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  • Mathematics In Europe - The landscape: areas
    are fairly young for example set theory Here we present nine disciplines algebra analysis applied mathematics including numerical analysis combinatorics foundations of mathematics mathematical logic and set theory geometry number theory number theory probability and statistics and topology The choice of the described disciplines does not reflect their importance the nine disciplines were chosen because it is possible to give a resonable description of them that is understandable for a general audience The nine disciplines and several of their subdisciplines or typical notions from them are represented graphically by the map of the Land of Middle Math below Please note that this is the author s view of the classification and the sizes of the regions on the map have no implications about the relative importance of the disciplines nor should any other similar conclusions be drawn from the style of the visual representation Still it was on purpose that no clear borders were drawn as many subdisciplines can be considered as jointly governed by two or more of the nine chosen disciplines For example graph theory can be considered as a part of topology as well as of combinatorics so in our map the Sea of graph theory is positioned on the border between Topologiath and Combinatorica If you want to know more about a specific mathematical discipline please click on the corresponding region of our Map of the Land of Middle Math below A much more detailed classification then the one described above is used by professional mathematicians all around the globe It is called the Mathematics Subjects Classification MSC It classifies mathematics in 63 classes and numerous subclasses Many of them have esoteric names such as Transcendental methods of algebraic geometry number 32J25 in MSC But even those sounding relatively harmless like Perfect graphs 05C17 in the

    Original URL path: http://mathematics-in-europe.eu/tr/42-information/landscape/75-the-landscape-areas (2013-11-18)
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  • Mathematics In Europe - The landscape: key concepts
    proofs Choosing the right method of proof can simplify the approach to a problem Here you can obtain information on the concepts of vector and dimension In considering the size of a set it is of particular importance whether it is finite or infinite These two categories of sets have fundamentally different properties Information about what is meant by existence in mathematics can be found here Problems involving finding maximum

    Original URL path: http://mathematics-in-europe.eu/tr/42-information/landscape/76-the-landscape-key-concepts (2013-11-18)
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  • Mathematics In Europe - Numbers
    Here we are talking about natural numbers integers rational numbers real numbers complex numbers see also the article of E Behrends from Five Minute Mathematics algebraic and transcendental numbers What of interest can be said about the number zero What of interest can be said about the number What of interest can be said about the base of the natural logarithm e What of interest can be said about the

    Original URL path: http://mathematics-in-europe.eu/tr/81-information/landscape-numbers/96-numbers (2013-11-18)
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  • Mathematics In Europe - The landscape: formulas
    formulas Ayrıntılar Kategori Landscape Some important formulas from mathematics and physics Pythagoras formula pdf Newton s universal law of gravity pdf Euler s most beautiful formula of mathematics pdf The prime number theorem of Gauß pdf Maxwell s equations of electrodynamics pdf Clausius equation of thermodynamics pdf Einstein I How changes time for a moving object pdf Einstein II E mc 2 pdf Einstein III The gravity equations pdf Heisenberg

    Original URL path: http://mathematics-in-europe.eu/tr/42-information/landscape/97-the-landscape-formulas (2013-11-18)
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  • Mathematics In Europe - Examples of current research
    overview of this vast field not only because of the large number of different subject areas but also because even the description of most problems requires advanced mathematical concepts and terminology Therefore as a first step we present a few select examples The following articles appeared originally in the Newsletter NL of the European Mathematical Society EMS They are republished here with permission of the EMS and the authors Blomer

