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STEFAN BANACH

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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.

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THE LIFE OF STEFAN BANACH Early years – born March 30, 1892 Gymnasium - tutoring mathematics is too sharp a tool to put into the hands of children; for training in logical thinking, there is nothing better than accusativus cum infinitivo and ablativus absolutus.

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THE 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)

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THE LIFE OF STEFAN BANACH Word War I- excused from military services Alfred Whitehead and Bernard Russells Principia Mathematica and Einsteins special and general theories of relativity Lectures at Jagiellonian University Hugo Steinhaus discovery Marriage - Lucja Braus in 1920

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THE 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 space To some degree, this dissertation brought functional analysis to independent life. Obstacles in formal process of obtaining Ph.D

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THE LIFE OF STEFAN BANACH 1922 – Banach received his habilitation, became a Professor Extraordinarius at Jan Kazimierz University Need for writing textbooks

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The Scottish Café

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What 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…

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The Scottish Café We 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

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The Scottish Book One of the most revered relics of the mathematical world A regular, ruled school notebook An unofficial communal scientific publication – anyone interested could write down problems to be solved and anyone could write his solutions.

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The Scottish Book Many 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 brandy 1972 – life goose was presented to Swedish mathematician Per Enflo.

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THE LIFE OF STEFAN BANACH Banachs unconventional behavior Superb teacher Author of elementary textbooks Soviet occupation – Dean of the Physical- Mathematical Faculty and Head of the Department of Mathematical Analysis

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THE LIFE OF STEFAN BANACH German invasion – 1941 Himmlers Extraordinary Pacification Action – 40 Polish scholars professors, writers, and other distinguished representatives of Lvov intelligentsia perished at the hands of the Nazis. Banachs pitiful physical condition – feeder of lice in the Rudolf Weigl Bacteriological Institute until July 1944

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THE LIFE OF STEFAN BANACH An offer of a chair at the Jagiellonian University Minister of Education Banach died on August in Lvov.

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Banachs Fixed Point Theorem

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A 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.

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Banach-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?

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Banach-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.)

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Banach-Tarski Paradox It is interesting to note that one corollary to This 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.

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Banach-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.

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Banach-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!

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Banach-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!

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Banach-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

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High School Connections Banachs biography Math Fun Facts Norm and vectors absolute value Banachs Contraction Principle Scottish Book- form of working with mathematically gifted students (classrooms blogs or websites)

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REFERENCES Roman Kaluza, Through a reporters eyes. The Life of Stefan Banach Su, Francis E., et al. "Banach-Tarski Paradox." Mudd Math Fun Facts.. Su, Francis E., et al. "Equidecomposability." Mudd Math Fun Facts..

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