Fields Medal Prize Winners (1998)


Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Rewriting Algebraic Fractions
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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Fields Medal Prize Winners Fields Medal Prize Winners (1998) Richard Ewen Borcherds (born 29 November 1959) He has been "Royal Society Research Professor" at the Department of Pure Mathematics and Mathematical Statistics at Cambridge University since 1996. Borcherds began his academic career at Trinity College, Cambridge before going as assistant professor to the University of California in Berkeley. He has been made a Fellow of the Royal Society, and has also held a professorship at Berkeley since 1993. Richard E. Borcherds will receive a medal for his work in the fields of algebra and geometry, in particular for his proof of the so-called Moonshine conjecture. This conjecture was formulated at the end of '70s by the British mathematicians John Conway and Simon Norton and presents two mathematical structures in such an unexpected relationship that the experts gave it the name "Moonshine". In 1989, Borcherds was able to cast some more light on the mathematical background of this topic and to produce a proof for the conjecture. The Moonshine conjecture provides an interrelationship between the so-called "monster-groups" and elliptical functions. These functions are used in the construction of wire-frame structures in two-dimensions, and can be helpful, for example, in chemistry for the description of molecular structures. Monster groups, in contrast, only seemed to be of importance in pure mathematicians. Groups are mathematical objects which can be used to describe the symmetry of structures. Expressed technically, they are a set of objects for which certain arithmetic rules apply (for example all whole numbers and their sums form a group). An important theorem of algebra says that all groups, however large and complicated they may seem, all consist of the same components - in the same way as the material world is made up of atomic particles. The "monster group" is the largest "sporadic, finite, simple" group - and one of the most bizarre objects in algebra. It has more elements than there are elementary particles in the universe (approx. 8 x 10^53). Hence the name "monster". In his proof, Borcherds uses many ideas of string theory - a surprisingly fruitful way a making theoretical physics useful for mathematical theory. Although still the subject of dispute among physicists, strings offer a way of explaining many of the puzzles surrounding the origins of the universe. They were proposed in the search for a single consistent theory which brings together various partial theories of cosmology. Strings have a length but no other dimension and may be open strings or closed loops. William Timothy Gowers (born 20 November 1963) He is lecturer at the Department of Pure Mathematics and Mathematical Statistics at Cambridge University and Fellow of Trinity College. From October 1998 he will be Rouse Professor of Mathematics. After studying through to doctorate level at Cambridge, Gowers went to University College London in 1991, staying until the end of 1995. In 1996 he received the Prize of the European Mathematical Society. William Timothy Gowers has provided important contributions to functional analysis, making extensive use of methods from combination theory. These two fields apparently have little to do with each other, and a significant achievement of Gowers has been to combine these fruitfully. Functional analysis and combination analysis have in common that many of their problems are relatively easy to formulate, but extremely difficult to solve. Gowers has been able to utilise complicated mathematical constructions to prove some of the conjectures of the Polish mathematician Stefan Banach (1892-1945), including the problem of "unconditional bases". Banach was an eccentric, preferring to spend his time in the café rather than in his office in the University of Lvov. In the twenties and thirties he filled a notebook with problems of functional analysis while sitting in the "Scottish Café", so that this later became known as the Scottish Book. Gower has made significant contribution above all to the theory of Banach spaces. Banach spaces are sets whose members are not numbers but complicated mathematical objects such as functions or operators. However, in a Banach space it is possible to manipulate these objects like numbers. This finds applications, for example, in quantum physics. A key question for mathematicians and physicists concerns the inner structure of these spaces, and what symmetry they show. Gowers has been able to construct a Banach space which has almost no symmetry. This construction has since served as a suitable counter-example for many conjectures in functional analysis, including the hyperplane problem and the Schr?der-Bernstein problem for Banach spaces. Gowers' contribution also opened the way to the solution of one of the most famous problems in functional analysis, the so-called "homogeneous space problem". A year ago, Gowers attracted attention in the field of combination analysis when he delivered a new proof for a theorem of the mathematician Emre Szemeredi which is shorter and more elegant than the original line of argument. Such a feat requires extremely deep mathematical understanding. Maxim Kontsevich (born 25 August 1964) He is a professor at the Institute des Hautes Etudes Scientific (I.H.E.S) in France and visiting professor at the Rutgers University in New Brunswick (USA). After studying at the Moscow University and beginning research at the "Institute for Problems of Information Processing", he gained a doctorate at the University of Bonn, Germany in 1992. He then received invitations to Harvard, Princeton, Berkeley and Bonn. Maxim Kontsevich has established a reputation in pure mathematics and theoretical physics, with influential ideas and deep insights. He has been influenced by the work of Richard Feynmann and Edward