the expanding role of mathematics in science and society

8/11/2019 The Expanding Role of Mathematics in Science and Society

1/71

1

School of Science

Indiana University-Purdue University Indianapolis

Carl C. Cowen, Dean

IUPUI School of SciencePresident, Mathematical Association of America

NCTM Annual Meeting, Anaheim

April 9, 2005

The Expanding Role of

Mathematics

in Science and Society


2/71

2

School of Science


Introduction

The Challenge and the Opportunity

Automatic Teller Machines:

The math of ATM security

CAT Scans and MRIs:

The math of medical diagnostics

SARS, HIV, Cancer, Parkinsons:

The math of epidemics and disease Sources for classroom materials


3/71

3

School of Science


Not educating enough people in science,technology, engineering and math

Need to diversify our math and science

workforce Middle school and high school are the

critical times

Spread message of opportunity to parentsand counselors

www.science.iupui.edu/ccowen/Careers.html


4/71

4

School of Science





critical times




5/71

5

School of Science





critical times




6/71

6

School of Science





critical times

Spread message of opportunity toparents and counselors



7/71

7

School of Science


Explosion in biological research andprogress

The mathematical sciences will be a part

Opportunity:few mathematical scientists are

biologically educated

few biological scientists aremathematically educated

Dr. Rita Colwell: We're not near the fulfillment of

biotechnology's promise. We're just on the cusp of it


8/71

8

School of Science


Explosion in biological research andprogress

The mathematical sciences will be a part

Opportunity:few mathematical scientists are

biologically educated

few biological scientists aremathematically educated

Report Bio2010: How biologists design, perform, and

analyze experiments is changing swiftly. Biological

concepts and models are becoming more quantitative


9/71


10/71

10

School of Science


ATM SecurityIncreasing use of electronic communications

in financial and other applications has

increased the need for encryption to ensureprivacy


11/71

11

School of Science


ATM SecurityIncreasing use of electronic communications

in financial and other applications has

increased the need for encryption to ensureprivacy

The most widely used encryption system in the

world is RSA, developed in 1976 by threemathematicians, Rivest, Shamir, and

Adelman.


12/71

12

School of Science



13/71

13

School of Science



14/71

14

School of Science



15/71

15

School of Science



16/71

16

School of Science



17/71

17

School of Science



18/71

18

School of Science



19/71

19

School of Science



20/71

20

School of Science



21/71

21

School of Science



22/71

22

School of Science



23/71

23

School of Science



24/71

24

School of Science



25/71

25

School of Science



26/71

26

School of Science



27/71

27

School of Science



28/71

28

School of Science



29/71

29

School of Science



30/71

30

School of Science



31/71

31

School of Science



32/71

32

School of Science



33/71

33

School of Science



34/71

34

School of Science



35/71

35

School of Science



36/71

36

School of Science



37/71

37

School of Science



38/71

38

School of Science



39/71

39

School of Science



40/71

40

School of Science



41/71

41

School of Science



42/71

42

School of Science



43/71

43

School of Science


CAT Scans and MRIsX-rays have been used for about a century in

medical diagnostics. They are high frequency,

high energy electromagnetic waves


44/71

44

School of Science




high energy electromagnetic waves likevisible light, except that the electromagnetic

waves of visible light are lower frequency and

lower energy.


45/71

45

School of Science




high energy electromagnetic waves likevisible light, except that the electromagnetic

waves of visible light are lower frequency and

lower energy.

Since we see differences of frequency in light

as color and differences of energy as

brightness, so you might say X-rays are very

bright and a different color from what we see!


46/71

46

School of Science


X-rays are so energetic that our bodies arenearly transparent to themlike glass is to

ordinary light. Bones are much less

transparent to X-rays than soft parts of our

bodies, indeed, each part of our bodies hasslightly different transparencies than other

parts.


47/71

47

School of Science


X-rays are so energetic that our bodies arenearly transparent to themlike glass is to

ordinary light. Bones are much less

transparent to X-rays than soft parts of our

bodies, indeed, each part of our bodies hasslightly different transparencies than other

parts.

Physicists would say that the rays areattenuated when they pass through dense

matter, and the amount of attenuation is

related to the properties of the matter they

are passing through.


48/71

48

School of Science


When X-rays strike film, the chemicals in thefilm are changedand the amount of the

change is related to the strength of the rays

that strike the film. When the film is

developed, an image appears that records theintensity of the X-rays reaching the film at

that spot.


49/71

49

School of Science


When X-rays strike film, the chemicals in thefilm are changedand the amount of the

change is related to the strength of the rays

that strike the film. When the film is

developed, an image appears that records theintensity of the X-rays reaching the film at

that spot.

The process is exactly analogous to the

creation of a black-and-white negative in

ordinary cameras thirty years agothe

bright areas of the picture are black, the dark

areas are white.


50/71

50

School of Science


X-ray film


51/71

51

School of Science


Just as we have replaced film in our cameraswith a light detector that creates a digital

image, in some situations X-ray film has been

replaced by X-ray detectors that create a

digital image.


