1. Mathematicians
Puzzling Analysis

2. Donna Hash Karen Perry Levonda Rutherford
Presented by:
Donna Hash
Karen Perry
Levonda Rutherford

3. David Hilbert
Invariant theory
Geometry
Integral equations

4. International Congress of Mathematics Paris, France 1900
Hilbert proposed 23 problems as a challenge for the next century of mathematicians

5. Hilbert’s work with integrals led directly to 20th century research in functional analysis

6. Henri Poincaré 1854-1912 The last universal mathematician
Strengths/interests
Astronomy
Mathematical physics
Theory of functions
Algebraic topology
Differential equations

7. Strongly believed that it was a mistake to try to axiomatize mathematics.

8. Differential Equations
1879 Doctoral Thesis
Differential Equations

9. Considered the originator of the theory of analytic functions of several complex variables

10. “It is by logic we prove, it is by intuition that we invent
“It is by logic we prove, it is by intuition that we invent.” ~Poincaré (1904)

11. René-Louis Baire 1874-1932 Studied under Poincaré, edited lectures
Wrote several analysis books
Théorie des nombres irrationels, des limites et de la continuité (1905)
Leçons sur les théories généales de l’analyse, 2 volumes ( )

12. Took a decisive step away from intuitive idea of functions and continuity Believed the theory of infinite sets was fundamental for rigorous real analysis

13. Doctoral Thesis
“Generally speaking, in the framework of ideas that here concern us, every problem in the theory of functions leads to certain questions in the theory of sets, and it is to the degree that these latter questions are resolved, that it is possible to solve the given problem more or less completely.”

14. Godfrey Harold Hardy 1877-1947 English mathematician
Responsible for awakening an interest in analysis in England
Lectured on Calculus of Variation
Pure mathematician

15. Mathematical Interests
Diophantine analysis
Summation of divergent series
Fourier series
Riemann zeta function
Distribution of primes
Integral equations
Additive theory of numbers
Inequalities
Waring’s problem

16. Presented a new proof of the prime number theorem
Number Theory
Presented a new proof of the prime number theorem

17. Hardy’s Law
Proportions of dominant and recessive genetic traits would be transmitted in a large mixed population
This was influential in blood group distribution

18. Collaborator
Great ability to write about mathematical insights with great clarity
John Littlewood, Ramanujan, Titchmarsh, Edmund Landau, Polya, E M Wright, and others.

19. Eccentricities
Hated photographs (only 5 are known to be in existence), mirrors, and war like activities
Loved cricket
He enjoyed making list from persons, living or deceased, for the perfect team

20. A Mathematicians Apology
Hardy’s book, written to give insight on how a mathematician thinks and to show the pleasure found in mathematics

21. John Edensor Littlewood
Co-authored many papers, articles, and books with Hardy
Improved the accuracy of anti-aircraft range tables
Discovered techniques which reduced the amount of work need to make accurate calculations

22. Work with Hardy The theory of series The Riemann zeta function
Inequalities
The theory of functions
Wrote a series of papers: Partitio numerorum using the Hardy-Littlewood-Ramanujan analytical method

23. 1938: Radio Research Board
Helped with nonlinear differential equations that were appearing in radio engineering

24. Main work/interest was in classical analysis
Shared Hardy’s enjoyment of cricket

25. Srinivasa Ramanujan One of India’s greatest mathematical geniuses
Self-educated, limited in his ability to present his work in a formal mathematical proof
Hardy helped with the formal presentation of written work

26. 1913 Sent a copy of his book, Orders of Infinity, to Hardy for review
Included worked out Riemann series, elliptic integrals, hypergeometric series and functional equations of zeta
In 1914 sailed to London to collaborate with Hardy

27. Contributions to Mathematics
Analytical theory of numbers
Elliptic functions
Continuous functions
Infinite series

28. Stefan Banach 1892-1945 Founded modern functional analysis
Theory of topological vector spaces
Measure theory, integration, and orthogonal series

29. 1920 Dissertation
Axiomatically defined a topic which today is referred to as the Banach Space

30. A Banach Space is a real or complex normed vector space that is complete as a metric space under the metric d(x,y) = ||x-y|| -O’Connor and Robertson

31. Studia Mathematica A mathematics journal with a focus on research in functional analysis and related topics.

32. Banach-Tarski Paradox
A ball can be divided into subsets and fitted back together forming two identical balls.
*A major contribution to work on axiomatic set theory

33. Felix Hauśdorff 1868-1942 Topology Set Theory
Creating a theory of topological and metric spaces
Set Theory
Introduced concept of partially ordered set

34. Felix Hauśdorff
Studied Gaussian law of errors, limit theorems, and the strong law of large numbers

35. 1907
Special types of ordinals, trying to prove Cantor’s continuum hypothesis

36. Proved results on the cardinality of Borel sets
1916
Proved results on the cardinality of Borel sets

37. 1919
Wrote a paper that included a proof that the dimensions of the middle third Canter Set is log2/log3, called Hausdorff dimension

38. Henri Léon Lebesgue 1875-1941 Formulated a theory for measure in 1901
Famous paper, “Sur une généralisation de l’intégrale définie”

39. Lebesgue Integral
Generalization of the Riemann integral that revolutionized calculus
At the end of the 1800’s analysis was limited to continuous functions

40. Lebesgue Integral
He gave the definition that generalizes the notion of the Riemann integral by allowing the inclusion of discontinuous functions

41. In other words, it extended the concept of the area below a curve to include many discontinuous functions

42. 1905
The problem: Fourier’s assumption for bounded functions did not always hold
The solution: Lebesgue was able to show that term by term integration of a uniformly bounded series of Lebesgue integrable function was always valid.
*O’Connor and Robertson

43. Fourier’s proof that if a function was representable by a trigonometric series then this series is necessarily its Fourier series, this satisfied the conditions of the original proof
*O’Connor and Robertson

44. The proof was now founded on a correct result regarding term by term integration of series. *O’Connor and Robertson

45. More Contributions Topology Potential theory Dirichlet problem
Calculus of variations
Set theory
Theory of surface area
Dimension theory

46. View about Generalizations
“Reduced to general theories, mathematics would be a beautiful form without content. It would quickly die.”
~Lebesgue

47. References for Project
Boyer, Carl B. A history of mathematics second edition. John Wiley and Sons, inc. New York, New York
Brabanec, Robert. Resources for the study of real analysis. The mathematical association of America, inc
Burkill, J. C. “Henri Lebesgue: Obituary notices of fellows of the royal society.” Notes and records of the royal society of London. Nov June
Edwards, Hostetler, Larson. Calculus of a single variable 7th edition. Houghton-Mifflin Company Boston, Ma
Grattan-Guinness, I. “The interest of G. H. Hardy, f. r. s., in the philosophy and the history of mathematics.” Notes and records of the royal society of London. V55:N3. Sept June 2008
O’Connor, J. J. and Robertson. MacTutor History of Mathematics. June 2008.
Titchmarsh, E.C. “Godfrey Harold Hardy: Obituary notes of fellows of the royal society.” Nov V6:N18. June org/stable/