Careers in the Mathematical
Sciences
Brian Marcus
Department of Mathematics
UBC
marcus@math.ubc.ca
Top 5 reasons to be a Math Major in
University
1. Free pizza at Math competition training
sessions.
2. Doing math homework is much more fun
than watching a good movie.
3. When you stare out into space, your friends
will think that you are deep in thought.
4. Math teachers tell really good jokes.
5. Math uses really cool Greek letters.
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
Mathematical Sciences
• Mathematics
• Computer Science
• Statistics
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, . . .)
Occupations in Mathematics
Resources
• Canadian Math Society(www.cms.math.ca)
-- Education: www.cms.math.ca/Education
• American Math Society(www.ams.org)
-- Employment: www.ams.org/employment
• Society for Industrial and Applied Mathematics
• American Statistical Association
• Association for Computing Machinery
 UBC (www.math.ubc.ca)
---Math Workshops:
www.math.ubc.ca/Schools/Workshop/index.shtml
 PIMS (www.pims.math.ca):
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
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
Simple Model
Building Things
Designing Antennas
• Goldstone tracking station – tracks deep space missions.
• Design required simulation of wind and heat loads.
Imaging
(reconstructing geometric objects from
imperfect information)
Film
Other applications
Disk Drive Technology
•
•
•
•
•
computers
music (CD, iPod)
video (DVD, PSP)
digital camera
pda (palm)
This IBM Disk Drive
was made in 1956.
Capacity: 5MB
Size: 50 24inch disks
Weight: 500 lbs
In 1998, IBM introduced the Microdrive.
Size: 1.1 inch diameter disk
Capacity: 170MB (1998)
6 GB (2005)
Optical Recording
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
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.
Boolean addition
 0 + 0 = 0,
0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 0
You can add three terms. Examples:
(1 + 0) + 1 = 1 + 1 = 0
(1 + 1) + 1 = 0 + 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 “
Hamming Code
Encode
x1 x2 x3 x4
by appending parity bits: x5 , x6 , x7
where:
x5  x1  x2  x3
x6  x1  x2  x4
x7  x1  x3  x4
So,
x1 x2 x3 x4
encodes to
x1 x2 x3 x4 x5 x6 x7
Example: encode 1110
x5 = 1+1+1=1, x6 = 1+1+0=0,
So, 1110 encodes to 1110100
x7 = 1+1+0=0
Hamming Codewords
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•
•
•
0000000
0001011
0010101
0011110
0100110
0101101
0110011
0111000
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•
•
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1000111
1001100
1010010
1011001
1100001
1101010
1110100
1111111
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
10
Encode:
1
• Assign any 4-bit string to positions
1,2,3,4
1
0
1
0
3
0
1
01
0
1
4
1
0
• Enforce even parity of clusters
7
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
Hamming Code Features
• Correction power: can correct 1 error in 7 bits.
• Efficiency: 4/7  .571
Data storage in the future
Data Transmission
Noise
Input
Message
Noisy
Output
CHANNEL
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.
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.
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.
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) )
 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 = 3 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
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
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
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
• 83429852989028591 = ???
Algebra
Quadratic Equation:
Quadratic Formula:
(2000 years old)
ax2  bx  c  0
 b  b  4ac
x
2a
2
Cubic, Quadratic Formulas: Involve 3rd, 4th roots
(500 years old)
Quintic Formula:
(200 years old)
No formula exists!
(using only +, -, x, /, roots)
Algebra
--- Numerical approximations,
--- Theory of solutions over other fields (e.g.,
Boolean)
• (past 80 years) Modern applications:
--- Quantum Physics
--- Error-correction coding
--- Google
There is no clear division between Pure and
Applied Mathematics!