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Indiana University-Purdue University Indianapolis

The Expanding Role of Mathematics in Science and Society

Carl C. Cowen, Dean

IUPUI School of Science

President, Mathematical Association of America

NCTM Annual Meeting, Anaheim

April 9, 2005

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Indiana University-Purdue University Indianapolis

• Introduction

• The Challenge and the Opportunity

• Automatic Teller Machines:

The math of ATM security

• CAT Scans and MRI’s:

The math of medical diagnostics

• SARS, HIV, Cancer, Parkinson’s:

The math of epidemics and disease

• Sources for classroom materials

2

Indiana University-Purdue University Indianapolis

- 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 parents and counselorswww.science.iupui.edu/ccowen/Careers.html

3

Indiana University-Purdue University Indianapolis

- 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 parents and counselorswww.science.iupui.edu/ccowen/Careers.html

4

Indiana University-Purdue University Indianapolis

- 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 parents and counselorswww.science.iupui.edu/ccowen/Careers.html

5

Indiana University-Purdue University Indianapolis

- 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 parents and counselorswww.science.iupui.edu/ccowen/Careers.html

6

Indiana University-Purdue University Indianapolis

- Explosion in biological research and progress
- The mathematical sciences will be a part
- Opportunity: few mathematical scientists are biologically educated few biological scientists are mathematically educated

Dr. Rita Colwell: “We're not near the fulfillment of biotechnology's promise. We're just on the cusp of it…”

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Indiana University-Purdue University Indianapolis

- Explosion in biological research and progress
- The mathematical sciences will be a part
- Opportunity: few mathematical scientists are biologically educated few biological scientists are mathematically educated

Report Bio2010: “How biologists design, perform, and analyze experiments is changing swiftly. Biological concepts and models are becoming more quantitative…”

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Indiana University-Purdue University Indianapolis

- Explosion in biological research and progress
- The mathematical sciences will be a part
- Opportunity: few mathematical scientists are biologically educated few biological scientists are mathematically educated

NSF/NIH: “Emerging areas transcend traditional academic boundaries and require interdisciplinary approaches that integrate biology, mathematics, and computer science.”

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Indiana University-Purdue University Indianapolis

ATM Security

Increasing use of electronic communications in financial and other applications has increased the need for encryption to ensure privacy

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Indiana University-Purdue University Indianapolis

ATM Security

Increasing use of electronic communications in financial and other applications has increased the need for encryption to ensure privacy

The most widely used encryption system in the world is RSA, developed in 1976 by three mathematicians, Rivest, Shamir, and Adelman.

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Indiana University-Purdue University Indianapolis

CAT Scans and MRI’s

X-rays have been used for about a century in medical diagnostics. They are high frequency, high energy electromagnetic waves

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Indiana University-Purdue University Indianapolis

CAT Scans and MRI’s

X-rays have been used for about a century in medical diagnostics. They are high frequency, high energy electromagnetic waves –– like visible light, except that the electromagnetic waves of visible light are lower frequency and lower energy.

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Indiana University-Purdue University Indianapolis

CAT Scans and MRI’s

X-rays have been used for about a century in medical diagnostics. They are high frequency, high energy electromagnetic waves –– like visible 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!

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Indiana University-Purdue University Indianapolis

X-rays are so energetic that our bodies are nearly transparent to them – like 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 has slightly different transparencies than other parts.

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Indiana University-Purdue University Indianapolis

X-rays are so energetic that our bodies are nearly transparent to them – like 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 has slightly different transparencies than other parts.

Physicists would say that the rays are attenuated when they pass through dense matter, and the amount of attenuation is related to the properties of the matter they are passing through.

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Indiana University-Purdue University Indianapolis

When X-rays strike film, the chemicals in the film are changed – and 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 the intensity of the X-rays reaching the film at that spot.

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Indiana University-Purdue University Indianapolis

When X-rays strike film, the chemicals in the film are changed – and 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 the intensity 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 ago – the bright areas of the picture are black, the dark areas are white.

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Indiana University-Purdue University Indianapolis

Just as we have replaced film in our cameras with 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.

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Indiana University-Purdue University Indianapolis

Just as we have replaced film in our cameras with 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!

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Indiana University-Purdue University Indianapolis

The trouble with X-rays photographs is that two-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-rays of the same site to get different perspectives on the situation.

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Indiana University-Purdue University Indianapolis

In CAT Scans (Computer Aided Tomography) MRI’s (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!

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Indiana University-Purdue University Indianapolis

In CAT Scans (Computer Aided Tomography) and MRI’s (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!

(Yes, that IS literally the integration we teach our calculus students, just a sophisticated use of it!)

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Indiana University-Purdue University Indianapolis

The pictures your doctor shows you are from slices 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 get different views of particular features of interest.

57

Indiana University-Purdue University Indianapolis

The pictures your doctor shows you are from slices 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 get different views of particular features of interest.

This new use of mathematics has revolutionized diagnostic medicine, yet few in our society recognize it as new mathematics helping all of us!

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Indiana University-Purdue University Indianapolis

SARS, HIV, Cancer, Parkinson’s

Increasingly, scientists are calling on mathematicians to help them understand the spread of disease or the mechanisms of disease by the creation of a mathematical model of the situation.

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Indiana University-Purdue University Indianapolis

SARS, HIV, Cancer, Parkinson’s

Increasingly, scientists are calling on mathematicians to help them understand the spread of disease or the mechanisms of disease by the creation of a mathematical model of the situation.

Just as an architect might create a cardboard model of a structure being designed in order to understand it better, scientists use mathematical models to understand problems

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Indiana University-Purdue University Indianapolis

The architect’s cardboard model is a smaller, simplified version of the real thing, and because it is smaller and simpler, it is easier to understand.

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Indiana University-Purdue University Indianapolis

The architect’s cardboard model is a smaller, simplified version of the real thing, and because it is smaller and simpler, it is easier to understand.

Scientists and mathematicians create mathematical models that are smaller and simpler than the real thing so that they will be easier to understand.

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Indiana University-Purdue University Indianapolis

The architect’s cardboard model is a smaller, simplified version of the real thing, and because it is smaller and simpler, it is easier to 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.

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Indiana University-Purdue University Indianapolis

The mathematical models used in epidemiology or in creating treatments of cancer or Parkinson’s 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.

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Indiana University-Purdue University Indianapolis

The mathematical models used in epidemiology or in creating treatments of cancer or Parkinson’s 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.

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Indiana University-Purdue University Indianapolis

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

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Indiana University-Purdue University Indianapolis

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

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Indiana University-Purdue University Indianapolis

Convergence

On-line Magazine that uses history to teach mathematics

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Indiana University-Purdue University Indianapolis

Mathematical Moments: explanations of math used in the modern world

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Indiana University-Purdue University Indianapolis

URL’s for the Sources

www.science.iupui.edu/ccowenwww.maa.org convergence.mathdl.org/ www.maa.org/news/columns.htmlwww.ams.org www.ams.org/ams/mathmoments.htmlwww.awm-math.org/www.math.buffalo.edu/mad/NAM/www.sacnas.org

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