Transcript Positively Expansive Maps and Resolution of Singularities

Fundamental Theorem of Algebra
and
Brouwer’s Fixed Point Theorem
(talk at Mahidol University)
Wayne Lawton
Department of Mathematics
National University of Singapore
[email protected]
http://www.math.nus.edu.sg/~matwml
Quadratic Polynomials
Muhammad ibn Musa al-Khwarizmi 780-850 Baghdad, Iraq
http://www-history.mcs.st-andrews.ac.uk/history/Biographies/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 mathematicians. 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.
Polynomials With Real Coefficients
Theorem 1. Degree P odd  P has a real root
Proof Express P as P(z)  z (1  z
where d is an odd positive integer and
d
d1
Q(z)  cd1z

d
Q(z))
 c1z  c0
d1
Triangle Inequality  | Q(z) |  Cmax {1,| z |
where C  d max{| c
|,...,| c |} hence
d1
}
0
z   C  Q(z)  0  P(z)  0
z  C  Q(z)  0  P(z)  0
Intermediate Value Theorem  P has a root in
[C,C]
Complex Numbers
http://en.wikipedia.org/wiki/Complex_number
http://www.math.toronto.edu/mathnet/question
Corner/complexorigin.html
1545 Cardan, used
notation
1777 Euler, used notation
1
i, i
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.
Polynomials With Complex Coefficients
Theorem 2. (Fundamental Theorem of Algebra)
Degree P > 0  P has a complex root
http://en.wikipedia.org/wiki/Fundamental_theorem_of_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 published 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.
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.
W0
W 1
W2
Proof of The Fundamental Theorem
It suffices to prove that the following assumption :
Assumption: d = Degree P > 0 and P has no zeros
leads, through logical deduction, to a contradiction.
For
R0
we construct the closed oriented curve
i
C R ()  P(R e )  C \{0}, [0, 2]
and define
WR  Wind C R () around 0.
d
Since
P(z)  z (1  z Q(z)) ,| R |  C  WR  d
WR
is continuous and integer valued hence constant,
d
contradicting the obvious
W0  Wind c0 around 0  0.
Proof of 2-dim Brouwer Fixed Point Theorem
Theorem 3. Every continuous function
2
2
2
f : D  { x  iy : x  y  1}  D
2
has a fixed point, p  D with f (p)  p  0.
Assume that f does not have a fixed point and construct
the continuous function g: D2  S1  boundary D2
2
as shown below, and for R  [0,1] construct the curve
i Then the following facts
C ()  g(Re )
R

p
f (p)

W[C0 (),0]  W[g(0),0]  0
i
W[C1(),0]  W[e ,0]  1
contradict continuity of W[CR (),0]
g(p)
Degree of a Function
Definitions M and N are manifolds with dimension
m and n and f : M  N is smooth then
p M
q N
is a regular point if
is a regular value if
rank Df (p)  n
1
p  f (q)  pis regular
Theorem 4 Set of regular values is an open subset of N
Theorem 5 (Implicit Function)
1
f (q)isan (m  n)
q N regular 
dimensional manifold.
Theorem 6 (Sard) Almost all points are regular.
Theorem 7
Degree f   pf 1 (q) sign det Df (p)
for m = n and q regular is independent of q.
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
Degree (f :  M  N)  0
Proof (Cobordism) q N regular  f 1 (q)
is a 1-dimensional manifold, hence is a union of circles
and line segments whose ends intersect the boundary
 M of M in opposite orientations at points
1
p (f :  M  N) (q)
and thus contribute a net sum of 0 to the degree.
Theorem 9 (Approximation Theory) Degrees
can be defined for continuous functions.
Brouwer’s Fixed Point Theorem
Theorem 10. Every continuous function
f : D  { (x1,..., x m ) : 
m
m
2
x
i 1 i
1}  D
m
has a fixed point.
Proof Assume not and construct g: Dm  Sm1   Dm
as done previously. Then
Theorem 8 
m1
Degree(g: S
m1
S
)0
contradicting the fact that
m1
Degree(g: S
m1
S
)  Degree(identity)  1.
Impact of Brouwer’s Fixed Point Theorem
Theorem 11. (Kakutani) If K  R
is compact and convex and f :K  Convex(K)
n
and is a function into the set of closed convex subsets
and the graph of f is closed, then there exists
p  K such that p  f (p)
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
Into (Much) Deeper Waters
Theorem 13. (Bott Periodicity) Extends winding
number concept to functions
2n 1
f :S
 GL(m,C), m  n
and shows that every f :S2n  GL(m,C), m  n
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 (selfdual solutions of Yang-Mills – e.g. universe)