Whats Math got to do with it: Mathematics at the frontier of science and technology

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Whats Math got to do with it?Mathematics at
the frontier of science and technology

• David Keyes
• Richard F. Barry Professor of Mathematics
  Statistics
• Adjunct Professor of Computer Science
• Old Dominion University
• adapted and augmented with permission from
  Professor Tony Chan
• Director, Institute for Pure and Applied
  Mathematicsand Dean of Sciences, UCLA

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Math Myths

• Math terrifying

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Myth 1
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Some other Math Myths

• Math terrifying
• Math static (Greeks, Newton,)
• Math ? other sciences
• Math solitary
• Math impractical major as career prep

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Math has an Image Problem

• Mathematicians smart, but from another world
• Actually, math lurks behind the curtain of
  popular subjects simulation, forecasting, data
  mining, networks, polling, design, optimization,
  synthetic environments
• Mathematicians have no monopoly on the practice
  of math nearly everyone in science and
  technology uses it

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Intellectual Foci of the Sciences
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Mathematics as Plumbing
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Math as the Hub of Science
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An Ancient Subject
Pythagorus of Samos 569-475 B.C. (?) musician,
geometer, first pure mathematician
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A Modern Subject
Karen Uhlenbeck, 1942 - University of
Texas partial differential equations and
mathematical physics
National Academy of Sciences National Medal of
Science, 2000
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Major Subfields of Math Sciences
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Math in Society and Technology
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Sir Michael Atiyah on Math
Mathematics is an art and a science, having
aesthetic and practical qualities. Modern
civilization rests firmly on a mathematical
foundation. On the other hand, the driving force
that created this great edifice is human
curiosity in its purest form.
The key principles of mathematics can be
described as analogy or abstraction.
Structural similarities between diverse areas can
be analyzed by concentrating on the essential
common features and ignoring the detail. For
example, wave motion is a term borrowed from the
sea, but common to the propagation of light,
radio, and sound, and susceptible to parallel
mathematical treatment.
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Why is Math Ubiquitous?

• Math allows description, analysis, and prediction
  (simulation) of quantitative systems
• Math exposes structures patterns of nature
• Math leverages wisdom, through abstraction
• Math gives expression to physical laws Newtons
  laws, Maxwells equations, Schrodingers
  equation, Einsteins relativity, etc.
• Math provides a lingua franca for scientific
  people across all cultures and eras
• Norbert Wiener The unreasonable effectiveness
  of math

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An Example of a Math Proof
Theorem There is no largest prime number.
Proof Assume there is a largest prime number.
List all the primes as p1, p2, ., pn. Let p
p1p2pn 1. Then p is prime because no
prime divides into it. Also, p gt pn. Hence,
contradiction!
E.g. Suppose 7 is largest prime. Then 2357
1 211 is also prime.
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An Example of Pure Math Fermats Last Theorm
The equation
n n n
X Y Z
has no integer solutions for n gt 2.
(Note n 2 has many solutions Pythagoras)
Pierre de Fermat
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Fermats Last Theorem Posed in 1630, proved
in1994
My wife has only known me while I have been
working on Fermat. I told her a few days after I
got married. I decided that I really only had
time for my problem and my family. When youre
talking to young children, theyre simply not
interested in Fermat.
Suddenly, totally unexpectedly, I had this
incredible revelation. It was the most important
moment of my working life It was so
indescribably beautiful, it was so simple and so
elegant, and I just stared in disbelief for
twenty minutes, then I walked around the
department. Id keep coming back to my desk to
see if it was still there it was still there!
The first seven years I worked on this problem I
loved every minute of it. There had been
setbacks, things which had seemed insurmountable,
but it was a private and personal battle...
Andrew J.Wiles Princeton Univ.
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Applied Math - Whats playing in classrooms and
labs near you?

• Functional Genomics
• Finance
• Cryptography
• Simulation of physical systems (e.g., airplanes,
  tokomaks, hurricanes)
• Simulation of discrete systems (e.g., traffic
  flow, networks, battlefields)
• Immersive visualization

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Functional Genomics
Slides from Molecular biology databases, by
Terry Speed, based on Chapter 2 of Post-genome
Informatics by Minoru Kanehisa, Oxford
University Press, 2000.
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Financial Mathematics

• Nova documentary about the Black-Merton-Scholes
  Formula. The film tells the fascinating story of
  the invention of the Black-Scholes Formula, a
  mathematical Holy Grail that forever altered
  the world of finance and earned its creators the
  1997 Nobel Prize in Economics.

