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<ul><li>Planned,  </li></ul><ul><li>Researched, </li></ul><ul><li>And Prepared by: </li></ul><ul><li>“ Trigonometers ” </l...
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Mathematicians
Luitzen Egbertus Jan Brouwer Born :  27 Feb 1881 in  Overschie  (now a suburb of Rotterdam), Netherlands Died :  2 Dec 196...
 
Biography
<ul><li>usually known by L.E.J. Brouwer form of his name with full initials, but he was known to his friends as Bertus, an...
<ul><li>Brouwer studied at the (municipal) University of Amsterdam where his most important teachers were Diederik Kortewe...
Korteweg
Gerrit Mannoury
<ul><li>In philosophy, his brainchild is  intuitionism , a revisionist foundation of mathematics.  </li></ul>
<ul><li>The implications are twofold.  </li></ul>
<ul><li>First, it leads to a form of constructive mathematics, in which large parts of classical mathematics are rejected....
<ul><li>Second, the reliance on a philosophy of mind introduces features that are absent from classical mathematics as wel...
<ul><li>Brouwer's principal students were Maurits Belinfante and Arend Heyting; the latter, in turn, was the teacher of An...
<ul><li>Brouwer's classes were also attended by Max Euwe, the later world chess champion, who published a game-theoretical...
<ul><li>Among Brouwer's assistants were Heyting, Hans Freudenthal, Karl Menger, and Witold Hurewicz, the latter two of who...
<ul><li>The most influential supporter of Brouwer's intuitionism outside the Netherlands at the time was, for a number of ...
Hermann Weyl
<ul><li>Brouwer seems to have been an independent and brilliant man of high moral standards, but with an exaggerated sense...
<ul><li>From 1914 to 1928, Brouwer was member of the editiorial board of the  Mathematische Annalen , and he was the found...
<ul><li>He was a member of, among others, the Royal Dutch Academy of Sciences, the Royal Society in London, the Preußische...
<ul><li>Brouwer received honorary doctorates from the universities of Oslo (1929) and Cambridge (1954), and was made Knigh...
<ul><li>Brouwer's archive is kept at the Department of Philosophy, Utrecht University, the Netherlands. An edition of corr...
Contribution
Luitzen Egbertus Jan Brouwer <ul><li>founded modern topology by establishing, for example, the  topological invariance of ...
<ul><li>gave the first correct definition of  dimension </li></ul>
Brouwer's Fix Point Theorem
<ul><li>Theorem 1    Every continuous mapping  f  of a closed  n -ball to itself has a fixed point. Alternatively, Let  be...
<ul><li>founded the doctrine of mathematical intuitionism, which views mathematics as the formulation of mental constructi...
Questions &  Answer
What is   topology ?
Topology <ul><li>the mathematical study of the properties that are preserved through deformations, twistings, and stretchi...
What is   intuitionism ?
<ul><li>views mathematics as a free activity of the mind, independent of any language or Platonic realm of objects, and th...
Andrey Nikolaevich Kolmogorov Born :  25 April 1903 in Tambov, Tambov province, Russia Died :  20 Oct 1987 in Moscow, Russia
 
Biography
<ul><li>Andrei Nikolaevich Kolmogorov 's parents were not married and his father took no part in his upbringing.  </li></ul>
<ul><li>His father  Nikolai Kataev , the son of a priest, was an agriculturist who was exiled.  </li></ul>
<ul><li>He returned after the Revolution to head a Department in the Agricultural Ministry but died in fighting in 1919.  ...
<ul><li>Kolmogorov's mother also, tragically, took no part in his upbringing since she died in childbirth at Kolmogorov's ...
<ul><li>His mother's sister, Vera Yakovlena, brought Kolmogorov up and he always had the deepest affection for her.  </li>...
<ul><li>In fact it was chance that had Kolmogorov born in Tambov since the family had no connections with that place.  </l...
<ul><li>Kolmogorov's mother had been on a journey from the Crimea back to her home in Tunoshna near Yaroslavl and it was i...
<ul><li>Kolmogorov's name came from his grandfather, Yakov Stepanovich Kolmogorov, and not from his own father. Yakov Step...
<ul><li>Kolmogorov graduated from Moscow State University in 1925 and then became a professor there in 1931.  </li></ul>
<ul><li>In 1939 he was elected to the Soviet Academy of Sciences, receiving the Lenin  Prize in 1965 and the Order of Leni...
Contribution
<ul><li>He laid the  mathematical foundations of   probability theory and the algorithmic theory of randomness  and made c...
<ul><li>All of these areas, and their interrelationships, underlie complex systems, as they are studied today. </li></ul>
<ul><li>His work on reformulating probability started with a 1933 paper in which he built up probability theory in a rigor...
<ul><li>Kolmogorov went on to study the motion of the planets and turbulent fluid flows, later publishing two papers in 19...
<ul><li>In 1954 he developed his work on dynamical systems in relation to planetary motion, thus demonstrating the vital r...
<ul><li>In 1965 he introduced the algorithmic theory of randomness via a measure of complexity, now referred to  Kolmogoro...
<ul><li>According to Kolmogorov, the complexity of an object is the length of the shortest computer program that can repro...
<ul><li>Random objects, in his view, were their own shortest description.  </li></ul>
<ul><li>Whereas, periodic sequences have low Kolmogorov complexity, given by the length of the smallest repeating &quot;te...
<ul><li>Kolmogorov's notion of complexity is a measure of randomness, one that is closely related to  Claude Shannon 's en...
<ul><li>Kolmogorov had many interests outside mathematics research, notable examples being the quantitative analysis of st...
Questions &  Answer
What place did he occupy among all the Soviet mathematicians in the number of foreign academies and scientific societies t...
First Place
Where did he graduate?
Moscow University
Albert Einstein Born : 14 March 1879 in Ulm, Württemberg, Germany Died : 18 April 1955 in Princeton, New Jersey, USA
 
Biography
<ul><li>Six weeks later the family moved to Munich, where he later on began his schooling at the Luitpold Gymnasium.  </li...
<ul><li>Later, they moved to Italy and Albert continued his education at Aarau, Switzerland and in 1896 he entered the Swi...
<ul><li>In 1901, the year he gained his diploma, he acquired Swiss citizenship and, as he was unable to find a teaching po...
<ul><li>In 1905 he obtained his doctor's degree.  </li></ul>
<ul><li>Unhappy with life in Berlin, his wife Mileva returned to Switzerland with their sons near the beginning of World W...
Contribution
<ul><li>Einstein worked as a professor of physics at universities in Prague and Zurich before moving to Berlin in 1914 wit...
<ul><li>He took a post at the Prussian Academy of Sciences, where he could continue his research and lecture at the Univer...
<ul><li>In 1915, Einstein perfected his general theory of relativity, summing up his theory with the mathematical equation...
<ul><li>His findings on relativity were published in  The Principle of Relativity ,  Sidelights on Relativity , and  The M...
<ul><li>In November 1919, the Royal Society of London announced that their experiment conducted during the solar eclipse o...
<ul><li>The implications of this announcement shook the world of science and earned Einstein the international acclaim he ...
What is Einstein's best-known equation?
E=mc²
According to this equation, any given amount of  WHAT  is equivalent to a certain amount of energy, and vice versa?
mass
Andre Weil Born : 6 May 1906 in Paris, France Died : 6 Aug 1998 in Princeton, New Jersey, USA
 
Biography
<ul><li>He studied  Sanskrit  as a child, loved to travel, taught at a Muslim university in India for two years (intending...
<ul><li>spent time in prison during World War II as a Jewish objector, was almost executed as a spy, escaped to America, a...
<ul><li>He once wrote: &quot;Every mathematician worthy of the name has experienced [a] lucid exaltation in which one thou...
Contribution
Algebraic Geometry  <ul><li>branch of mathematics which combines techniques of abstract algebra, especially commutative al...
<ul><li>occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as...
Weil’s Conjectures <ul><li>some highly-influential proposals from the late 1940s by André Weil on the generating functions...
<ul><li>Weil proved a special case of the Riemann hypothesis </li></ul>
<ul><li>he contributed, at least indirectly, to the recent proof of &quot;Fermat's last Theorem;&quot;  </li></ul>
<ul><li>He also worked in group theory, general and algebraic topology, differential geometry, sheaf theory, representatio...
