Title: Whats Math got to do with it: Mathematics at the frontier of science and technology
1Whats 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
2Math Myths
3Myth 1
4 Some other Math Myths
- Math terrifying
- Math static (Greeks, Newton,)
- Math ? other sciences
- Math solitary
- Math impractical major as career prep
5Math 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
6Intellectual Foci of the Sciences
7Mathematics as Plumbing
8Math as the Hub of Science
9An Ancient Subject
Pythagorus of Samos 569-475 B.C. (?) musician,
geometer, first pure mathematician
10A Modern Subject
Karen Uhlenbeck, 1942 - University of
Texas partial differential equations and
mathematical physics
National Academy of Sciences National Medal of
Science, 2000
11Major Subfields of Math Sciences
12Math in Society and Technology
13Sir 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.
14Why 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
15An 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.
16An 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
17Fermats 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.
18Applied 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
19Functional Genomics
Slides from Molecular biology databases, by
Terry Speed, based on Chapter 2 of Post-genome
Informatics by Minoru Kanehisa, Oxford
University Press, 2000.
20Financial 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.
21Akamai
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
22 Image Processing Inpaintings
23Synthetic Images
24Whats Math got to do with it
Picture courtesy of Doug Roble, Digital Domains.
25When 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
26Simulation Complements Experimentation
Experiments prohibited or impossible
Experiments difficult to instrument
Experiments dangerous
Environment global climate groundwater flow
Experiments expensive
Experiments controversial
Scientific Simulation
27Example Aerodynamics
Airflows over wing models can be computed and
visualized
1999 Gordon Bell Prize
Anderson (NASA), Keyes Kaushik (ODU), Gropp
Smith (Argonne)
28Example Bioinformatics
Proteins fold in a way that minimizes
configuration energy
M. Wagner (ODU)
29Example 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.
30Wind-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
31Wind-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
32Mathematics 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
33Sample 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
34 Radiation Diffusion Equations
- Photon Energy equation
- Material Energy equation
- where
35 How to Read an Equation!
36Grid 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
37Early Time Material Energy Contours
38Later Material Energy Contours
39Final Material Energy Contours
40How does this simulation work?
Construct grid of triangles
41An 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)
42Another Marshak Obstacle Problem
Z.02
Z.1
Z.01
Z.05
Simulation courtesy of Linda Stals, ODU, 2001
43Movies of Temperature and Grid
44Old 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
45Advances 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.
46On Algorithms and Architectures
"I would rather have today's algorithms on
yesterday's computers than vice versa."
Phillipe Toint (contemporary)
47On 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
48(No Transcript)
49 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
50Do 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
51Math Myths Revisited
- Math fear
- Math static (Greeks, Newton,)
- Math ? other sciences
- Math solitary
- Math impractical as a career
52Math Myths Debunked
- Math joyous
- Math dynamic (internet, genomics, )
- Math at the frontiers of other sciences
- Math collaborative
- Math foundational for evolving careers
53URLs 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
54Math 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)
55Scientific 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)
56Final 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
57Mathematics is Empowering!
See you in class! http//www.math.odu.edu