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Akamai
Communication Networks
MIT Professor of Applied Mathematics Tom
Leighton, who had an office down the hall from
web guru Mr. Berners-Lee, was intrigued by a
challenge from the latter. ... Leighton
recognized that a solution to web congestion
could be found in applied mathematics and
algorithms. Leighton and graduate student Lewin
were joined by other scientists with expertise in
computer science and data networking to develop
the mathematical algorithms necessary to handle
the dynamic routing of content which became a
5 billion NASDAQ company.
CUSTOMERS Abercrombie Fitch Adobe Apple C-SPAN
CBS Chicago Sun-Times CNN Interactive iVillage.com
KBkids.com LA Times Lands' End Lycos Monster.com
New York Times Oxygen Media Reuters Times
Company Digital Washington Post Yahoo
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Image Processing Inpaintings

• Scratch Removal

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Synthetic Images
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Whats Math got to do with it
Picture courtesy of Doug Roble, Digital Domains.
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When to Use Simulations?
SCIENTIFIC
ENTERTAINMENT
Physical Phenomena
Physical or Artificial Phenomena
Experiments
Effects Shots
Expensive or Impossible
Inexpensive
Inexpensive
Just do it!
Just do it!
Simulations
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Simulation Complements Experimentation
Experiments prohibited or impossible
Experiments difficult to instrument
Experiments dangerous
Environment global climate groundwater flow
Experiments expensive
Experiments controversial
Scientific Simulation
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Example Aerodynamics
Airflows over wing models can be computed and
visualized
1999 Gordon Bell Prize
Anderson (NASA), Keyes Kaushik (ODU), Gropp
Smith (Argonne)
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Example Bioinformatics
Proteins fold in a way that minimizes
configuration energy
M. Wagner (ODU)
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Example Wildland Firespread
Fires at the wildland-urban interface can be
simulated, leading to strategies for planning
preventative burns, fire control, and evacuation
Joint work between ODU, Sandia National Lab, and
TRW, Inc.
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Wind-driven Fire Simulation
For this example a pair of elliptical fires, one
with an unburnt island, merge and evolve under a
wind from the west
Simulation by Vivien Mallet, Ecole Centrale de
Lyon
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Wind-driven Fire Simulation
For this example an originally elliptical fire
evolves under a wind that is originally from the
west and eventually turns to be from the south
Simulation by Vivien Mallet, Ecole Centrale de
Lyon
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Mathematics Statistics at ODU

• One of about 25-30 primarily Applied and
  Computational Departments in the US
• 28 faculty for about 40 majors and 30 graduate
  students (and thousands of student clients from
  other departments!)
• 3 UG majors Mathematics, Statistics, Mathematics
  Education
• Core department in new Computational Sciences
  Engineering Initiative

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Sample Research Projects in Math Stat at ODU

• Human radiation risk abatement in spacecraft
• Parallel computer algorithms for optically thick
  radiation transport
• Wildland firespread modeling
• Prediction of noise from ducted fan-jet engines
• Digital image compression
• Survival analysis of heart transplants
• Prostate cancer bio-markers

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Radiation Diffusion Equations

• Photon Energy equation
• Material Energy equation
• where

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How to Read an Equation!
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Grid for Marshak Wave Problem
Region of low atomic number
Simulation courtesy of Dimitri Mavriplis, ICASE,
2000
Wave impinging from left side
Region of high atomic number
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Early Time Material Energy Contours
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Later Material Energy Contours
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Final Material Energy Contours
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How does this simulation work?
Construct grid of triangles
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An Undergraduate can do this!

• Mathematical modeling (335)
• Geometry (basic)
• Multivariate calculus (311)
• Partial differential equations (401)
• Computer programming (CS 250)
• Data structures and algorithms (CS361)
• Linear algebra (316U)

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Another Marshak Obstacle Problem
Z.02
Z.1
Z.01
Z.05
Simulation courtesy of Linda Stals, ODU, 2001
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Movies of Temperature and Grid
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Old Dominion Leading 17M Dept of EnergyTOPS
Project
http//www.er.doe.gov/feature_articles_2001/august
/SCIAC/SciDAC_announcement.htm http//web.odu.edu/
webroot/orgs/IA/university_news.nsf/articles/08142
001044418PM
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Advances in Computing not just Hardware

• For the solution of important differential
  equations the progress made through better
  methods from 1945 to 1978 exceeds the progress
  made through faster computers by a factor of 40

• Factoring a large integer using modern
  mathematical techniques would be 120,000 times
  faster than using techniques from the 1970s. So
  if it takes one day to do the problem now, it
  would have taken over 300 years then.