    Original URL path: http://mathematics-in-europe.eu/tr/41-information/research/280-examples-of-current-research (2013-11-18)
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  • Mathematics In Europe - The Millenium Prize Problems
    have been checked and all lie on the critical line This and some other results seem to support the hypothesis but there are also some arguments against it But as yet there is no proof of the Riemann s hypothesis or its negation Figure 3 The behaviour of the real and imaginary part of the Riemann zeta function on the critical line The Birch and Swinnerton Dyer Conjecture Another open problem in number theory that was chosen as one of the Seven is the Birch and Swinnerton Dyer conjecture stated in the 1960s In a simplified version it states that if a function associated to an elliptic curve E and denoted by L E is zero at point 1 L E 1 0 then the curve E has infinitely many rational points J Tate as stated on PlanetMath said about this conjecture This remarkable conjecture relates the behavior of a function L at a point where it is not at present known to be defined to the order of a group which is not known to be finite It may be puzzling to an outsider that a conjecture about a curve is something from number theory the discipline dealing with properties of integers and more generally rational numbers and not a conjecture in mathematical analysis But elliptic curves are objects from algebraic geometry of particular importance to number theory A famous example is that proving statements about them was a crucial step in the probably most famous mathematical theorem the Fermat s last theorem Elliptic curves over rational numbers can be thought of as curves with equations of the form x 3 ax b y 2 where a and b are rational numbers and the left side has no repeated factors every cubic polynomial like the one on the left can be written as a product of three factors of the form x c A few examples of elliptic curves over rational numbers are shown in figure 4 Rational points on elliptic curves are points with both coordinates rational It is not easy to find out if an elliptic curve has finitely or infinitely many rational points To an elliptic curve E one can associate an L function L E a function closely connected to the already mentioned Riemann function Birch and Swinnerton Dyer noticed that in all the examples they studied the answer is coded in the function L E if L E 1 0 then there are infinitely many rational points on the elliptic curve E The full conjecture is more algebraic formulated in terms of groups Namely if you take two rational points from a curve and define their sum by summing the corresponding coordinates e g 1 2 0 3 1 0 2 3 1 5 then it can be proven that the sum is also a rational point on the same curve Algebraically stated the rational points of the curve E form an Abelian group with respect to addition This group can be described as a number r of copies of the group Z of integers and the number r is called the rank of the curve E The Birch and Swinnerton Dyer conjecture says the rank of the curve E is the order of 1 as the zero of L E a part of the conjecture is also an analytic formula about L E Recall here that the order of a zero c of a function f its multiplicity k is defined as a zero f c 0 such that f x x c k is defined arround c and has value 0 at c but f x x c k 1 is not defined at c e g 1 is a zero of order 2 of the polynomial x x 1 2 x 3 So far the conjecture has been proven only in some special cases but as with Riemann s hypothesis there are many numerical calculations that support the possible truth of this conjecture Figure 4 Elliptic curves x 3 x 1 y 2 x 3 x y 2 x 3 x 1 y 2 and x 3 x 2 y 2 Navier Stokes Equations After the two number theoretic prize problems we present two problems from mathematical physics that were chosen among the great Seven The first of these two concerns the solutions of the Navier Stokes equations They are two partial differential equations describing the motion of fluid substances i e gases and liquids The problem with them is that they are not fully understood although they were formulated in the 19th century The equations describe how physical laws affect the state of a fluid for various positions and moments Most people that ever travelled by plane experienced turbulences points in space where the air flow was highly irregular Turbulences can be described as vortices where each vortex gives rise to another smaller one this one creates another etc If these transition from one small vortex to another happen to fast the fluid flow becomes higly irregular instead of being a nice smooth one The question is can one predict a turbulence knowing some initial data The unknown functions in the Navier Stokes equations are the velocity and the pressure of the fluid both functions of position and time For a specified viscosity of the fluid and a given external force also depending on position and time they become a concrete system of partial differential equations If one knows the velocity vectors of the fluid at all points at a given moment the initial conditions for the equations their solution would describe all of its future behaviour provided the solution exists So far exact solutions are known only in some simple cases that are not very interesting For example if a sea was at rest in some initial moment and no forces act upon it it will rest forever In more realistic cases that are of interest for example for the design of aircrafts one has to find approximate solutions Still even if one never will know how to find an exact solution it would be important to know that one exists Even more one wants good solutions This means that if the initial conditions are smooth if the initial velocity and the external force are smooth enough the million dollar problem is to prove or disprove that then the solutions to the Navier Stokes equations are also smooth i e do not have singular points where the fluid flow becomes turbulent and that they have bounded kinetic energy Yang Mills theory The other of the mathphysical millenium problems comes from quantum theory Most people nowadays have heard of quantum mechanics and elementary particles like quarks As written on the official CMI page on this prize problem The laws of quantum physics stand to the world of elementary particles in the way that Newton s laws of classical mechanics stand to the macroscopic world Quantum physics is highly mathematical and it is very important for the theory to be usable that the laws are not only consistent with observations but also mathematically correct In the 1950s Yang and Mills formulated a theory for describing elementary particles Now there are several versions of this theory Although experiments support these theories the mathematical foundations remain unclear In particular in order to use it succesfully it should be able to mathematically predict a so called mass gap observed in experiments although the elementary quantum particles travel with the speed of light as waves they have positive mass Namely the energy of the empty space is zero but as soon as one elementary particle enters it the energy must have at least some positive value According to Einstein s famous formula E mc 2 c is the speed of light in vacuum so if the energy of the space when one particle enters is E one could assign a positive mass m E c 2 to the particle Besides this mass gap there are other properties of elementary particles that have been observed in experiments and confirmed by numerical simulations but are not thoroughly understood from the theoretical point So one can say that this prize problem is about establishing a mathematically rigorous Yang Mills theory that can predict a mass gap P vs NP The fifth problem we describe comes from theoretical computer science and is easy to state is P NP Compared to the previous ones the explanation of what this problem asks is much simpler although the answer so far seems as hard to be obtained as in the previous cases Here P and NP denote classes of problems that ask for a yes or no answer Informally asking if P NP is the same as asking if for every yes or no problem for which the answer is easy to check an NP problem