<ul><li>He invented several new concepts including vector bundles, and uniform space.  </li></ul>
<ul><li>His work has found applications in particle physics and string theory.  </li></ul>
<ul><li>He is considered to be one of the most influential of modern mathematicians.  </li></ul>
Questions  &  Answer
What branches of mathematics are combined  in algebraic geometry?
Algebra & Geometry
What is the symbol for  theta ?
θ
Richard Dedekind Born : 6 Oct 1831 in Braunschweig, duchy of Braunschweig (now Germany) Died : 12 Feb 1916 in Braunschweig...
 
Biography
<ul><li>Much of his education took place in Brunswick as well, where he first attended school and then, for two years, the...
<ul><li>In 1850, he transferred to the University of Göttingen, a center for scientific research in Europe at the time.  <...
<ul><li>Carl Friedrich Gauss, one of the greatest mathematicians of all time, taught in Göttingen, and Dedekind became his...
<ul><li>He wrote a dissertation in mathematics under Gauss, finished in 1852.  </li></ul>
<ul><li>As was customary, he also wrote a second dissertation ( Habilitation ), completed in 1854, shortly after that of h...
<ul><li>Dedekind stayed in Göttingen for four more years, as an unsalaried lecturer ( Privatdozent ).  </li></ul>
<ul><li>During that time he was strongly influenced by P.G. Lejeune-Dirichlet, another renowned mathematician in Göttingen...
<ul><li>(Later, Dedekind did important editorial work for Gauss, Dirichlet, and Riemann.) In 1858, he moved to the Polytec...
<ul><li>He returned to Brunswick in 1862, where he became professor at the local university and taught until his retiremen...
<ul><li>In this later period, he published most of his major works.  </li></ul>
<ul><li>He also had further interactions with important mathematicians; thus, he was in correspondence with Georg Cantor, ...
<ul><li>He stayed in his hometown until the end of his life, in 1916.  </li></ul>
Contribution
<ul><li>developed a major redefinition of irrational numbers in terms of arithmetic concepts.  </li></ul>
<ul><li>Although not fully recognized in his lifetime, his treatment of the ideas of the infinite and of what constitutes ...
<ul><li>While teaching, Dedekind developed the idea that both rational and irrational numbers could form a continuum (with...
<ul><li>He said that an irrational number would then be that boundary value that separates two especially constructed coll...
Dedekind Cut <ul><li>concept advanced in 1872 by the German mathematician Richard Dedekind that combines an arithmetic for...
<ul><li>He reasoned that the real numbers form an ordered  continuum , so that any two numbers  x  and  y  must satisfy on...
<ul><li>He postulated a cut that separates the continuum into two subsets, say  X  and  Y , such that if  x  is any member...
<ul><li>If the cut is made so that  X  has a largest rational member or  Y  a least member, then the cut corresponds to a ...
<ul><li>If, however, the cut is made so that  X  has no largest rational member and  Y  no least rational member, then the...
<ul><li>Dedekind developed his arithmetical rendering of irrational numbers in 1872 in his  Stetigkeit und Irrationale Zah...
<ul><li>He also proposed, as did the German mathematician Georg Cantor, two years later, that a set—a collection of object...
<ul><li>By supplementing the geometric method in analysis, Dedekind contributed substantially to the modern treatment of t...
Questions  &  Answer
What is an  irrational number ?
Irrational Number <ul><li>any real number that cannot be expressed as the quotient of two integers </li></ul>
Who was his teacher in the University of Göttingen?
Carl Friedrich Gauss
Stefan Banach Born :  30 March 1892 in Kraków, Austria-Hungary (now Poland) Died:  31 Aug 1945 in Lvov, (now Ukraine)
 
Biography
<ul><li>son of Stefan Greczek, a tax official and, possibly, of Katarzyna Banach who was in fact his mother remains uncert...
<ul><li>Banach was brought up in Krakow by Franciszka Plowa and received his early education from a French intellectual, J...
<ul><li>In 1902, Banach finished primary school in Krakow and began his secondary education at the Henryk Sienkiewicz Gymn...
<ul><li>According to one of his colleagues, Banach was very good in mathematics and natural sciences, but was not interest...
<ul><li>He finished the Gymnasium in 1910 without distinction.   </li></ul>
<ul><li>As he felt that nothing new can be discovered in mathematics, he chose to study engineering at the Lwow Polytechni...
<ul><li>His father did not want to support him financially, so he supported himself probably by tutoring.  </li></ul>
<ul><li>During this period, he frequently left Lwow to build roads, but also attended mathematics lectures at the Jagiello...
<ul><li>In the spring of 1916, he met in Krakow, by chance, Steinhaus, a mathematician who just got a position at Lwow Uni...
<ul><li>Steinhaus, impressed by young Banach's talent for mathematics, told him about a problem he couldn't solve.  </li><...
<ul><li>Banach helped Steinhaus and the ensuing paper they wrote together was published in Krakow in 1918.  Since then Ban...
<ul><li>Also thanks to Steinhaus he met Lucja Braus, whom he married in Zakopane in 1920.   </li></ul>
<ul><li>In the same year, Banach became an assistant to Lomnicki, a professor of mathematics at Lwow Polytechnic.   </li><...
<ul><li>Lomnicki served as Banach's major advisor for his doctoral thesis.   </li></ul>
<ul><li>In 1922, the Jan Kazimierz University in Lwow awarded Banach his habilitation (a degree allowing to teach at the u...
<ul><li>In July 1922, he was appointed Extraordinary Professor.   </li></ul>
<ul><li>In 1924, he was promoted to Ordinary Professor (Full Professor).  </li></ul>
<ul><li>He spent the academic year 1924-25 in Paris.   </li></ul>
<ul><li>In 1929, he started, with Steinhaus, the journal Studia Mathematica.  </li></ul>
<ul><li>In 1931, he started co-editing, together with Steinhaus, Knaster, Kuratowski, Mazurkiewicz and Sierpinski, a serie...
<ul><li>In 1936, Banach gave a plenary address at the International Congress of Mathematicians in Oslo, Norway.  </li></ul>
<ul><li>From 1927 until 1934, he wrote some joint papers with Kuratowski.   </li></ul>
<ul><li>He also worked with Ulam and Turowicz.  </li></ul>
<ul><li>In 1939, Banach was elected President of the Polish Mathematical Society.  </li></ul>
<ul><li>After the Soviets invaded Lwow in 1939, Banach was allowed to continue to hold his chair at the university and he ...
<ul><li>Famous Soviet mathematicians Sobolev and Aleksandrov visited Banach in Lwow in 1940.   </li></ul>
<ul><li>In the same period, Banach attended conferences in the Soviet Union.  </li></ul>
<ul><li>He was in Kiev when Germany invaded the Soviet Union, but he returned immediately to his family in Lwow.  </li></ul>
<ul><li>He was arrested, but after few weeks he was released.  </li></ul>
<ul><li>He also survived the Nazi slaughter of Polish university professors.  </li></ul>
<ul><li>His advisor Lomnicki was among those who perished.  </li></ul>
<ul><li>From the end of 1941 through the remainder of the Nazi occupation (July 1944), Banach worked feeding lice in Prof....
<ul><li>After the Soviets reentered Lwow, Banach contacted his Soviet friend Sobolev, who wrote about this encounter: &quo...
<ul><li>He remained the same sociable, cheerful and extraordinarily well-meaning and charming Stefan Banach whom I had see...
<ul><li>That is how he remains in my memory: with a great sense of humor, an energetic human being, a beautiful soul and a...
Contribution
<ul><li>Founded the  important modern mathematical field of functional analysis  and made major contributions to the  theo...
<ul><li>In addition, he contributed to  measure theory ,  integration ,  the theory of sets   and orthogonal series .  </l...
Questions  &  Answer
What is the  measure theory  about?
Measure Theory <ul><li>the study of measures.  </li></ul><ul><li>generalizes the intuitive notions of length, area, and vo...
What is a  vector space ?
Vector Space <ul><li>a mathematical structure formed by a collection of vectors </li></ul>
Bernhard Riemann Born : 17 Sept 1826 in Breselenz, Hanover (now Germany) Died : 20 July 1866 in Selasca, Italy
 
Biography
<ul><li>His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars....