From Numerical Methods, Software, and
Analysis, 1983, by John R. Rice, Purdue
University. From The future of integer
factorization, 1995, by Andrew Odlyzko, ATT
Bell Laboratories.
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On Algorithms and Architectures
"I would rather have today's algorithms on
yesterday's computers than vice versa."
Phillipe Toint (contemporary)
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On Experimental Mathematics
"There will be opened a gateway and a road to a
large and excellent science into which minds more
piercing than mine shall penetrate to recesses
still deeper." Galileo (1564-1642) on
experimental mathematics
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Journeys begin with a single step

• 142 courses in 24 departments throughout the ODU
  catalog have Math Stat prerequisites
• Fall 2001 3,300 students in 94 sections
• Approximately 80 of registrations are at the 100
  or 200 level
• The other 20 of the students have 80 of the fun

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Do New Math Win a Million Bucks!

• The Clay Mathematics Institute recently created
  the Millenium Prize see http//www.claymath.org/p
  rize_problems
• Seven problems that have defied solution for ages
  are singled out for 1 million in prize money
  each (Poincaré conjecture, Riemann hypothesis,
  etc.)
• The proof of Fermats last theorem (from 1630) in
  1994 has reinvigorated such quests

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Math Myths Revisited

• Math fear
• Math static (Greeks, Newton,)
• Math ? other sciences
• Math solitary
• Math impractical as a career

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Math Myths Debunked

• Math joyous
• Math dynamic (internet, genomics, )
• Math at the frontiers of other sciences
• Math collaborative
• Math foundational for evolving careers

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URLs for Followup

• This talk www.math.odu.edu/keyes/math_outreach.pp
  t
• ODU Math Stat Department www.math.odu.edu
• American Mathematical Society www.ams.org
• Society for Industrial and Applied Math
  www.siam.org
• Mathematics Awareness Activities
  http//mathforum.com/mam/01/manifoldoc.html

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Math Needs Your Support as a Citizen-Taxpayer

• ...
• As the former NIH director, Harold Varmus
  indicated, the biological research is dependent
  on continuing breakthroughs in basic science-math
  research. Over the last 7 years (from fiscal 1994
  through 2001) we have increased the NIH research
  budget from 11.544 billion to 19.729 billion.
  That is an increase of 71 since 1994. By
  contrast we have increased the NSF research
  budget in the same period only from 2.472
  billion to 3.134 billion or an increase of 27.
  This will ultimately lead to an atrophying of our
  investment in math, physics, chemistry and other
  basic knowledge and then to a decline in our
  national security, in our economic growth, and in
  our ability to do medical research. Our current
  economy is a reflection of past investments in
  scientific research (the computer chip and the
  internet are only two examples of government
  funded progress).
• ...

Newt Gingrich Letter (9/28/2000)
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Scientific Progress is Intertwined with
Mathematical Progress

• ...
• Scientists can wage an effective war on disease
  only if we--as a nation and as a scientific
  community--harness the energies of many
  disciplines, not just biology and medicine. The
  allies must include mathematicians, physicists,
  engineers and computer and behavioral scientists.
  I made this case repeatedly during my tenure as
  director of NIH, and the NIH has made significant
  efforts to boost its support of these areas. But
  in the long run, it is essential to provide
  adequate budgets for the agencies that
  traditionally fund such work and train its
  practitioners. Moreover, this will encourage the
  interagency collaboration that fuels
  interdisciplinary science. Only in this way will
  medical research be optimally poised to continue
  its dazzling progress.
• The writer is president of Memorial
  Sloan-Kettering Cancer Center and a
• former director of the National Institutes of
  Health.
• He received the Nobel Prize in Medicine in 1989.

Harold Varmus Letter (10/4/2000)
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Final Quotations
It is quite possible that mathematics was
invented in the ancient Middle East to keep track
of tax receipts and grain stores. How odd that
out of this should come a subtle scientific
language that can effectively describe and
predict the most arcane aspects of the Universe.
Isaac Asimov
The book of the Universe lies continually open
to mans gaze, yet none can hope to comprehend it
who has not first mastered the language. This
language is mathematics. Galileo
The nations may be divided by everything else,
but they all share a single body of mathematics.
Isaac Asimov
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Mathematics is Empowering!
See you in class! http//www.math.odu.edu