52/71

52

School of Science


Just as we have replaced film in our cameraswith a light detector that creates a digital

image, in some situations X-ray film has been

replaced by X-ray detectors that create a

digital image.

This has been physics and engineering so

far, but we are now ready to add the MATH!


53/71

53

School of Science


The trouble with X-rays photographs is thattwo-dimensional images are created, so that it

is often hard to tell which piece of a broken

bone is on top of the other. It is for this

reason that doctors often take several X-raysof the same site to get different perspectives

on the situation.


54/71

54

School of Science


In CAT Scans (Computer Aided Tomography)MRIs (Magnetic Resonance Imaging), and

similar diagnostic technologies, images are

made from thousands, even millions of

different perspectives

and mathematics is used to integrate the

information from all these perspectives into a

single THREE-dimensional digital image!


55/71

55

School of Science


In CAT Scans (Computer Aided Tomography)and MRIs (Magnetic Resonance Imaging),

and similar diagnostic technologies, images

are made from thousands, even millions of

different perspectives

and mathematics is used to integratethe

information from all these perspectives into a

single THREE-dimensional digital image!

(Yes, that IS literally the integrationwe teach

our calculus students, just a sophisticated use

of it!)


56/71

56

School of Science


detectors


57/71

57

School of Science


The pictures your doctor shows you are fromslices of the three-dimensional digital image.

Sometimes, they have computers to show you

the whole three dimensional image on the

screen and then turn it or slice it to getdifferent views of particular features of

interest.


58/71

58

School of Science


The pictures your doctor shows you are fromslices of the three-dimensional digital image.

Sometimes, they have computers to show you

the whole three dimensional image on the

screen and then turn it or slice it to getdifferent views of particular features of

interest.

This new use of mathematics hasrevolutionized diagnostic medicine, yet few in

our society recognize it as new mathematics

helping all of us!


59/71

59

School of Science


SARS, HIV, Cancer, ParkinsonsIncreasingly, scientists are calling on

mathematicians to help them understand the

spread of disease or the mechanisms ofdisease by the creation of a mathematical

model of the situation.


60/71

60

School of Science


SARS, HIV, Cancer, ParkinsonsIncreasingly, scientists are calling on

mathematicians to help them understand the

spread of disease or the mechanisms ofdisease by the creation of a mathematical

model of the situation.

Just as an architect might create a cardboardmodel of a structure being designed in order

to understand it better, scientists use

mathematical models to understand problems


61/71

61

School of Science


The architects cardboard model is a smaller,simplified version of the real thing, and

because it is smaller and simpler, it is easier to

understand.


62/71

62

School of Science




understand.

Scientists and mathematicians create

mathematical models that are smaller and

simpler than the real thing so that they will be

easier to understand.


63/71

63

School of Science




understand.

Scientists and mathematicians create

mathematical models that are smaller and

simpler than the real thing so that they will be

easier to understand.

We often use geometric models of our houses

when to help us compute the amount carpet

we need to buy to cover our floors.


64/71

64

School of Science


The mathematical models used inepidemiology or in creating treatments of

cancer or Parkinsons are often differential

equations models, matrix models, or models

from discrete (sometimes called finite) math.Moreover, they often must incorporate

statistical tools into the models, because

biological processes are rarely deterministic.


65/71

65

School of Science


The mathematical models used inepidemiology or in creating treatments of

cancer or Parkinsons are often differential

equations models, matrix models, or models

from discrete (sometimes called finite) math.Moreover, they often must incorporate

statistical tools into the models, because

biological processes are rarely deterministic.

These models are used to predict the way

the real situations develop, and then used to

understand how to modify those outcomes.


66/71

66

School of Science


Main Points Sophisticated math is everywhere around us

New mathematics is being created every day

and it is being used to improve our lives We must use our influence to help counselors,

parents, and students understand the amazing

variety of careers that can be built on an

education in math and science

We must dramatically increase participation

of women and other underrepresented groups

in college and post-graduate math and science


67/71

67

School of Science


Sources Mathematical Association of America

American Mathematical Society

Association for Women in Mathematics

National Association of Mathematicians

SACNAS Society for the Advancement of

Chicanos and Native Americans in Science

h l i


68/71

68

School of Science


On-line

Magazine

that uses

history toteach

mathematics

Convergence

S h l f S i


69/71

69

School of Science


Mathematical Moments:

explanations of math used

in the modern world

S h l f S i


70/71

70

School of Science


MAA Online - Columns:

S h l f S i


71/71

School of Science


www.science.iupui.edu/ccowen

www.maa.org

convergence.mathdl.org/www.maa.org/news/columns.html

www.ams.org

www.ams.org/ams/mathmoments.html

www.awm-math.org/

www.math.buffalo.edu/mad/NAM/

www.sacnas.org

URLs for the Sources

the expanding role of mathematics in science and society

Documents