The Expanding Role of Mathematics in Science and Society
Transcript of The Expanding Role of Mathematics in Science and Society
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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
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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
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School of Science
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 parentsand counselors
www.science.iupui.edu/ccowen/Careers.html
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School of Science
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 parentsand counselors
www.science.iupui.edu/ccowen/Careers.html
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School of Science
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 parentsand counselors
www.science.iupui.edu/ccowen/Careers.html
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School of Science
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 toparents and counselors
www.science.iupui.edu/ccowen/Careers.html
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Indiana University-Purdue University Indianapolis
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
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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
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Indiana University-Purdue University Indianapolis
ATM SecurityIncreasing use of electronic communications
in financial and other applications has
increased the need for encryption to ensureprivacy
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Indiana University-Purdue University Indianapolis
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.
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CAT Scans and MRIsX-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 MRIsX-rays have been used for about a century in
medical diagnostics. They are high frequency,
high energy electromagnetic waves likevisible light, except that the electromagnetic
waves of visible light are lower frequency and
lower energy.
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CAT Scans and MRIsX-rays have been used for about a century in
medical diagnostics. They are high frequency,
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!
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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.
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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.
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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.
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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.
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X-ray film
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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.
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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!
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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.
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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!
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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!)
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detectors
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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.
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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!
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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.
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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
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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.
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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.
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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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.
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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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.
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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.
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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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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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On-line
Magazine
that uses
history toteach
mathematics
Convergence
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Mathematical Moments:
explanations of math used
in the modern world
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MAA Online - Columns:
S h l f S i
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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