Mathematicians Puzzling Analysis 1900-1940

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

David Hilbert 1862-1943 Invariant theory Geometry Integral equations

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

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

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

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

Differential Equations 1879 Doctoral Thesis Differential Equations

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

“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)

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 (1907-1908)

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

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.”

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Studia Mathematica 1929 A mathematics journal with a focus on research in functional analysis and related topics.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

References for Project Boyer, Carl B. A history of mathematics second edition. John Wiley and Sons, inc. New York, New York. 1991. Brabanec, Robert. Resources for the study of real analysis. The mathematical association of America, inc. 2004. Burkill, J. C. “Henri Lebesgue: Obituary notices of fellows of the royal society.” Notes and records of the royal society of London. Nov 1944. June 23 2008. http://www.jstor.org/stable/768841. Edwards, Hostetler, Larson. Calculus of a single variable 7th edition. Houghton-Mifflin Company Boston, Ma. 2002. 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 2001. http://www.jstor.org/stable/531950. June 2008 O’Connor, J. J. and Robertson. MacTutor History of Mathematics. http://www-history.mcs.st-andrews.ac.uk/Biographies/ June 2008. Titchmarsh, E.C. “Godfrey Harold Hardy: Obituary notes of fellows of the royal society.” Nov 1949. V6:N18. June 23 2008. http://www.jstor .org/stable/768934.