1. Careers in the Mathematical Sciences
Brian Marcus
Department of Mathematics
UBC

2. Top 5 reasons to be a Math Major in University
Free pizza at Math competition training sessions.
Doing math homework is much more fun than watching a good movie.
When you stare out into space, your friends will think that you are deep in thought.
Math teachers tell really good jokes.
Math uses really cool Greek letters.

3. More serious reasons to be a Math Major
You like precision, logic, computation, and solving puzzles.
You want to use mathematics to solve real world problems
You want to invent and discover new mathematics
You want a rewarding career

4. Mathematical Sciences
Mathematics
Computer Science
Statistics

5. Employment in Mathematical Sciences
Education: Secondary and Elementary Schools, Colleges and Universities
Research Labs: Government, Industry
Development in Business and Industry (computer, communications, finance, defense, environmental, aerospace, engineering, film, biomedical, . . .)

6. Occupations in Mathematics

10. Resources Canadian Math Society(www.cms.math.ca)
-- Education:
American Math Society(
-- Employment:
Society for Industrial and Applied Mathematics
American Statistical Association
Association for Computing Machinery
UBC (
---Math Workshops:
PIMS (

25. Applied Mathematics
Applied Mathematics uses math as a tool and develops new applications of math to Science and Engineering:
Mathematical Models of Complex systems (Weather, Heart)
Design tools for building things (Aircraft, Space Vehicles, Robots)
Algorithms for:
Reconstructing images (MRI scans, photos of Mars)
Predicting the stock market
Storing and retrieving data accurately (music on iPod)
Encrypting information securely.
Statistics
Designing clinical trials for new medications, medical procedures
Predicting the course of epidemics, natural disasters

26. Mathematical Modeling
Concept: Model a physical system using equations and make conclusions based on numerical simulation.
Benefits compared to experimenting with a
real physical system:
Less expensive
More Feasible

28. Simple Model

31. Building Things

33. Designing Antennas
Goldstone tracking station – tracks deep space missions.
Design required simulation of wind and heat loads.

35. Imaging (reconstructing geometric objects from imperfect information)

38. Film

43. Other applications

46. Disk Drive Technology computers music (CD, iPod) video (DVD, PSP)
digital camera
pda (palm)

47. This IBM Disk Drive was made in 1956.
Capacity: 5MB
Size: 50 24inch disks
Weight: 500 lbs

48. In 1998, IBM introduced the Microdrive.
Size: 1.1 inch diameter disk Capacity: 170MB (1998)
6 GB (2005)

50. Optical Recording

52. Mathematics used in data recording
Sampling Theory: How to represent a continuous wave as a sequence of 0 and 1 bits
Trigonometry and Calculus: How to focus the laser on circular tracks and adjust the speed of the rotating disk
Algebra: How to correct errors: dust, scratches, imperfections in disk surface, electronics noise

53. Error Correction Coding
Idea:Append redundancy so that you can correct errors
Simple Example: Repetition Code
Encode:
write 0 as 000
write 1 as 111
Decode data by reading 3 bits at a time. If only one bit is in error, then you can correct the error by majority vote.
Error correction power: can correct 1 error in 3 bits
Efficiency of code: 1/3
Idea: Exchange increased reliability for increased data density by writing data on a smaller scale.

54. Boolean addition 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 0
0 + 0 = 0, = 1, = 1, = 0
You can add three terms. Examples:
(1 + 0) + 1 = = 0
(1 + 1) + 1 = = 1
a+b+c = 0 if a,b,c, have an even number of 1’s
a+b+c = 1 if a,b,c, have an odd number of 1’s
a+b+c is viewed as a “parity bit “

55. Hamming Code x x x Encode by appending parity bits: where:
So, encodes to
Example: encode 1110 x x x
= 1+1+1=1, = 1+1+0=0, = 1+1+0=0 5 6 7
So, 1110 encodes to

