1. Fundamental Theorem of Algebra and Brouwer’s Fixed Point Theorem (talk at Mahidol University) Wayne Lawton Department of Mathematics National University of Singapore matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml

2. Quadratic Polynomials http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Al-Khwarizmi.html Muhammad ibn Musa al-Khwarizmi 780-850 Baghdad, Iraq 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.quadratic equationproportion 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 forEuclid

3. 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

4. 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 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.

5. Polynomials With Complex Coefficients 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 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 forGauss Argand 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. Theorem 2. (Fundamental Theorem of Algebra) Degree P > 0  P has a complex root

6. Winding Number http://en.wikipedia.org/wiki/Winding_number planeplane around a given pointpoint is an integer representing the totalinteger 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.orientationnegative The winding number of closed oriented curve in thecurve

7. Proof of The Fundamental Theorem Assumption: d = Degree P > 0 and P has no zeros It suffices to prove that the following assumption : leads, through logical deduction, to a contradiction. Forwe construct the closed oriented curve and define Since is continuous and integer valued hence constant, contradicting the obvious

8. Proof of 2-dim Brouwer Fixed Point Theorem has a fixed point, Theorem 3. Every continuous function Assume that f does not have a fixed point and construct the continuous function as shown below, and forconstruct the curve Then the following facts contradict continuity of

9. Degree of a Function Definitions M and N are manifolds with dimension m and n and f : M  N is smooth then Theorem 5 (Implicit Function) is a regular point if is a regular value if dimensional manifold. Theorem 6 (Sard) Almost all points are regular. Theorem 4 Set of regular values is an open subset of N Theorem 7 for m = n and q regular is independent of q.

10. 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.

11. Brouwer’s Fixed Point Theorem has a fixed point. Theorem 10. Every continuous function Proof Assume not and construct as done previously. Then Theorem 8  contradicting the fact that

12. Impact of Brouwer’s Fixed Point Theorem is compact and convex and Theorem 11. (Kakutani) If and is a function into the set of closed convex subsets and the graph of f is closed, then there exists http://en.wikipedia.org/wiki/John_Forbes_Nash 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 ! Theorem 12. (Leray-Schauder) Infinite dimesions http://www.ams.org/notices/200003/mem-leray.pdf

13. 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)