<ul><li>His mother died before her children had reached adulthood.  </li></ul>
<ul><li>Riemann was the second of six children, shy, and suffered from numerous nervous breakdowns.  </li></ul>
<ul><li>Riemann exhibited exceptional mathematical skills, such as fantastic calculation abilities, from an early age, but...
<ul><li>During 1840, Riemann went to Hanover to live with his grandmother and attend lyceum (middle school).  </li></ul>
<ul><li>After the death of his grandmother in 1842, he attended high school at the Johanneum Lüneburg.  </li></ul>
<ul><li>In high school, Riemann studied the Bible intensively, but he was often distracted by mathematics.  </li></ul>
<ul><li>To this end, he even tried to prove mathematically the correctness of the Book of Genesis.  </li></ul>
<ul><li>His teachers were amazed by his adept ability to solve complicated mathematical operations, in which he often outs...
<ul><li>In 1846, at the age of 19, he started studying philology and theology in order to become a priest and help with hi...
<ul><li>During the spring of 1846, his father (Friedrich Riemann), after gathering enough money to send Riemann to univers...
<ul><li>He was sent to the renowned University of Göttingen, where he first met Carl Friedrich Gauss, and attended his lec...
<ul><li>In 1847, Riemann moved to Berlin, where Jacobi, Dirichlet, Steiner, and Eisenstein were teaching.  </li></ul>
<ul><li>He stayed in Berlin for two years and returned to Göttingen in 1849.  </li></ul>
<ul><li>Bernhard Riemann held his first lectures in 1854, which founded the field of Riemannian geometry and thereby set t...
<ul><li>In 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen....
<ul><li>Although this attempt failed, it did result in Riemann finally being granted a regular salary.  </li></ul>
<ul><li>In 1859, following Dirichlet's death, he was promoted to head the mathematics department at Göttingen.  </li></ul>
<ul><li>He was also the first to suggest using dimensions higher than merely three or four in order to describe physical—a...
<ul><li>In 1862 he married Elise Koch and had a daughter.  </li></ul>
<ul><li>Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866.  </li></ul>
<ul><li>He died of tuberculosis during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore) w...
<ul><li>Meanwhile, in Göttingen his housekeeper tidied up some of the mess in his office, including much unpublished work....
<ul><li>Riemann refused to publish incomplete work and some deep insights may have been lost forever. </li></ul>
Contribution
<ul><li>Riemannian geometry </li></ul><ul><li>Riemann surface </li></ul><ul><li>Riemann integral </li></ul><ul><li>Riemann...
Riemannian Geometry <ul><li>the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a...
Riemannian Geometry <ul><li>This gives in particular local notions of angle, length of curves, surface area, and volume.  ...
Riemannian Geometry <ul><li>From those some other global quantities can be derived by integrating local contributions.  </...
Riemann Surface <ul><li>a one-dimensional complex manifold </li></ul>
<ul><li>can be thought of as &quot;deformed versions&quot; of the complex plane: locally near every point they look like p...
<ul><li>For example, they can look like a sphere or a torus or a couple of sheets glued together.  </li></ul>
Riemann Integral <ul><li>the first rigorous definition of the integral of a function on an interval.  </li></ul>
<ul><li>While the Riemann integral is unsuitable for many theoretical purposes, it is one of the easiest integrals to defi...
<ul><li>Some of these technical deficiencies can be remedied by the Riemann–Stieltjes integral, and most of them disappear...
<ul><li>a method for approximating the total area underneath a curve on a graph, otherwise known as an integral </li></ul>...
<ul><li>may also be used to define the integration operation </li></ul>
Riemann–Liouville Integral <ul><li>associates with a real function  ƒ  :  R  ->  R  another function  I α ƒ  of the same k...
<ul><li>The integral is a manner of generalization of the repeated antiderivative of  ƒ  in the sense that for positive in...
Riemann Zeta Function <ul><li>a prominent function of great significance in number theory because of its relation to the d...
Riemann Zeta Function <ul><li>also has applications in other areas such as physics, probability theory, and applied statis...
Riemann Hypothesis <ul><li>a conjecture about the distribution of the zeros of the Riemann zeta-function stating that all ...
<ul><li>The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fi...
Riemannian Metric <ul><li>a generalization of the first fundamental form of a surface in three-dimensional Euclidean space...
Question  &  Answer
Who asked his student Riemann to prepare a Habilitationsschrift on the foundations of geometry?
Carl Friedrich Gauss
Jules Henri Poincare Born : 29 April 1854 in Nancy, Lorraine, France Died : 17 July 1912 in Paris, France
 
Signature
Biography
<ul><li>a French mathematician, living at the turn of the century, who made many fundamental contributions to mathematics ...
<ul><li>In the natural sciences he is best  appreciated for his highly original work on celestial mechanics.  </li></ul>
<ul><li>Residence :  France </li></ul><ul><li>Nationality :  French </li></ul><ul><li>Fields :  Mathematician and physicis...
<ul><li>Institutions :  </li></ul><ul><ul><li>Corps des Mines </li></ul></ul><ul><ul><li>Caen University  </li></ul></ul><...
<ul><li>Alma mater: </li></ul><ul><ul><li>Lycée Nancy </li></ul></ul><ul><ul><li>École Polytechnique </li></ul></ul><ul><u...
Contribution
<ul><li>Poincaré conjecture </li></ul><ul><li>Topology </li></ul><ul><li>Special relativity </li></ul>
<ul><li>Poincaré–Hopf Theorem </li></ul><ul><li>Poincaré duality </li></ul><ul><li>Poincaré–Birkhoff–Witt theorem </li></u...
<ul><li>Hilbert–Poincaré series </li></ul><ul><li>Poincaré metric </li></ul><ul><li>Rotation number </li></ul><ul><li>Coin...
<ul><li>Chaos theory </li></ul><ul><li>Sphere-world </li></ul><ul><li>Poincaré–Bendixson theorem </li></ul>
<ul><li>Poincaré–Lindstedt method </li></ul><ul><li>Poincaré Recurrence </li></ul>
<ul><li>RAS Gold Medal  (1900) </li></ul><ul><li>Sylvester Medal  (1901) </li></ul><ul><li>Matteucci Medal  (1905) </li></...
Poincaré Conjecture <ul><li>states that every simply connected closed three-manifold is homeomorphic to the three-sphere (...
<ul><li>More colloquially, the conjecture says that the three-sphere is the only type of bounded three-dimensional space p...
<ul><li>the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of ob...
<ul><li>generalizes Galileo's principle of relativity–that all uniform motion is relative, and that there is no absolute a...
<ul><li>– from mechanics to all the laws of physics, including both the laws of mechanics and of electrodynamics, whatever...
<ul><li>incorporates the principle that the speed of light is the same for all inertial observers regardless of the state ...
<ul><li>illustrated by the special case of the Hairy ball theorem, which simply states that there is no smooth vector fiel...
<ul><li>states that if  M  is an  n -dimensional oriented closed manifold (compact and without boundary), then the  k th c...
<ul><li>the product of canonical monomials in  Y  can be reduced uniquely to a linear combination of canonical monomials b...
<ul><li>allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of defini...
<ul><li>Such bounds are of great importance in the modern, direct methods of the calculus of variations. </li></ul>
<ul><li>a formal power series in one indeterminate, say  t , where the coefficient of  tn  gives the dimension (or rank) o...
<ul><li>the metric tensor describing a two-dimensional surface of constant negative curvature.  </li></ul>Poincaré Metric
<ul><li>It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces.  <...
<ul><li>an invariant of homeomorphisms of the circle.  </li></ul>Rotation Number
<ul><li>It was first defined by Henri Poincaré in 1885, in relation to the precession of the perihelion of a planetary orb...
<ul><li>Poincaré later proved a theorem characterizing the existence of periodic orbits in terms of rationality of the rot...
<ul><li>can be used to distinguish topological spaces.  </li></ul>Betti Number
<ul><li>Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing...
Chaos Theory <ul><li>an area of inquiry in mathematics, physics, and philosophy studying the behavior of dynamical systems...
Sphere-world <ul><li>The idea of a sphere-world was constructed by Henri Poincaré while pursuing his argument for conventi...
