is
approximately
1
√
2π
∞
√
n/2
e−x2/2
dx.
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38. Probability: A Few Puzzles
Spectral distribution 1:
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39. Probability: A Few Puzzles
Spectral distribution 2:
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