METU Department of Mathematics Math 304 History of Math Concepts Spring 2013

Final

May 23 2013, 17:30 – 19:00 150 points in 19 questions

Name and the student number:

Q 1. (6 pts) What was the contribution of Weierstrass to the analysis ? Which example constructed by Weierstrass surprised his contemporary mathematicians ? Which outstanding students of him do you know ?

Q 2. (5 pts) Who proved that numbers π and e are irrational ? Who proved that they are transcendent ?

Q 3. (5 pts) Which discovery related to cubic surfaces was made in XIX-th century ? Who is related to this discovery and how ?

Q 4. (5 pts) What is the Cayley-Klein model of a hyperbolic plane ? What are the ”lines” in these two models ? What is the distance between points ?

Q 5. (6 pts) When did Cantor publish his theory and what was its content ? Who opposed his theory, and who supported ?

Q 6. (5 pts) What was the reason of a crisis of foundations of mathematics ? When did it happen ?

Q 7. (5 pts) When and where was the International Congress of Mathematicians at which Hilbert presented his problem list ? How many problems ? Why did they turn out to be so important ?

Q 8. (5 pts) Who published works related to special relativity theory before Einstein ? When ?

Q 9. (5 pts) When and where N.Bourbaki group of mathematicians appeared ? Whom of its members do you know ?

Q 10. (10 pts) Who proposed the following concepts and notation notation i = \sqrt{−1} to use the space of higher dimension to describe the universe operations of “divergence” and “gradient” equation f_{xx} + f_{yy} + f_{zz} = 0 theory of Chaos quaternions ideals and rings vector space Mathematical Induction Nullstellensatz

Q 11. (5 pts) Which Math discoveries are related to the following authors Gaspard Wessel de Moivre Sophus Lie

Q 12. (6 pts) Who are the authors of the research works or presentations titled “Analytical Mechanics” “Elements of Geometry” “A concise outline of the foundations of Geometry” “On the number of primes less than a given magnitude” “On the hypothesis that underlines the Geometry” “Geometric Calculus”

Q 13. (18 pts) Explain briefly the following mathematical notions or statements Bezout Theorem Cramer’s Paradox Cauchy-Binet Formula Four-color Problem Prime number theorem Golbach Conjecture Dedekind section Transcendental numbers Poincare Conjecture

Q 14. (16 pts) Who in 17-19 centuries greatly contributed to the following fields of mathematics

Probability Theory Graph Theory Solution of algebraic equations Logic Abstract Algebra Linear Algebra Projective Geometry Hyperbolic Geometry

Q 15. (3 pts) In which Journals did Euler publish his papers ?

Q 16. (5 pts) When and in which journals were published for the first time the discoveries of

Abel Galois Lobachevski

Q 17. (6 pts) Give several names of mathematicians representing Italian School of Algebraic Geometry Q 18. (20 pts) Describe the Math interests and achiements of the following mathematicians (indicating the titles of their works and years of publication).

Poincare

Hilbert

Dedekind

Hamilton

Klein

Q 19. (14 pts) Which of the great mathematicians are famous for writing a large number of publications ?

Who was called ”the last universalist” ? Who ”ends up the Golden Age of Geometry” according to Bourbaki ? Who was known as a ”Prince of mathematicians” ? Who was the first and the second president of London Math. Society ? Who was the founder of the Italian school of algebraic geometry ? About whom Gauss has written “I regard this geometer as a genius of the first

order”

Who headed the Math branch of Berlin Academy of Science after Euler ? Who was known as the “father of differential geometry” ? Why Kovalevskaya had difficulty in finding a university position ? Who helped

her to find a position and at which university ?

Who were known among the most inspiring teachers ? Who is known as “the father of topology” (algebraic topology) ? Who proposed axioms for Natural Numbers ? Who studies 3-body Problem ?