<ul><li>offered a thought experiment about a sphere with strange properties. </li></ul>
Poincaré–Bendixson Theorem <ul><li>tells that the fate of any bounded solution of a differential equation in the is to con...
Poincaré–Lindstedt Method <ul><li>Provides us with a consistent perturbation scheme that identifies the frequency correcti...
<ul><li>gives us an approximation for the displacement x(t) </li></ul>
Poincaré Recurrence <ul><li>states that certain systems will, after a sufficiently long time, return to a state very close...
Question  &  Answer
What was founded by Jules Henri Poincare  through his innovations and is the mathematical theory of dynamical systems?
Qualitative Dynamics
What new mathematical tool was used by Jules Henri Poincare to attempt to answer a very longstanding question “ Is the sol...
Topology
<ul><li>At the end of the 19th century this question was re-posed by  King Oscar II of Sweden  with a cash prize promised ...
King Oscar II of Sweden
<ul><li>In attacking the problem Poincare limited his sights to the restricted problem of just three bodies moving under t...
<ul><li>He won the prize with his publication of &quot;On The Problem of Three Bodies and the Equations of Equilibrium&quo...
<ul><li>But through this investigation Poincare came to understand that infinitely complicated behaviors could arise in si...
<ul><li>Without the benefit of computers, only through his mathematical insight and his calculation abilities, he was able...
David Hilbert Born : January 23, 1862) Königsberg  or  Wehlau  (today  Znamensk, Kaliningrad Oblast ),  Province of Prussi...
 
Biography
<ul><li>Residence :  Germany </li></ul><ul><li>Nationality :  German </li></ul><ul><li>Fields :  Mathematician and Philoso...
David Hilbert <ul><li>the first of two children and only son of Otto and Maria Therese (Erdtmann) Hilbert </li></ul>
<ul><li>born in either  Königsberg  (according to Hilbert's own statement) or in  Wehlau  (today Znamensk, Kaliningrad Obl...
<ul><li>In the fall of 1872, he entered the  Friedrichskolleg Gymnasium  (the same school that Immanuel Kant had attended ...
<ul><li>but after an unhappy duration he transferred (fall 1879) to and graduated from (spring 1880) the more science-orie...
<ul><li>Upon graduation he enrolled (autumn 1880) at the University of Königsberg, the &quot;Albertina&quot;. </li></ul>
<ul><li>In the spring of 1882, Hermann Minkowski (two years younger than Hilbert and also a native of Königsberg but so ta...
<ul><li>&quot;Hilbert knew his luck when he saw it. In spite of his father's disapproval, he soon became friends with the ...
<ul><li>In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius, i.e., an associate professor.  </li></ul>
<ul><li>An intense and fruitful scientific exchange between the three began and especially Minkowski and Hilbert would exe...
<ul><li>Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled </li></ul>
<ul><li>Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen   </li></ul><ul><li>(&qu...
<ul><li>Hilbert remained at the University of Königsberg as a professor from 1886 to 1895.  </li></ul>
<ul><li>In 1892, Hilbert married  Käthe Jerosch  (1864–1945), &quot;the daughter of a Konigsberg merchant, an outspoken yo...
<ul><li>While at Königsberg they had their one child  Franz Hilbert  (1893–1969).  </li></ul>
<ul><li>In 1895, as a result of intervention on his behalf by Felix Klein he obtained the position of Chairman of Mathemat...
<ul><li>His son Franz would suffer his entire life from an (undiagnosed) mental illness, his inferior intellect a terrible...
<ul><li>Sadly, Minkowski — Hilbert's &quot;best and truest friend” — would die prematurely of a ruptured appendix in 1909....
Contribution
<ul><li>Hilbert's basis theorem </li></ul><ul><li>Hilbert's axioms </li></ul><ul><li>Hilbert's problems </li></ul><ul><li>...
Hilbert's basis Theorem <ul><li>states that every ideal in the ring of multivariate polynomials over a field is finitely g...
<ul><li>This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as th...
<ul><li>a set of 20 (originally 21) assumptions proposed by David Hilbert in 1899, as the foundation for a modern treatmen...
Hilbert's Problems <ul><li>a set of (originally) unsolved problems in mathematics proposed by Hilbert.  </li></ul>
<ul><li>Of the 23 total appearing in the printed address, ten were actually presented at the Second International Congress...
Hilbert's Program <ul><li>a proposed solution to the foundational crisis of mathematics, when early attempts to clarify th...
Einstein–Hilbert Action <ul><li>he action that yields the Einstein's field equations through the principle of least action...
Hilbert Space <ul><li>an abstract vector space possessing the structure of an inner product that allows length and angle t...
<ul><li>David Hilbert was a German mathematician, recognized as one of the most influential and universal mathematicians o...
<ul><li>He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the a...
<ul><li>He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis. </li></ul>
<ul><li>Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers.  </li></ul>
<ul><li>A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set th...
<ul><li>Hilbert and his students contributed significantly to establishing rigor and some tools to the mathematics used in...
<ul><li>He is also known as one of the founders of proof theory, mathematical logic and the distinction between mathematic...
Question  &  Answer
<ul><li>In what areas did David Hilbert discovered and developed a broad range of fundamental ideas in many areas? </li></ul>
<ul><li>Invariant theory and the axiomatization of geometry </li></ul>
<ul><li>David Hilbert is also known as one of the founders of what? </li></ul>
<ul><li>proof theory, mathematical logic and the distinction between mathematics and metamathematics </li></ul>
Alexander Grothendieck Born:  March 28, 1928 in Berlin, Germany
 
Biography
<ul><li>Alexander Grothendieck is considered one of the greatest mathematicians of the 20th century.  </li></ul>
<ul><li>Residence : France </li></ul><ul><li>Nationality : Stateless </li></ul><ul><li>Field : Mathematician </li></ul>
Alexander Grothendieck <ul><li>born in Berlin to anarchist parents: a Russian father from an ultimately Hassidic family,  ...
<ul><li>and a mother from a German Protestant family,  Johanna &quot;Hanka&quot; Grothendieck ; both of his parents had br...
<ul><li>At the time of his birth Grothendieck's mother was married to  Johannes Raddatz , a German journalist, and his bir...
<ul><li>The marriage was dissolved in 1929 and Shapiro/Tanaroff acknowledged his paternity, but never married Hanka Grothe...
<ul><li>Grothendieck lived with his parents until 1933 in Berlin.  </li></ul>
<ul><li>At the end of that year, Shapiro moved to Paris, and Hanka followed him the next year.  </li></ul>
<ul><li>They left Grothendieck in the care of Wilhelm Heydorn, a Lutheran Pastor and teacher in Hamburg where he went to s...
<ul><li>During this time, his parents fought in the Spanish Civil War.  </li></ul>
Contribution
<ul><li>Algebraic Geometry </li></ul><ul><li>Homological Algebra </li></ul><ul><li>Functional Analysis </li></ul>
Algebraic Geometry <ul><li>a branch of mathematics which combines techniques of abstract algebra, especially commutative a...
Homological Algebra <ul><li>branch of mathematics which studies homology in a general algebraic setting.  </li></ul>
Functional Analysis <ul><li>the branch of mathematics, and specifically of analysis, concerned with the study of vector sp...
<ul><li>He is noted for his mastery of abstract approaches to mathematics, and his perfectionism in matters of formulation...
<ul><li>In particular, he demonstrated the ability to derive concrete results using only very general methods.  </li></ul>
<ul><li>Relatively little of his work after 1960 was published by the conventional route of the learned journal, circulati...
<ul><li>He is the subject of many stories and some misleading rumors concerning his work habits and politics, his confront...
Question  &  Answer
<ul><li>Grothendieck   is most famous for his revolutionary advances in what? </li></ul>
<ul><li>Algebraic Geometry </li></ul>
<ul><li>Grothendieck has also made major contributions to what? </li></ul>
<ul><li>Algebraic Topology ,  Number Theory ,  Category Theory ,  Galois Theory ,  Descent Theory ,  Commutative Homologic...
Joseph Louis Lagrange Born:  25 January 1736) Turin, Piedmont Died:  10 April 1813 (aged 77) Paris, France
 
Biography
<ul><li>Residence: </li></ul><ul><li>Piedmont </li></ul><ul><li>France </li></ul><ul><li>Prussia </li></ul><ul><li>Nationa...
