Exercise problems for Math 304

1. What is Newton’s Law of Gravity? Besides its clarity, why is this physics theory different from its contemprary theories?
2. What are the philosophical implications of Law of Gravity?
3. Who are the inventors of Calculus? Which active field of mathematics do you think is related to the Calculus? Why?
4. What are the infinite series studied for?
5. How was the infinitesmality defined at first? What is the mature definition? Who provided this definition in 19th century?
6. If possible, apply Newton’s iteration method to find the root of the equation given by arctanx + lnx = 0 by using the initial value of 1.
7. What century did the French revolution take place? At the time of French revolution mathematics made great and fast progress. What is the reason for this?
8. Who is Galois? What did he do in mathematics?
9. How was the relationship between Galois and Cauchy? Why?
10. How did Galois die? At what age?
11. What is Riemann’s Zeta function? Why is this functon important?
12. State the Riemann hypothesis.
13. Who initiated the study of set theory in 19th century?
14. What was the intention of Cantor and Dedekind when they were founding set theory?
15. What is Russel’s Paradox? What was it put for in the first place?
16. What is the use of Zermelo-Frankel axiom system together with the ’Axiom of choice’?
17. What is ’Axiom of choice’?
18. What is the definition of countability for a set?
19. Prove that the set of real numbers is not countable (uncountable).
20. Name and state the hypothesis which is lying (together with the ZF axiom system) in the foundation of today’s analysis.
21. Construct a NOWHERE dense, uncountable subset of the real numbet set.
22. Who is the founder of the 20th century (combinatorial) topology? What is the name of his manuscript?
23. What is Euler characteristic? Who used the Euler characteristic as an invariant for the classification of closed surfaces upto homoemorphism?
24. How are the closed smooth surfaces with constant scalar curvature classified?
25. Show that if a closed surface has constant Gaussion (scalar) curvature of zero then it is a torus.
26. What are the ideas of Poincare on rigor and intuition?
27. What is the importance of David Hilbert for the mathematics of the 20th century?
28. Was he succesful? Why? (Hint: No!)
29. Who is Godel?
30. What did Godel prove?
31. When did Hitler take the government in Germany? When was the the Nazi regime established?
32. Before the World War II, who was the leading country in the world? How did USA become a pioneer in sciences and technology?
33. After the French revolution there had been great advances in sciences. In 20th century, there was two big scale wars in the world. What is the relation between these events?