Connections Between Mathematics and Biology

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School of Science 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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School of Science Indiana University-Purdue
University Indianapolis
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 parents and
  counselors www.science.iupui.edu/ccowen/Careers.h
  tml

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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 parents and
  counselors www.science.iupui.edu/ccowen/Careers.h
  tml

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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 parents and
  counselors www.science.iupui.edu/ccowen/Careers.h
  tml

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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 parents and
  counselors www.science.iupui.edu/ccowen/Careers.h
  tml

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School of Science 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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School of Science 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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School of Science 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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School of Science 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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School of Science 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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School of Science Indiana University-Purdue
University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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University Indianapolis
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School of Science Indiana University-Purdue
University Indianapolis
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School of Science Indiana University-Purdue
University Indianapolis
CAT Scans and MRIs X-rays have been used for
about a century in medical diagnostics. They are
high frequency, high energy electromagnetic waves

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School of Science Indiana University-Purdue
University Indianapolis
CAT Scans and MRIs 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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School of Science Indiana University-Purdue
University Indianapolis
CAT Scans and MRIs 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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School of Science 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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School of Science 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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School of Science 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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School of Science 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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School of Science Indiana University-Purdue
University Indianapolis
X-ray film
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School of Science 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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School of Science 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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School of Science 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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School of Science Indiana University-Purdue
University Indianapolis
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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School of Science Indiana University-Purdue
University Indianapolis
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
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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School of Science Indiana University-Purdue
University Indianapolis
detectors
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School of Science 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.
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School of Science 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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School of Science Indiana University-Purdue
University Indianapolis
SARS, HIV, Cancer, Parkinsons 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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School of Science Indiana University-Purdue
University Indianapolis
SARS, HIV, Cancer, Parkinsons 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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School of Science Indiana University-Purdue
University Indianapolis
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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School of Science Indiana University-Purdue
University Indianapolis
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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School of Science Indiana University-Purdue
University Indianapolis
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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School of Science Indiana University-Purdue
University Indianapolis
The mathematical models used in epidemiology
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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School of Science Indiana University-Purdue
University Indianapolis
The mathematical models used in epidemiology
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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School of Science 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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School of Science 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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School of Science Indiana University-Purdue
University Indianapolis
Convergence
On-line Magazine that uses history to teach
mathematics
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School of Science Indiana University-Purdue
University Indianapolis
Mathematical Moments explanations of math
used in the modern world
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School of Science Indiana University-Purdue
University Indianapolis
MAA Online - Columns
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School of Science Indiana University-Purdue
University Indianapolis
URLs for the Sources
www.science.iupui.edu/ccowenwww.maa.org
convergence.mathdl.org/ www.maa.org/news/colu
mns.htmlwww.ams.org www.ams.org/ams/mathmomen
ts.htmlwww.awm-math.org/www.math.buffalo.edu/mad
/NAM/www.sacnas.org
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