<ul><li>Fields: </li></ul><ul><li>Mathematics </li></ul><ul><li>Mathematical physics </li></ul><ul><li>Institutions:   Éco...
<ul><li>born, of French and Italian descent (a paternal great grandfather was a French army officer who then moved to Turi...
<ul><li>His father, who had charge of the Kingdom of Sardinia's military chest, was of good social position and wealthy,  ...
<ul><li>but before his son grew up he had lost most of his property in speculations, and young Lagrange had to rely on his...
<ul><li>He was educated at the college of Turin, but it was not until he was seventeen that he showed any taste for mathem...
<ul><li>Alone and unaided he threw himself into mathematical studies;  </li></ul>
<ul><li>at the end of a year's incessant toil he was already an accomplished mathematician, and was made a lecturer in the...
Contribution
<ul><li>Analytical Mechanics </li></ul><ul><li>Celestial Mechanics </li></ul><ul><li>Mathematical Analysis </li></ul><ul><...
Analytical Mechanics <ul><li>a term used for a refined, highly mathematical form of classical mechanics, constructed from ...
Celestial Mechanics <ul><li>branch of astronomy that deals with the motions of celestial objects.  </li></ul>
Mathematical Analysis <ul><li>branch of pure mathematics most explicitly concerned with the notion of a limit, whether the...
Number Theory <ul><li>branch of pure mathematics concerned with the properties of numbers in general, and integers in part...
Question  &  Answer
<ul><li>What theorem stated by Bachet did he proved without justification? </li></ul>
<ul><li>He proved the theorem, stated by Bachet without justification, that every positive integer is the sum of four squa...
<ul><li>What equations did he reduced to their canonical forms? </li></ul>
<ul><li>He reduced the equations of the quadrics (or conicoids) to their canonical forms. </li></ul>
Adrien-Marie Legendre Born:  18 September 1752) Paris ,  France Died:  10 January 1833 (aged 80) Paris ,  France
 
Biography
<ul><li>Residence   France </li></ul><ul><li>Nationality   French </li></ul>
<ul><li>Institutions   </li></ul><ul><li>École Militaire </li></ul><ul><li>Alma mater   </li></ul><ul><li>Collège Mazarin ...
<ul><li>born in Paris (or possibly, in Toulouse, depending on sources) on 18 September 1752 to a wealthy family.  </li></ul>
<ul><li>He was given a top quality education at the Collège Mazarin in Paris, defending his thesis in physics and mathemat...
<ul><li>From 1775 to 1780 he taught at the École Militaire in Paris, and from 1795 at the École Normale, and was associate...
<ul><li>In 1782, he won the prize offered by the Berlin Academy for his treatise on projectiles in resistant media, which ...
<ul><li>In 1783 he became an  adjoint of the   Académie des Sciences , and an  associé  in 1785.  </li></ul>
<ul><li>During the French Revolution, in 1793, he lost his private fortune, but with the help of his wife, Marguerite-Clau...
<ul><li>In 1795 he became one of the six members of the mathematics section of the reconstituted Académie des Sciences, na...
<ul><li>In 1824, as a result of refusing to vote for the government candidate at the Institut National, Legendre was depri...
<ul><li>He died in Paris on 9 January 1833, after a long and painful illness.  </li></ul>
<ul><li>Legendre's widow made a cult of his memory, carefully preserving his belongings.  </li></ul><ul><li>Upon her death...
Contribution
<ul><li>Legendre Transformation  </li></ul><ul><li>Elliptic Functions </li></ul><ul><li>Éléments de géométrie  </li></ul>
<ul><li>Legendre Symbol  </li></ul><ul><li>Least Squares Method </li></ul>
Legendre Transformation  <ul><li>an operation that transforms one real-valued function of a real variable into another.  <...
Elliptic Functions <ul><li>a function defined on the complex plane that is periodic in two directions (a doubly periodic f...
Éléments de géométrie  <ul><li>published in 1794 and was the leading elementary text on the topic for around 100 years.  <...
Éléments de géométrie <ul><li>greatly rearranged and simplified many of the propositions from Euclid's Elements to create ...
Legendre Symbol  <ul><li>served as the prototype for innumerable higher power residue symbols </li></ul>
Least Squares Method <ul><li>a procedure to determine the best fit line to data; the proof uses simple calculus and linear...
Question  &  Answer
<ul><li>What were his works that were brought to perfection by others? </li></ul>
<ul><li>His work on roots of polynomials inspired Galois theory; Abel's work on elliptic functions was built on Legendre's...
<ul><li>What method did he developed that has broad application in linear regression, signal processing, statistics, and c...
<ul><li>He developed the least squares method, which has broad application in linear regression, signal processing, statis...
Aryabhata Born:  476 in Kusumapura (now Patna), India Died:  550 in India
Statue of Aryabhata on the grounds of IUCAA, Pune.  As there is no known information regarding his appearance, any image o...
Biography
<ul><li>Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematic...
<ul><li>While there is a tendency to misspell his name as &quot;Aryabhatta&quot; by analogy with other names having the &q...
<ul><li>every astronomical text spells his name thus, including Brahmagupta's references to him &quot;in more than a hundr...
<ul><li>Furthermore, in most instances &quot;Aryabhatta&quot; does not fit the metre either. </li></ul>
<ul><li>Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga, when he was 23 years ol...
<ul><li>This corresponds to 499 CE, and implies that he was born in 476 CE. </li></ul>
<ul><li>Aryabhata provides no information about his place of birth.  </li></ul>
<ul><li>The only information comes from Bhāskara I, who describes Aryabhata as āśmakīya, &quot;one belonging to the aśmaka...
<ul><li>While aśmaka was originally situated in the northwest of India, it is widely attested that, during the Buddha's ti...
<ul><li>Aryabhata is believed to have been born there. However, early Buddhist texts describe Ashmaka as being further sou...
<ul><li>while other texts describe the Ashmakas as having fought Alexander, which would put them further north. </li></ul>
Contribution
<ul><li>Place value system and zero  </li></ul><ul><li>Pi as irrational  </li></ul><ul><li>Mensuration and trigonometry  <...
<ul><li>Indeterminate equations  </li></ul><ul><li>Algebra  </li></ul><ul><li>Aryabhatiya </li></ul>
Place value system and zero  <ul><li>first seen in the 3rd century Bakhshali Manuscript, was clearly in place in his work;...
<ul><li>he certainly did not use the symbol, but French mathematician Georges Ifrah argues that knowledge of zero was impl...
Pi as irrational  <ul><li>Aryabhata worked on the approximation for Pi (π), and may have come to the conclusion that π is ...
<ul><li>In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes: </li></ul>
<ul><li>chaturadhikam śatamaśṭaguṇam dvāśaśṭistathā sahasrāṇām Ayutadvayaviśkambhasyāsanno vrîttapariṇahaḥ. </li></ul>
<ul><li>&quot;Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a d...
<ul><li>This implies that the ratio of the circumference to the diameter is  ((4+100)×8+62000)/20000 = 3.1416 , which is a...
Mensuration and trigonometry  <ul><li>In Ganitapada 6, Aryabhata gives the area of a triangle as </li></ul>
<ul><ul><li>tribhujasya phalashariram samadalakoti bhujardhasamvargah </li></ul></ul>
<ul><li>that translates to:  </li></ul><ul><ul><li>&quot; for a triangle, the result of a perpendicular with the half-side...
Indeterminate Equations <ul><li>A problem of great interest to Indian mathematicians since ancient times has been to find ...
Algebra   <ul><li>In Aryabhatiya Aryabhata provided elegant results for the summation of series of squares and cubes: </li...
<ul><li>and   </li></ul>
Aryabhatiya <ul><li>Direct details of Aryabhata's work are therefore known only from the Aryabhatiya. </li></ul>
<ul><li>The name &quot;Aryabhatiya&quot; is due to later commentators. </li></ul>
<ul><li>The text consists of the 108 verses and 13 introductory verses, and is divided into four pādas or chapters: </li><...
<ul><li>Gitikapada : (13 verses):  </li></ul><ul><ul><li>large units of time—kalpa,  </li></ul></ul><ul><ul><li>manvantra,...
<ul><li>Ganitapada  (33 verses):  </li></ul><ul><ul><li>covering mensuration   </li></ul></ul><ul><ul><li>arithmetic and g...
