Transcript The P=NP problem - New Mexico State University

A brief history of Mathematics
Before the Ancient Greeks:
• Egyptians and Babylonians (c. 2000 BC):
• Knowledge comes from “papyri”
• Rhind Papyrus
Babylonian Math
• Main source: Plimpton 322
• Sexagesimal (base-sixty) originated with ancient
Sumerians (2000s BC), transmitted to
Babylonians … still used —for measuring time,
angles, and geographic coordinates
Greek Mathematics
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Thales (624-548)
Pythagoras of Samos (ca. 580 - 500 BC)
Zeno: paradoxes of the infinite
410- 355 BC- Eudoxus of Cnidus (theory
of proportion)
• Appolonius (262-190): conics/astronomy
• Archimedes (c. 287-212 BC)
Archimedes, Syracuse
Euclid (c 300 BC), Alexandria
Ptolemy (AD 83–c.168), Roman Egypt
• Almagest: comprehensive treatise on geocentric
astronomy
• Link from Greek to Islamic to European science
Al-Khwārizmī (780-850), Persia
• Algebra, (c. 820): first book
on the systematic solution
of linear and quadratic
equations.
• he is considered as the
father of algebra:
• Algorithm: westernized
version of his name
Leonardo of Pisa (c. 1170 – c. 1250)
aka Fibonacci
• Brought Hindu-Arabic
numeral system to Europe
through the publication of his
Book of Calculation, the Liber
Abaci.
• Fibonacci numbers,
constructed as an example in
the Liber Abaci.
Cardano, 1501 —1576)
• illegitimate child of Fazio Cardano, a friend of
Leonardo da Vinci.
• He published the solutions to the cubic and
quartic equations in his 1545 book Ars Magna.
• The solution to one particular case of the cubic,
x3 + ax = b (in modern notation), was
communicated to him by Niccolò Fontana
Tartaglia (who later claimed that Cardano had
sworn not to reveal it, and engaged Cardano in a
decade-long fight), and the quartic was solved by
Cardano's student Lodovico Ferrari.
John Napier (1550 –1617)
• Popularized use of the (Stevin’s)
decimal point.
• Logarithms: opposite of powers
• made calculations by hand much
easier and quicker, opened the way
to many later scientific advances.
• “MirificiLogarithmorumCanonisDesc
riptio,” contained 57 pages of
explanatory matter and 90 of tables,
• facilitated advances in astronomy
and physics
Galileo Galilei (1564-1642)
• “Father of Modern Science”
• Proposed a falling body in a vacuum
would fall with uniform acceleration
• Was found "vehemently suspect of
heresy", in supporting Copernican
heliocentric theory … and that one may
hold and defend an opinion as probable
after it has been declared contrary to
Holy Scripture.
René Descartes (1596 –1650)
• Developed “Cartesian
geometry” : uses algebra to
describe geometry.
• Invented the notation using
superscripts to show the
powers or exponents, for
example the 2 used in x2 to
indicate squaring.
Blaise Pascal (1623 –1662)
• important contributions to the
construction of mechanical
calculators, the study of fluids,
clarified concepts of pressure
and.
• wrote in defense of the scientific
method.
• Helped create two new areas of
mathematical research:
projective geometry (at 16) and
probability theory
Pierre de Fermat (1601–1665)
• If n>2, then
a^n + b^n = c^n has
no solutions in
non-zero integers
a, b, and c.
Sir Isaac Newton (1643 – 1727)
• conservation of momentum
• built the first "practical" reflecting telescope
• developed a theory of color based on
observation that a prism decomposes white light
into a visible spectrum.
• formulated an empirical law of cooling and
studied the speed of sound.
• And what else?
• In mathematics:
• development of the calculus.
• demonstrated the generalised binomial theorem,
developed the so-called "Newton's method" for
approximating the zeroes of a function....
Euler (1707 –1783)
• important discoveries in calculus…graph
theory.
• introduced much of modern
mathematical terminology and notation,
particularly for mathematical analysis,
• renowned for his work in mechanics,
optics, and astronomy.
• Euler is considered to be the preeminent
mathematician of the 18th century and
one of the greatest of all time
David Hilbert (1862 –1943)
• Invented or developed a broad
range of fundamental ideas, in
invariant theory, the
axiomatization of geometry,
and with the notion of Hilbert
space
John von Neumann ) (1903 –1957)
major contributions set theory, functional
analysis, quantum mechanics, ergodic
theory, continuous geometry, economics
and game theory, computer science,
numerical analysis, hydrodynamics and
statistics, as well as many other
mathematical fields.
Regarded as one of the foremost
mathematicians of the 20th century
Jean Dieudonné called von Neumann "the
last of the great mathematicians.”
Norbert Wiener (1894-1964)
.
• American theoretical and applied
mathematician.
• pioneer in the study of stochastic and
noise processes, contributing work
relevant to electronic engineering,
electronic communication, and control
systems.
• founded “cybernetics,” a field that
formalizes the notion of feedback and
has implications for engineering,
systems control, computer science,
biology, philosophy, and the
organization of society.
Claude Shannon (1916 –2001)]
• famous for having founded
“information theory” in 1948.
• digital computer and digital circuit
design theory in 1937
• demonstratedthat electrical
application of Boolean algebra
could construct and resolve any
logical, numerical relationship.
• It has been claimed that this was
the most important master's thesis
of all time
What does the future hold?
• Applications..
• Biology and Cybernetics
Clay Millenium Prizes
• Birch and Swinnerton-Dyer Conjectureif ζ(1) is equal to 0, then there are
an infinite number of rational points (solutions), and conversely, if ζ(1) is
not equal to 0, then there is only a finite number of such points. The
Hodge conjecture asserts that for particularly nice types of spaces called
projective algebraic varieties, the pieces called Hodge cycles are actually
(rational linear) combinations of geometric pieces called algebraic cycles.
• Navier-Stokes Equationhe challenge is to make substantial progress
P vswhich
NP Problem
toward a mathematical theory
will unlock the secrets hidden in the
Navier-Stokes equations.
• P vs NP Problem
• Poincaré Conjecture
• The Riemann hypothesis asserts that all interesting solutions of the
equation
• ζ(s) = 0
• Yang-Mills and Mass Gap