56. Hamming Codewords

57. The Hamming codewords are the bit-strings of length 7 such that
each cluster A,B,C has even parity: A B 6 2 5 1 1 1 1 1 Ø
Encode: 1 1 1 1 •
Assign any 4-bit string to positions 1 1 4 3
1,2,3,4 1 •
Enforce even parity of clusters 7 Ø
Assume at most one error is made. C
Bit position of error : Clusters with odd parity 1:
A,B,C 2:
A,B 3:
A,C 4:
B,C 5: A 6: B 7: C

59. Hamming Code Features Correction power: can correct 1 error in 7 bits.
Efficiency: 4/

60. Data storage in the future

61. Data Transmission
Noise
Input Message
Noisy
Output
CHANNEL

63. Public key cryptography
Each agent has two keys:
Private key which he/she keeps secret.
Public key which everyone knows.
Agent A encrypts a message by using Agent B’s public key.
Agent B decrypts the message using his private key.

64. What makes this work?
There is a mathematical relation between the public and private keys, which involves two large prime factors of a large number.
It is nearly impossible to derive the private key from the public key.
In order to “break the code,” you must factor a large number.

67. RSA method P and Q are large prime numbers. N = PQ
Agent B’s public key: E, N
Agent B’s private key: D, N
Mathematical relation:
Remainder of (DE)/((P-1)(Q-1)) is 1.
Main point: if you do not know P and Q, it is nearly impossible to derive D from E and N.

68. Encryption/Decryption
Agent A encrypts any message
M = 0, 1, . . ., N – 1 as:
S = remainder of (M^E)/N
Agent B decrypts S as:
T = remainder of (S^D)/N
Fact: T = M
(because DE = 1 mod (P-1)(Q-1) )

69. Agent B’s public key:
3, 33
Agent A encrypts any message M = 0,1,2, . . ., 32 as
S = Remainder of (M^3)/33.
Agent B’s private key:
7,33
Agent B decrypts S as:
Remainder of (S^7)/33 = M
Why?
33 = x 11
20 = (3-1) x (11 -1)
Remainder of (7 X 3)/20 is 1.
It follows that:
Remainder of (S^7)/33 =
Remainder of M^(7 x 3 )/33 = M

70. Pure Mathematics used in Applications
Algebra
- Pure theory developed from study of solutions to systems of equations
- Surprising applications, such as error-correction codes for data recording and telecommunications
Number Theory
- Pure theory developed from factoring numbers into prime numbers
- Surprising applications to cryptography used in secret intelligence and computer security

71. Number Theory Problems
Proof of Infinitude of Primes
Algorithm to generate primes
(recent development)
Largest known prime
Fermat’s Last Theorem
Open Problem: Variations on Fermat

72. Number Theory
A prime number is a number whose only divisors are 1 and itself.
5 is prime, but 4 is not prime since 2x2=4.
The first primes are: 2,3,5,7,11,13,17,19,23,29,31
Sample questions in Number Theory:
How many prime numbers are there?
-- Infinitely many
How many numbers p are there such that both p and p +1 are prime? (p, p+1)
-- (2,3)
How many numbers p are there such that both p and p +2 are prime? (p, p+2)
Examples: (3,5), (5,7), (11,13), (17,19), (29,31)
--- Unknown (Twin Primes Conjecture)
Is there an efficient method to factor numbers into prime numbers?
4 = 2x2, 30 = 5x3x2
= ???

73. Algebra = + c bx ax Quadratic Equation: Quadratic Formula:2 = + c bx ax
Quadratic Equation:
Quadratic Formula:
(2000 years old)
Cubic, Quadratic Formulas:
(500 years old)
Involve 3rd, 4th roots
Quintic Formula: No formula exists!
(200 years old) (using only +, -, x, /, roots)

74. Algebra --- Numerical approximations,
--- Theory of solutions over other fields (e.g., Boolean)
(past 80 years) Modern applications:
--- Quantum Physics
--- Error-correction coding
--- Google

75. There is no clear division between Pure and
Applied Mathematics!