<ul><li>Kalakriyapada  (25 verses):  </li></ul><ul><ul><li>different units of time  </li></ul></ul><ul><ul><li>a method fo...
<ul><li>Golapada  (50 verses):  </li></ul><ul><ul><li>Geometric/trigonometric aspects of the celestial sphere  </li></ul><...
Question  &  Answer
<ul><li>What were his most famous works? </li></ul>
<ul><li>His most famous works are the Aryabhatiya (499 CE, when he was 23 years old) and the Arya-siddhanta.  </li></ul>
<ul><li>Aryabhatiya  consists of the 108 verses and 13 introductory verses, and is divided into four pādas or chapters. Wh...
<ul><li>Gitikapada  (13 verses) </li></ul><ul><li>Ganitapada  (33 verses) </li></ul><ul><li>Kalakriyapada  (25 verses) </l...
Jakob Steiner Born:  18 March 1796 in Utzenstorf, Switzerland Died:  1 April 1863 in Bern, Switzerland
 
Biography
<ul><li>Jakob Steiner's parents were Anna Barbara Weber (1757-1832) and Niklaus Steiner (1752-1826).  </li></ul>
<ul><li>Anna and Niklaus were married on 28 January 1780 and they had eight children.  </li></ul>
<ul><li>Jakob was the youngest of the children and spent his early years helping his parents with the small farm and busin...
<ul><li>He did not learn to read and write until he was 14 but he then proved invaluable  </li></ul><ul><li>As a child he ...
<ul><li>At the age of 18, against the wishes of his parents, he left home to attend Johann Heinrich Pestalozzi's school at...
<ul><li>The fact that Steiner was unable to pay anything towards his education at the school was not a problem, for Pestal...
<ul><li>Pestalozzi's school had a very significant effect on Steiner's attitude both to the teaching of mathematics and al...
<ul><li>Pestalozzi's school had a very significant effect on Steiner's attitude both to the teaching of mathematics and al...
<ul><li>In the autumn of 1818, Steiner left Yverdom and travelled to Heidelberg where he earned his living giving private ...
<ul><li>He attended lectures at the Universities of Heidelberg on combinatorial analysis, differential and integral calcul...
<ul><li>Also at this time he became interested in mechanics and he wrote three unpublished manuscripts on the topic in 182...
<ul><li>At Easter 1821 he left Heidelberg and travelled to Berlin, where again he supported himself with a very modest inc...
<ul><li>He had no formal teaching qualifications so he decided that he needed to sit the necessary examinations to allow h...
<ul><li>He was not completely successful for after taking the necessary examinations in Berlin he was only awarded a restr...
<ul><li>His problem was not in mathematics but in the other subjects which were examined such as history and literature.  ...
<ul><li>This restricted license was, however, sufficient to allow him to be appointed to the Werder Gymnasium in Berlin.  ...
<ul><li>At first he received good reports on his teaching but he fell out with the director of the school, Dr Zimmermann. ...
<ul><li>Steiner, who was a firm believer in Pestalozzi's methods of teaching, used those methods in the classroom.  </li><...
<ul><li>Zimmermann claimed that these were only suitable for elementary courses, and Steiner was dismissed in the autumn o...
<ul><li>The official reason was that his teaching was receiving criticism but the real reason was clearly his desire to us...
<ul><li>Again he took up private tutoring to earn enough money to allow him to attend courses at the University of Berlin,...
<ul><li>In 1825 Steiner was appointed as an assistant master at the Technical School of Berlin.  </li></ul>
Contribution
<ul><li>Synthetic Geometry </li></ul><ul><li>Systematische Entwickelung der Abhängigkeit geometrischer Gestalten von einan...
<ul><li>Die geometrischen Constructionen ausgeführt mittels der geraden Linie und eines festen Kreises </li></ul><ul><li>V...
Synthetic Geometry <ul><li>the branch of geometry which makes use of theorems and synthetic observations to draw logical c...
Systematische Entwickelung der Abhängigkeit geometrischer Gestalten von einander   Éléments de géométrie
<ul><li>He laid the foundation of modern synthetic geometry.  </li></ul>
<ul><li>He introduces what are now called the geometrical forms (the row, flat pencil etc.), and establishes between their...
Die geometrischen Constructionen ausgeführt mittels der geraden Linie und eines festen Kreises
<ul><li>He shows, what had been already suggested by J. V. Poncelet, how all problems of the second order can be solved by...
Vorlesungen über synthetische Geometrie
<ul><li>published posthumously at Leipzig by C. F. Geiser and H. Schroeter in 1867 </li></ul>
Question  &  Answer
<ul><li>Steiner's mathematical work was mainly confined to what field of mathematics? </li></ul>
<ul><li>Geometry </li></ul>
<ul><li>He has been considered _______________since Apollonius of Perga. </li></ul>
<ul><li>the greatest pure geometer genius </li></ul>
John Brehaut Wallis Born:  23 Nov 1616 in Ashford, Kent, England Died:  28 Oct 1703 in Oxford, England
John Wallis
Biography
<ul><li>an English mathematician who is given partial credit for the development of modern calculus.  </li></ul>
<ul><li>Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal court.  </li></ul>
<ul><li>He is also credited with introducing the symbol  ∞  for infinity.  </li></ul><ul><li>Asteroid 31982 Johnwallis  wa...
<ul><li>born in Ashford, Kent, the third of five children of Reverend John Wallis and Joanna Chapman.  </li></ul>
<ul><li>initially educated at a local Ashford school, but moved to James Movat's school in Tenterden in 1625 following an ...
<ul><li>Wallis was first exposed to mathematics in 1631, at Martin Holbeach's school in Felsted; he enjoyed maths, but his...
<ul><li>As it was intended that he should be a doctor, he was sent in 1632 to Emmanuel College, Cambridge.  </li></ul>
<ul><li>While there, he kept an act on the doctrine of the circulation of the blood; that was said to have been the first ...
<ul><li>His interests, however, centered on mathematics.  </li></ul>
<ul><li>He received his Bachelor of Arts degree in 1637, and a Master's in 1640, afterwards entering the priesthood.  </li...
<ul><li>From 1643-49, he served as a non-voting scribe at the Westminster Assembly.  </li></ul>
<ul><li>Wallis was elected to a fellowship at Queens' College, Cambridge in 1644, which he however had to resign following...
<ul><li>Throughout this time, Wallis had been close to the Parliamentarian party, perhaps as a result of his exposure to H...
<ul><li>He rendered them great practical assistance in deciphering Royalist dispatches.  </li></ul>
<ul><li>The quality of cryptography at that time was mixed; despite the individual successes of mathematicians such as Fra...
<ul><li>Most ciphers were ad-hoc methods relying on a secret algorithm, as opposed to systems based on a variable key.  </...
<ul><li>Returning to London — he had been made chaplain at St Gabriel Fenchurch, in 1643 — Wallis joined the group of scie...
<ul><li>He was finally able to indulge his mathematical interests, mastering William Oughtred's Clavis Mathematicae in a f...
<ul><li>He soon began to write his own treatises, dealing with a wide range of topics, continuing throughout his life. </l...
<ul><li>Wallis joined the moderate Presbyterians in signing the remonstrance against the execution of Charles I, by which ...
<ul><li>In spite of their opposition he was appointed in 1649 to be the Savilian Chair of Geometry at Oxford University, w...
<ul><li>Besides his mathematical works he wrote on theology, logic, English grammar and philosophy, and he was involved in...
<ul><li>Although William Holder had earlier taught a deaf man Alexander Popham to speak ‘plainly and distinctly, and with ...
<ul><li>Wallis later claimed credit for this, leading Holder to accuse Wallis of 'rifling his Neighbours, and adorning him...
Contribution
<ul><li>Opera Mathematica I </li></ul><ul><li>Analytical Geometry </li></ul><ul><li>Integral Calculus </li></ul><ul><li>Al...
Opera Mathematica I <ul><li>In his Opera Mathematica I (1695) Wallis introduced the term &quot;continued fraction.&quot; <...
Analytical Geometry <ul><li>In 1655, Wallis published a treatise on conic sections in which they were defined analytically...
Analytical Geometry <ul><li>This was the earliest book in which these curves are considered and defined as curves of the s...
