3STEFAN BANACH Founded the important modern mathematical field of functional analysis and made major contributions to the theory of topological vector spaces. In addition, he contributed to measure theory, integration, the theory of sets and orthogonal series.
4THE LIFE OF STEFAN BANACH Early years – born March 30, 1892Gymnasium - tutoring“mathematics is too sharp a tool to put intothe hands of children; for training in logical thinking, there is nothing better than accusativus cum infinitivo and ablativus absolutus.”
5THE LIFE OF STEFAN BANACH Lvov Polytechnic“since mathematics was so highly developed, it would be impossible to do anything new in this discipline…”1914 – half-diploma examinations (freshman and sophomore years)
6THE LIFE OF STEFAN BANACH Word War I- excused from military servicesAlfred Whitehead and Bernard Russell’sPrincipia Mathematica and Einstein’s special and general theories of relativityLectures at Jagiellonian UniversityHugo Steinhaus’ discoveryMarriage - Lucja Braus in 1920
8THE LIFE OF STEFAN BANACH 1920 – doctoral dissertation: “On operations on abstract sets and their applications to integral equations”Introduces an abstract object that later came to be called a Banach spaceTo some degree, this dissertation brought functional analysis to independent life.Obstacles in formal process of obtaining Ph.D
9THE LIFE OF STEFAN BANACH 1922 – Banach received his habilitation, became a Professor Extraordinarius at Jan Kazimierz UniversityNeed for writing textbooks
11The Scottish CaféWhat the Cafés of Montmartre did for the arts of Fin-de-Siècle Paris, the Scottish Café did for mathematics in Lvov.Incredibly fruitful collaboration of a group of unusually gifted and original minds.…tiny tables with marble tops were extremely useful as tablets to be covered with mathematical formulas. At first the owner was not overly enthusiastic…
12The Scottish CaféWe have to regretfully state that many valuable results of Banach and of his school were lost (…) as a result of lack of pedantry among members of the school and, first of all, Banach himself.Scottish Café – a phenomenon of teamwork in unorthodox places that led to joint solution of research problems
13The Scottish BookOne of the most revered relics of the mathematical worldA regular, ruled school notebookAn unofficial communal scientific publication – anyone interested could write down problems to be solved and anyone could write his solutions.
14The Scottish BookMany problems from Scottish Book played a significant role the development of functional analysis and other branches of mathematics.Prizes - life goose, bottle of wine, flask of brandy1972 – life goose was presented to Swedish mathematician Per Enflo.
16THE LIFE OF STEFAN BANACH Banach’s unconventional behaviorSuperb teacherAuthor of elementary textbooksSoviet occupation – Dean of the Physical-Mathematical Faculty and Head of the Department of Mathematical Analysis
17THE LIFE OF STEFAN BANACH German invasion – 1941Himmler’s Extraordinary Pacification Action – 40 Polish scholars professors, writers, and other distinguished representatives of Lvov intelligentsia perished at the hands of the Nazis.Banach’s pitiful physical condition – feeder of lice in the Rudolf Weigl Bacteriological Institute until July 1944
18THE LIFE OF STEFAN BANACH An offer of a chair at the Jagiellonian UniversityMinister of EducationBanach died on August in Lvov.
20A theorem stating that if a mapping ƒ of a metric space E into itself is a contraction, then there exists a unique element x of E such that ƒ(x) = x. Also known as contraction mapping principle.
21Banach-Tarski Paradox Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using rigid motions only, form TWO solid balls, EACH THE SAME SIZE AND SHAPE as the original?So why can't you do this in real life, say, with a block of gold?
22Banach-Tarski Paradox If matter were infinitely divisible (which it is not) then it might be possible. But the pieces involved are so "jagged" and exotic that they do not have a well-defined notion of volume, or measure, associated to them. In fact, what the Banach-Tarski paradox shows is that no matter how you try to define "volume" so that it corresponds with our usual definition for nice sets, there will always be "bad" sets for which it is impossible to define a "volume"! (Or else the above example would show that 2 = 1.)
23Banach-Tarski Paradox It is interesting to note that one corollary toThis paradox is that you can take a sphere, cut it into n pieces, remove some of the pieces, and reassemble the remaining pieces back into the original sphere without missing anything. Obviously it is not possible with a physical sphere; but it is possible with mathematical spheres (which are infinitely divisible), if the Axiom of Choice is assumed.
24Banach-Tarski Paradox An alternate version of this theorem says: it is possible to take a solid ball the size of a pea, and by cutting it into a FINITE number of pieces, reassemble it to form A SOLID BALL THE SIZE OF THE SUN.You might want to say that mathematics in this case reveals to us that we must be very careful about how we define things (like volumes) that seem very intuitive to us.
25Banach-Tarski Paradox First of all, if we didn't restrict ourselves to rigid motions, it would be more believable. For instance, you can take the interval [0,1], stretch it to twice its length and cut it into 2 pieces each the same as the original interval. Secondly, if we didn't restrict ourselves to a finite number of pieces, it would be more believable, too: the cardinality of the number of points in one ball is the same as that of two balls!
26Banach-Tarski Paradox Let A be a unit circle, and let B be a unit circle with one point X missing (called a "deleted circle"). Are sets A and B equidecomposable? Consider set B and let U be the subset consisting of all points that are a positive integer number of radians clockwise from X along the circle. This is a countable infinite set (the irrationality of Pi prevents two such points from coinciding). Let set V be everything else.If you pick set U up and rotate it counterclockwise by one radian, something very interesting happens. The deleted hole at X gets filled by the point 1 radian away, and the point at the (n-1)-th radian gets filled by the point at the n-th radian. Every point vacated gets filled, and in addition, the empty point at X gets filled too!
27Banach-Tarski Paradox Thus, B may be decomposed into sets U and V, which after this reassembling, form set A, a complete circle!This elementary example forms the beginnings of the idea of how to accomplish the Banach-Tarski paradox
28High School Connections Banach’s biographyMath Fun FactsNorm and vector’s absolute valueBanach’s Contraction Principle“Scottish Book”- form of working with mathematically gifted students (classroom’s blogs or websites)
29REFERENCESRoman Kaluza, Through a reporter’s eyes. The Life of Stefan BanachSu, Francis E., et al. "Banach-Tarski Paradox." Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>.Su, Francis E., et al. "Equidecomposability." Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>.