it is true that the answer is also easy to determine a P problem Obviously every P problem is a NP problem but it is not known if the NP class is a stricly larger class of problems than the P class If somebody would prove that P NP or find an NP problem that is not P he or she would get a million dollars The solution would have major consequences not only in computer science but also in other mathematical disciplines and in biology and cryptography For example the safety of currently used cryptosystems in economic and internet transactions depends on not knowing a polynomial time algorithm to break the code If P NP were proven by a constructive method it would mean that the current cryptosystems would lose their safety and would have to be replaced To make it a little bit more precise we have to say what is meant by easy in the statement above it means relatively fast in the sense that the number of the steps in the algorithm for getting the answer is not too large The official term for this fastness is polynomial time An algorithm is said to be polynomial time if the number of steps it performs i e its running time is upper bound by a polynomial depending only on the algorithm itself and the size of the input For example all basic arithmetic operations can be done in polynomial time Examples of problems that are NP but not known if they are also P include the following The subset sum problem given a set of integers does the sum of some non empty subset equal exactly zero For a chosen subset it is easy to check if the sum of its elements is zero but to find such a set can be very difficult and it is not known if there is a polynomial time algorithm for the solution The party problem Say you want to organize a wedding and plan to invite many of your 500 aquantances but the largest hall the restaurant can offer has only 100 places Since you cannot invite them all you consider their mutual relations and notice that there are some that won t come if you invite somebody else from the list The problem is to decide how to fill as many of the places on your wedding feast without inviting any two incompatible guests If you take a concrete guest list of 100 of them it is easy to check if there are any two incompatible members of the list But it is not easy no polynomial time algorithm is known for this to find a list satisfying the condition The Hodge Conjecture The last two million dollar problems come from topology Topology is the mathematical discipline studying questions like is something connected in one piece and has it holes and how many It is often said that topology is rubber geometry since a topologists like a geometrist studies spatial objects also in higher dimensional spaces but unlike a geometrist who is interested in sizes and shapes a topologist is interested in the properties that would remain unchanged if the object was stretched twisted or otherwise deformed without cutting or glueing like it was made of rubber The Hodge conjecture stated by William Hodge in 1950 concerns projective algebraic varieties An algebraic variety is the set of solutions of a system of polynomial equations Simple examples of algebraic varieties are lines and planes in space and also circles and spheres One can consider algebraic varieties as generalisations of algebraic curves to higher dimensions where an algebraic curve in the plane is defined as the set of all zeros of a polynomial in two variables e g a line circle ellipse hyperbola parabola Every polynomial equation with n unknowns can be interpreted as an n 1 dimensional surface in an n dimensional space since every point in an n dimensional space has n coordinates and each equation relating the coordinates takes away one level of freedom i e lessens the dimension for one Thus every algebraic variety can be though of as an intersection of surfaces in higher dimensions For example consider our usual three dimensional space Every point can be described by three independent coordinates x y z The equation x 2 y 2 z 2 1 leaves us with two degrees of freedom since if we select values for x and y the corresponding values for z are not arbitrary any more but depends on the choice of x and y The preceding equation is the equation of the unit sphere a two dimensional algebraic variety If we intersect it by the x y plane we get a circle a one dimensional variety If one considers such a variety embedded in a projective space it is called a projective algbraic variety a projective space can be though of like a normal space with all the points with proportional coordinates identified e g in the real projective space points 1 2 0 2 4 0 and 3 6 0 would be considered the same One can take more generally complex instead of real coordinates to obtain complex projective spaces Now it is not easy to imagine what an algebraic variety could look like but differential and integral calculus can be done on algebraic varieties in much the same way as it is done usually Formally one talks about cohomology theories The Hodge conjecture says that complex projective algebraic varieties can be glued from pieces called Hodge cycles that are always algebraic cycles sums of subvarieties multiplied each by a rational number This conjecture is important because The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations quoted from the official CMI page So far some special cases have been proven The Poincaré Theorem Very recently the story about a mathematician refusing his prize of one million dollars awarded to him for the solution of some problem filled the pages of media worlwide I wrote some problem because probably noone where noone is taken as synonimous to non professional who read the news remembers the name of the problem the Poincaré conjecture let alone its meaning and most also forgot the name of the mathematician Grigori Perelman This is not the first case this great Russian mathematician refused a prize he already did so in 2006 when he refused the Fields medal often described as the Nobel prize for mathematics It is queer is it that most media were more interested in what they consider a peculiar behaviour of a genius than in the problem he solved And this problem was not only one of the hardest open problems in mathematics but its solution has also some deep implications on the shape of space The Poincaré conjecture was formulated by the French mathematician Henri Poincaré at the end of the 19th century Poincaré is often described as the last great universal mathematician and one of the founders of algebraic topology His conjecture concerns spheres A sphere in a space is the set of all points at some fixed distance from a given point If we take the distance as 1 and the point as the origin we notice that in the one dimensional case the line the sphere consists of two disconnected points it is then called a 0 sphere because a point has dimension 1 In a two dimensional space a plane we get the 1 sphere a circle and in the three dimensional space a 2 sphere is just the usual sphere note that by sphere we mean just the boundary without the interior As already said in topology objects are equivalent the official term is homeomorphic if they can be continuously deformed one into the other so in topological sense an ellipsoid the boundary of a cube and the surface of our earth are also 2 spheres Note that a 1 sphere is a boundary of a full circle and a 2 sphere is a boundary of a full ball but that spheres themselves have no boundaries If someone is puzzled about the 2 sphere not having any boundary if you are confined to live on it there is no point where you cannot go further without falling of in whatever direction on the Earth you move you won t fall of Figure 5 A sphere and a torus are examples of 2 manifolds Now imagine that you live a long time ago How could you know that you live on the surface of a 2 sphere and not on a giant torus or pretzel You could try to walk arround in circles in such a way that every time you come to your starting point you reduce the diameter of your walking tour If you are on a 2 sphere sooner or later your diameter would shrink to zero your circles have continously shrunk to a point On the other side if you re on a torus there are circular tours that