Analytical Geometry <ul><li>It helped to remove some of the perceived difficulty and obscurity of René Descartes' work on ...
Integral Calculus <ul><li>Arithmetica Infinitorum , the most important of Wallis's works, was published in 1656.  </li></ul>
Integral Calculus <ul><li>In this treatise the methods of analysis of Descartes and Cavalieri were systematised and extend...
Integral Calculus <ul><li>He begins, after a short tract on conic sections, by developing the standard notation for powers...
Integral Calculus <ul><li>Leaving the numerous algebraic applications of this discovery, he next proceeds to find, by inte...
Algebra <ul><li>In 1685 Wallis published Algebra, preceded by a historical account of the development of the subject, whic...
Algebra <ul><li>The second edition, issued in 1693 and forming the second volume of his Opera, was considerably enlarged. ...
Algebra <ul><li>This algebra is noteworthy as containing the first systematic use of formulae.  </li></ul>
Geometry <ul><li>He is usually credited with the proof of the Pythagorean theorem using similar triangles. </li></ul>
Geometry <ul><li>However, Thabit Ibn Qurra (AD 901), an Arab mathematician, had produced a generalization of the Pythagore...
Geometry <ul><li>It is a reasonable conjecture that Wallis was aware of Thabit's work </li></ul>
Geometry <ul><li>Wallis was also inspired by the works of Islamic mathematician Sadr al-Tusi, the son of Nasir al-Din al-T...
Calculator <ul><li>One aspect of Wallis's mathematical skills has not yet been mentioned, namely his great ability to do m...
Calculator <ul><li>One night he calculated the square root of a number with 53 digits in his head. </li></ul>
Calculator <ul><li>In the morning he dictated the 27 digit square root of the number, still entirely from memory.  </li></ul>
Question  &  Answer
Who is given partial credit for the development of modern calculus?
John Wallis
What did Wallis introduce in his Opera Mathematica I (1695)
&quot;continued fraction&quot;
Carl Friedrich Gauss Born:  30 April 1777 in Brunswick, Duchy of Brunswick (now Germany) Died:  23 Feb 1855 in Göttingen, ...
John Wallis
Biography
<ul><li>born on April 30, 1777 in Braunschweig, in the Electorate of Brunswick-Lüneburg, now part of Lower Saxony, Germany...
<ul><li>christened and confirmed in a Catholic church near the school he had attended as a child.  </li></ul>
<ul><li>There are several stories of his early genius. According to one, his gifts became very apparent at the age of thre...
<ul><li>Another famous story has it that in primary school his teacher, J.G. Büttner, tried to occupy pupils by making the...
<ul><li>The young Gauss reputedly produced the correct answer within seconds, to the astonishment of his teacher and his a...
<ul><li>Gauss's presumed method was to realize that pairwise addition of terms from opposite ends of the list yielded iden...
<ul><li>However, the details of the story are at best uncertain; some authors, such as Joseph Rotman in his book A first c...
<ul><li>As his father wanted him to follow in his footsteps and become a pastor, he was not supportive of Gauss's schoolin...
<ul><li>Gauss was primarily supported by his mother in this effort and by the Duke of Braunschweig, who awarded Gauss a fe...
<ul><li>Mathematicians including Jean le Rond d'Alembert had produced false proofs before him, and Gauss's dissertation co...
<ul><li>Ironically, by today's standard, Gauss's own attempt is not acceptable, owing to implicit use of the Jordan curve ...
<ul><li>However, he subsequently produced three other proofs, the last one in 1849 being generally rigorous.  </li></ul>
<ul><li>His attempts clarified the concept of complex numbers considerably along the way. </li></ul>
<ul><li>Gauss predicted correctly the position at which it could be found again, and it was rediscovered by Franz Xaver vo...
<ul><li>Zach noted that &quot;without the intelligent work and calculations of Doctor Gauss we might not have found Ceres ...
<ul><li>Though Gauss had been up to that point supported by the stipend from the Duke, he doubted the security of this arr...
<ul><li>Thus he sought a position in astronomy, and in 1807 was appointed Professor of Astronomy and Director of the astro...
<ul><li>The discovery of Ceres by Piazzi on 1 January 1801 led Gauss to his work on a theory of the motion of planetoids d...
<ul><li>Piazzi had only been able to track Ceres for a couple of months, following it for three degrees across the night s...
<ul><li>Then it disappeared temporarily behind the glare of the Sun.  </li></ul>
<ul><li>Several months later, when Ceres should have reappeared, Piazzi could not locate it: the mathematical tools of the...
<ul><li>Gauss, who was 23 at the time, heard about the problem and tackled it. After three months of intense work, he pred...
<ul><li>In the process, he so streamlined the cumbersome mathematics of 18th century orbital prediction that his work—publ...
<ul><li>It introduced the Gaussian gravitational constant, and contained an influential treatment of the method of least s...
<ul><li>Gauss was able to prove the method in 1809 under the assumption of normally distributed.  </li></ul>
<ul><li>The method had been described earlier by Adrien-Marie Legendre in 1805, but Gauss claimed that he had been using i...
<ul><li>Gauss was a prodigious mental calculator.  </li></ul>
<ul><li>Reputedly, when asked how he had been able to predict the trajectory of Ceres with such accuracy he replied, &quot...
<ul><li>The questioner then wanted to know how he had been able to look up so many numbers from the tables so quickly. &qu...
<ul><li>In 1818 Gauss, putting his calculation skills to practical use, carried out a geodesic survey of the state of Hano...
<ul><li>To aid in the survey, Gauss invented the heliotrope, an instrument that uses a mirror to reflect sunlight over gre...
<ul><li>Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it.  </li></ul>
<ul><li>This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that...
<ul><li>His friend Farkas Wolfgang Bolyai with whom Gauss had sworn &quot;brotherhood and the banner of truth&quot; as a s...
<ul><li>Bolyai's son, János Bolyai, discovered non-Euclidean geometry in 1829; his work was published in 1832.  </li></ul>
<ul><li>This unproved statement put a strain on his relationship with János Bolyai (who thought that Gauss was &quot;steal...
<ul><li>Letters by Gauss years before 1829 reveal him obscurely discussing the problem of parallel lines.  </li></ul>
<ul><li>Waldo Dunnington, a life-long student of Gauss, successfully proves in Gauss, Titan of Science that Gauss was in f...
<ul><li>In 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledg...
<ul><li>They constructed the first electromagnetic telegraph in 1833, which connected the observatory with the institute f...
<ul><li>Gauss ordered a magnetic observatory to be built in the garden of the observatory, and with Weber founded the magn...
<ul><li>He developed a method of measuring the horizontal intensity of the magnetic field which has been in use well into ...
<ul><li>Gauss died in Göttingen, Hannover (now part of Lower Saxony, Germany) in 1855 and is interred in the cemetery Alba...
<ul><li>Two individuals gave eulogies at his funeral, Gauss's son-in-law Heinrich Ewald and Wolfgang Sartorius von Walters...
<ul><li>His brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams and the cerebral are...
<ul><li>Highly developed convolutions were also found, which in the early 20th century was suggested as the explanation of...
Contribution
<ul><li>Degaussing </li></ul><ul><li>Gauss  </li></ul><ul><li>Gauss–Bonnet theorem  or  Gauss–Bonnet formula </li></ul><ul...
<ul><li>Gauss–Jordan elimination </li></ul><ul><li>Gauss–Kronrod quadrature formula </li></ul><ul><li>Heptadecagon </li></...
Degaussing <ul><li>process of decreasing or eliminating an unwanted magnetic field.  </li></ul>
Gauss <ul><li>The, abbreviated as  G , is the cgs unit of measurement of a magnetic field B (which is also known as the &q...
Gauss–Bonnet theorem  or  Gauss–Bonnet formula <ul><li>an important statement about surfaces which connects their geometry...
Gauss–Codazzi–Mainardi equations <ul><li>are fundamental equations in the theory of embedded hypersurfaces in a Euclidean ...
Gauss–Jordan elimination <ul><li>a version of Gaussian elimination that puts zeros both above and below each pivot element...
Gauss–Kronrod quadrature formula <ul><li>method for numerical integration (calculating approximate values of integrals).  ...
