Fundamental Theorem of Algebra and Brouwers Fixed Point Theorem talk at Mahidol University

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Fundamental Theorem of Algebraand Brouwers
Fixed Point Theorem(talk at Mahidol University)

• Wayne Lawton
• Department of Mathematics
• National University of Singapore
• matwml_at_nus.edu.sg
• http//www.math.nus.edu.sg/matwml

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Quadratic Polynomials
Muhammad ibn Musa al-Khwarizmi  780-850 Baghdad,
Iraq

• http//www-history.mcs.st-andrews.ac.uk/history/Bi
  ographies/Al-Khwarizmi.html

Muslim mathematician and astronomer. He lived in
Baghdad during the golden age of Islamic science
and, like Euclid, wrote mathematical books that
collected and arranged the discoveries of earlier
mathe-maticians. His Al-Kitab al-mukhtasar
fihisab al-jabr wa'l-muqabala (The Compendious
Book on Calculation by Completion and Balancing)
is a compilation of rules for
Solving linear and quadratic equations, as well
as problems of geometry and proportion. Its
translation into Latin in the 12th century
provided the link between the great Hindu and
Arab mathematicians and European scholars. A
corruption of the book's title resulted in the
word algebra a corruption of the author's own
name resulted in the term algorithm.
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Polynomials With Real Coefficients

• Theorem 1. Degree P odd ? P has a real root

Proof Express P as
where d is an odd positive integer and
Triangle Inequality ?
where
hence
Intermediate Value Theorem ? P has a root in
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Complex Numbers

• http//en.wikipedia.org/wiki/Complex_number

http//www.math.toronto.edu/mathnet/questionCorner
/complexorigin.html
1545 Cardan, used notation
1777 Euler, used notation
Wessel in 1797 and Gauss in 1799 used the
geometric interpretation of complex numbers as
points in a plane, which made them somewhat more
concrete and less mysterious.
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Polynomials With Complex Coefficients
Theorem 2. (Fundamental Theorem of Algebra)
Degree P gt 0 ? P has a complex root
http//en.wikipedia.org/wiki/Fundamental_theorem_o
f_algebra
At the end of the 18th century two new proofs
were published which did not assume the existence
of roots. One of them, due to James Wood and
mainly algebraic, was published in 1798 and it
was totally ignored. Wood's proof had an
algebraic gap. The other one was publish- ed by
Gauss in 1799 and it was mainly geometric, but it
had a topological gap. A rigorous proof was
published by Argand in 1806 it was here that,
for the first time, the fundamental theorem of
algebra was stated for polynomials with complex
coefficients, rather than just real coefficients.
Gauss produced two other proofs in 1816 and
another version of his original proof in 1849.
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Winding Number

• http//en.wikipedia.org/wiki/Winding_number

The winding number of closed oriented curve in the
plane around a given point is an integer
representing the total number of times that
curve travels counterclockwise around the point.
It depends on the orientation of the curve, and
is negative if the curve travels around the point
clockwise.
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Proof of The Fundamental Theorem
It suffices to prove that the following
assumption
Assumption d Degree P gt 0 and P has no zeros
leads, through logical deduction, to a
contradiction.
For
we construct the closed oriented curve
and define
Since
is continuous and integer valued hence constant,
contradicting the obvious
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Proof of 2-dim Brouwer Fixed Point Theorem
Theorem 3. Every continuous function
has a fixed point,
Assume that f does not have a fixed point and
construct
the continuous function
as shown below, and for
construct the curve
Then the following facts
contradict continuity of
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Degree of a Function
Definitions M and N are manifolds with dimension
m and n and f M ? N is smooth then
is a regular point if
is a regular value if
Theorem 4 Set of regular values is an open subset
of N
Theorem 5 (Implicit Function)
dimensional manifold.
Theorem 6 (Sard) Almost all points are regular.
Theorem 7
for m n and q regular is independent of q.
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Degree of a Function
Theorem 8 M and N are manifolds with dimension m
and n m-1 and f M ? N is smooth then
Proof (Cobordism)
is a 1-dimensional manifold, hence is a union of
circles and line segments whose ends intersect
the boundary
in opposite orientations at points
of
and thus contribute a net sum of 0 to the degree.
Theorem 9 (Approximation Theory) Degrees can be
defined for continuous functions.
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Brouwers Fixed Point Theorem
Theorem 10. Every continuous function
has a fixed point.
Proof Assume not and construct
as done previously. Then
Theorem 8 ?
contradicting the fact that
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Impact of Brouwers Fixed Point Theorem
Theorem 11. (Kakutani) If
is compact and convex and
and is a function into the set of closed convex
subsets and the graph of f is closed, then there
exists
Corollary. (Nash) Every n-person noncooperative
convex game has an equilibrium solution.
John Nash shared the 1994 Nobel in Economics for
this observation, perhaps his simplest result !
http//en.wikipedia.org/wiki/John_Forbes_Nash
Theorem 12. (Leray-Schauder) Infinite dimesions
http//www.ams.org/notices/200003/mem-leray.pdf
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Into (Much) Deeper Waters
Theorem 13. (Bott Periodicity) Extends winding
number concept to functions
and shows that every
can be morphed into a constant function.
Theorem 14. (Atiyah-Singer Index) Shows that the
analytical index ( essentially different
solutions of an elliptic system of linear partial
differential equations) equals the topological
index of its symbol (Fourier transform).
Corollaries Grothendieck-Riemann-Roch,
Gauss-Bonnet, String Theory, Instantons
(self-dual solutions of Yang-Mills e.g.
universe)