    Original URL path: http://mathematics-in-europe.eu/tr/41-information/research/140-the-millenium-prize-problems (2013-11-18)
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  • Mathematics In Europe - "Mathematics inside"
    many cases there ought to be a label stating Mathematics Inside Here are a few examples Data compression 1 a pdf 1 291 kB data compression 2 Without mathematical techniques today s CD and MP3 players would be unthinkable Security against unwanted eavesdroppers Cryptography is based on mathematical methods Computer aided tomography The three dimensional views of the body used in medical diagnosis and treatment would be impossible without mathematical methods for image reconstruction Mathematics in finance a pdf 1 235 kB Since the advent of derivatives higher mathematics has become an essential aspect of banking Today banks and investment firms employ thousands of mathematicians Climate a pdf 1 2 4 MB How will climate change in the coming years Mathematical models predict various scenarios based on different sets of assumptions Quantum computing a pdf 1 166 kB cf also articles written by N Crato and by E Behrends What will be the next revolution in computing Mathematicians are studying a model called the quantum computer The mathematics of elections a pdf 130 kB This article by L Modica was originally published in Italian in Xla Tangente in 2011 a pdf 600 kB Quadratic functions in elementary economy an article

    Original URL path: http://mathematics-in-europe.eu/tr/48-information/research-mathinside/144-mathematics-inside (2013-11-18)
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  • Mathematics In Europe - Philosophy of mathematics: an overview
    terms and concepts denote abstract mathematical objects that exist in a separate mathematical world that is independent of the work of mathematicians According to the Platonists the task of the mathematician consists in determining the properties of and relationships among mathematical objects and thereby increasing our knowledge of mathematical reality Empiricism The philosophy of empiricism whose origins are to be found in the Anglo Saxon philosophy of the seventeenth century is determined by an epistemological idea Empiricists maintain that to perceive is the same as to experience in the sense that all knowledge comes through experience and only through experience This idea is opposed above all to the philosophy of rationalism which denies to the senses any contribution to knowledge and declares rational understanding as the only means of obtaining truth The expression of mathematics represents a problem for traditional empiricism because here we are dealing with an area of knowledge that is commonly seen as independent of experience The empiricists countered this by asserting that mathematical theorems describe quite general features of empirical reality In this sense they agree most closely with the assertions of physics differentiating themselves from these only in their greater generality Kant s Philosophy of Mathematics Kant s philosophy is based on the idea that all knowledge of empirical reality is a product of intuition on the one hand and certain active products of reason on the other According to Kant reason arranges the material of experience according to certain principles such as space and time which are not gathered from experience but precede it to the extent that they make experience possible in the first place We become aware of such principles of reasoning among which Kant includes the theorems of pure mathematics by reflecting on the conditions for the possibility of experience The

    Original URL path: http://mathematics-in-europe.eu/tr/43-information/history-philosophy/81-philosophy-of-mathematics-an-overview (2013-11-18)
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