Heptadecagon <ul><li>a seventeen-sided polygon  </li></ul>
Modular Arithmetic <ul><li>a.k.a  clock arithmetic </li></ul><ul><li>a system of arithmetic for integers, where numbers &q...
Question  &  Answer
Who awarded Gauss a fellowship to the Collegium Carolinum?
<ul><li>Duke of Braunschweig  </li></ul>
What is the book in which Gauss made important contributions to number theory?
<ul><li>Disquisitiones Arithmeticae </li></ul>
Takakazu Seki Born:  March 1642 in Fujioka, Kozuke, Japan Died:  5 Dec 1708 in Edo (now Tokyo), Japan
John Wallis
Biography
<ul><li>born into a samurai warrior family. However at an early age he was adopted by a noble family named  Seki Gorozayem...
<ul><li>The name by which he is now known, Seki, derives from the family who adopted him rather than from his natural pare...
<ul><li>Seki was an infant prodigy in mathematics.  </li></ul>
<ul><li>He was self-educated in mathematics having been introduced to the topic by a servant in the household who, when Se...
<ul><li>Seki soon built up a library of Japanese and Chinese books on mathematics and became acknowledged as an expert.  <...
<ul><li>He was known as  'The Arithmetical Sage ', a term which is carved on his tombstone, and soon had many pupils. </li...
<ul><li>He was born to the Uchiyama clan, a subject of Ko-shu han, and later adopted into the Seki family, a subject of th...
<ul><li>While in Ko-shu han, he was involved in a surveying project to produce a reliable map of his employer's land.  </l...
<ul><li>He spent many years in studying 13th century Chinese calendars to replace the less accurate one used in Japan at t...
Contribution
<ul><li>Wasan </li></ul><ul><li>Sangaku </li></ul><ul><li>Rectification of the Circle  </li></ul><ul><li>Calculation of Pi...
Wasan Geometry <ul><li>algebra with numerical method, polynomial interpolation and their applications, indeterminate integ...
Sangaku <ul><li>Japanese geometrical puzzles in Euclidean geometry on wooden tablets created during the Edo period (1603–1...
Rectification of the Circle  <ul><li>Constructing an ideal straight line equal in length to the circumference of the circl...
Calculation of Pi <ul><li>most useful for accelerating the convergence of a sequence that is converging linearly </li></ul>
Question  &  Answer
What is the another term for early Japanese mathematics?
<ul><li>Wasan </li></ul>
What was the process that Seki used when he obtained a value for π that was correct to the 10th decimal place?
<ul><li>&quot;Aitken's delta-squared process&quot; </li></ul>
Archimedes Born:  287 BC in Syracuse, Sicily (now Italy) Died:  212 BC in Syracuse, Sicily (now Italy)
John Wallis
Biography
<ul><li>born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a colony of Magna Graecia.  </li></ul>
<ul><li>The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for ...
<ul><li>In The Sand Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing is known.  <...
<ul><li>Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracus.  </li></ul>
<ul><li>A biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details o...
<ul><li>It is unknown, for instance, whether he ever married or had children.  </li></ul>
<ul><li>During his youth Archimedes may have studied in Alexandria, Egypt, where Conon of Samos and Eratosthenes of Cyrene...
<ul><li>He referred to Conon of Samos as his friend, while two of his works (The Method of Mechanical Theorems and the Cat...
<ul><li>Archimedes died c. 212 BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus c...
<ul><li>According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the c...
<ul><li>A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish wor...
<ul><li>The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a lesser-known account ...
<ul><li>According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thou...
<ul><li>General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable scientific as...
<ul><li>A sphere has 2/3 the volume and surface area of its circumscribing cylinder.  </li></ul>
<ul><li>A sphere and cylinder were placed on the tomb of Archimedes at his request. </li></ul>
<ul><li>The last words attributed to Archimedes are &quot; Do not disturb my circles &quot; (Greek:  μή μου τούς κύκλους τ...
<ul><li>This quote is often given in Latin as &quot; Noli turbare circulos meos ,&quot; but there is no reliable evidence ...
<ul><li>The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a sphere an...
<ul><li>Archimedes had proven that the volume and surface area of the sphere are two thirds that of the cylinder including...
<ul><li>In 75 BC, 137 years after his death, the Roman orator Cicero was serving as quaestor in Sicily.  </li></ul>
<ul><li>He had heard stories about the tomb of Archimedes, but none of the locals was able to give him the location.  </li...
<ul><li>Eventually he found the tomb near the Agrigentine gate in Syracuse, in a neglected condition and overgrown with bu...
<ul><li>Cicero had the tomb cleaned up, and was able to see the carving and read some of the verses that had been added as...
<ul><li>The standard versions of the life of Archimedes were written long after his death by the historians of Ancient Rom...
<ul><li>The account of the siege of Syracuse given by Polybius in his Universal History was written around seventy years a...
<ul><li>It sheds little light on Archimedes as a person, and focuses on the war machines that he is said to have built in ...
Contribution
<ul><li>Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus.  </li></ul>
<ul><li>In Measurement of a Circle, Archimedes gives the value of the square root of 3 as lying between 265⁄153 (approxima...
<ul><li>In The Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is 4...
<ul><li>In The Sand Reckoner, Archimedes set out to calculate the number of grains of sand that the universe could contain...
<ul><li>He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle. </li...
Question  &  Answer
What did Archimedes use to approximate the value of π?
<ul><li>Method of Exhaustion </li></ul>
What did Archimedes prove about the area of the circle?
<ul><li>He proved that it is equal to π multiplied by the square of the radius of the circle. </li></ul>
Bhaskara Achārya Born:  1114   Died:  1185
John Wallis
Biography
<ul><li>a.k.a.  Bhaskara II  and  Bhaskara Achārya  (&quot;Bhaskara the teacher&quot;)  </li></ul><ul><li>an Indian mathem...
<ul><li>He was born near Bijjada Bida (in present day Bijapur district, Karnataka state, South India) into the Deshastha B...
<ul><li>head of an astronomical observatory at Ujjain, the leading mathematical centre of ancient India.  </li></ul><ul><l...
<ul><li>It has been recorded that his great-great-great-grandfather held a hereditary post as a court scholar, as did his ...
<ul><li>His father Mahesvara was as an astrologer, who taught him mathematics, which he later passed on to his son Loksamu...
<ul><li>Loksamudra's son helped to set up a school in 1207 for the study of Bhāskara's writings. </li></ul>
<ul><li>Bhaskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th...
<ul><li>His main works were the  Lilavati  (dealing with arithmetic),  Bijaganita  (Algebra) and  Siddhanta   Shiromani  (...
Contribution
<ul><li>A proof of the Pythagorean theorem by calculating the same area in two different ways and then canceling out terms...
<ul><li>In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations.  </li></ul>
<ul><li>Solutions of indeterminate quadratic equations (of the type ax² + b = y²).  </li></ul>
<ul><li>Integer solutions of linear and quadratic indeterminate equations (Kuttaka). The rules he gives are (in effect) th...
<ul><li>A cyclic Chakravala method for solving indeterminate equations of the form ax² + bx + c = y. The solution to this ...
<ul><li>His method for finding the solutions of the problem x² − ny² = 1 (so-called &quot;Pell's equation&quot;) is of con...
<ul><li>Solutions of Diophantine equations of the second order, such as 61x² + 1 = y². This very equation was posed as a p...
<ul><li>Solved quadratic equations with more than one unknown, and found negative and irrational solutions.  </li></ul>
<ul><li>Preliminary concept of mathematical analysis.  </li></ul>
<ul><li>Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.  </li><...
<ul><li>Conceived differential calculus, after discovering the derivative and differential coefficient.  </li></ul>
<ul><li>Stated Rolle's theorem, a special case of one of the most important theorems in analysis, the mean value theorem. ...
<ul><li>Calculated the derivatives of trigonometric functions and formulae. </li></ul>
<ul><li>In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric resul...
Question  &  Answer
Who taught him mathematics?
<ul><li>His father, Mahesvara </li></ul>
What do you call of his method for finding the solutions of the problem x² − ny² = 1?
<ul><li>Pell's equation </li></ul>
Blaise Pascal Born:  June 19, 1623  Died:  August 19, 1662
John Wallis
Biography
<ul><li>a French mathematician, physicist, and religious philosopher.  </li></ul><ul><li>a child prodigy who was educa
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