Witten. Kontsevich is an expert in the so-called "string theory" and quantum field theory. He made his name with contributions to four problems of geometry. He was able to prove a conjecture of Witten and demonstrate the mathematical equivalence of two models of so-called quantum gravitation. The quantum theory of gravity is an intermediate step towards a complete unified theory. It harmonises physical theories of the macrocosm (mass attraction) and the microcosm (forces between elementary particles). Another result of Kontsevich relates to knot theory. Knots mean exactly the same thing for mathematicians as for everyone else, except that the two ends of the rope are always jointed together. A key question in knot theory is, which of the various knots are equivalent? Or in other words, which knots can be twisted and turned to produce another knot without the use of scissors? This question was raised at the beginning of the 20th century, but it is still unanswered. It is not even clear which knots can be undone, that is converted to a simple loop. Mathematicians are looking for ways of classifying all knots. They would be assigned a number or function, with equivalent knots having the same number. Knots which are not equivalent must have different numbers. However, such a characterisation of knots has not yet been achieved. Kontsevich has found the best "knot invariant" so far. Although knot theory is part of pure mathematics, there seem to be scientific applications. Knot structures occur in cosmology, statistical mechanics and genetics. Curtis T. McMullen (born 21 May 1958) He is visiting professor at Harvard University. He studied in Williamstown, Cambridge University and Paris before gaining a doctorate in 1985 at Harvard. He lectured at various universities before becoming professor at the University of California in Berkeley. Since 1998 he has taught at Harvard. The Fields Medal is his tenth major award. In 1998 he has been elected to the American Academy of Arts and Sciences. Curtis T. McMullen is being awarded a medal primarily in recognition of his work in the fields of geometry and "complex dynamics", a branch of the theory of dynamic systems, better known perhaps as chaos theory. McMullen has made contributions in numerous fields of mathematics and fringe areas. He already provided one important result in his doctoral thesis. The question was how to calculate all the solutions of an arbitrary equation. For simple equations it is possible to obtain the solutions by simple rearrangement. For most equations, however it is necessary to use approximation. One well-known form is the "Newton method" - already known in a rudimentary form in ancient times. For second-degree polynomials this provides very good results without exception. A key question therefore was whether a comparable method - which happened not to have been discovered - also existed for equations of higher degrees. Curtis T. McCullen's conclusion was that there is definitely no such universal algorithm for equations above degree three; only a partially applicable method is possible. For degree-three equations he developed a "new" Newtonian method and could thus completely solve the question of approximation solutions. A further result of McMullen relates to the Mandelbrot set. This set describes dynamic systems which can be used to model complicated natural phenomena such as weather or fluid flow. The point of interest is where a system drifts apart and which points move towards centres of equilibrium. The border between these two extremes is the so-called Julia set, named after the French mathematician Gaston Julia, who laid the foundations for the theory of dynamic systems early in the twentieth century. The Mandelbrot set shows the parameters for which the Julia set is connected, i.e. is mathematically attractive. This description is very crude, but a better characterisation of the boundary set was not available. Curtis T. McMullen made a major advance, however, when he showed that it is possible to decide in part on the basis of the Mandelbrot set which associated dynamic system is "hyperbolic" and can therefore be described in more detail. For these systems a well-developed theory is available. McMullen's results were suspected already in the sixties, but nobody had previously been able to prove this exact characterisation of the Julia set. Andrew J. Wiles awarded the "IMU silver plaque" Andrew J. Wiles (born 11 April 1953) He is Professor of Mathematics at Princeton University. Since 1995 he has also been a member of the Institute for Advanced Study (IAS). Wiles studied in England at Cambridge University before going to America as assistant professor at Harvard in 1974. In 1982 he became professor in Princeton. His fields of research are number theory and arithmetic geometry. The British mathematician Andrew J. Wiles has been honoured with the "IMU silver plaque" at 18 August. The chairman of the Fields Medals Committee, Yuri Manin, presented him with this award during the opening ceremony of the International Congress of Mathematicians in the Berlin International Congress Centre. This world congress of mathematicians, the largest and most important one worldwide, is taking place until 25th August. Some 3500 mathematicians from all over the world will be visiting Berlin to participate. During the congress, which is held every four years, four Fields Medals are awarded to outstanding mathematicians under the age of forty. In view of their significance the Fields Medals are often dubbed the "Nobel Prize of mathematics". Four years ago, Andrew J. Wiles was a hot favourite for an award, since in 1993 he had presented a proof of Fermat's Last Theorem - one of the most famous mathematical puzzles, which had remained unsolved for more than 350 years. Shortly afterwards, however, colleagues found a gap in the proof which Wiles was only able to close up a year later. But this was too late for the Fields Medal, because Wiles was then over the age limit of forty. With its special tribute, the International