UNIT 22 PURE AND APPLIED MATHEMTICS

1. Read the text and say whether the following statements are true or false:

1. The theorem known as the Poincare conjecture was first postulated by French mathe­matician Henri Poincare 150 years ago.

2. It was very easy to prove the Poincare conjecture theorem.

3. Perelman’s work withstood two years of scrutiny before he became eligible for the prize.

                         Perelman proved Poincare’s Conjecture

  The Poincare conjecture had a $1-million reward on offer for its proof: it is one of seven such “Millenium Problems” singled out in 2000 by the Clay Mathematics Institute in Cambridge, Mass.

  The branch of mathematics that studies manifolds is topology. Among the funda­mental questions topologists can ask about 3-manifolds are: What is the sim­plest type of 3-manifold, the one with the least complicated structure? Does it have many cousins that are equally simple, or is it unique? What kinds of 3-manifolds are there?

The answer to the first of those ques­tions has long been known: a space called the 3-sphere is the simplest compact 3-manifold. (Noncompact manifolds can be thought of as being infinite or having an edge. Hereafter only com­pact manifolds are being considered.) The other two questions have been up for grabs for a century but have been answered in 2002 by Grigori ("Grisha") Perelman, a Russian mathematician who has proved a theorem known as the Poincare conjecture.

  First postulated by French mathematician Henri Poincare exactly 100 years ago, the conjecture holds that the 3-sphere is unique among 3-manifolds; no other 3-manifold shares the properties that make it so simple. The 3-manifolds that are more complicated than the 3-sphere have boundaries that you can run up against like a brick wall, or multiple connections from one region to another, like a path through the woods that splits and later rejoins. The Poincare conjecture states that the 3-sphere is the only compact 3-manifold that lacks all those com­plications. Any three-dimensional object that shares those properties with the sphere can therefore be morphed into the same shape as a 3-sphere; so far as topol­ogists are concerned, the object is just an­other copy of the 3-sphere. Perelman's proof also answers the third of the above mentioned ques­tions: it completes work that classifies all the types of 3-manifolds that exist.

It takes some mental gymnastics to imagine what a 3-sphere is like – it is not simply a sphere in the everyday sense of the word. But it has many properties in common with the 2-sphere, which we are all familiar with: If you take a spherical balloon, the rubber of the balloon forms a 2- sphere. The 2-sphere is two-dimensional because only two coordinates – latitude and longitude – are needed to specify a point on it. Also, if you take a very small disk of the balloon and examine it with a magnifying glass, the disk looks a lot like one cut from a flat two-dimensional plane of rubber. It just has a slight curvature. To a tiny insect crawling on the balloon, it would seem like a flat plane. Yet if the insect travelled far enough in what would seem to it to be a straight line, eventually it would arrive back at its starting point.

Similarly, a gnat in a 3-sphere – or a person in one as big as our universe! – perceives itself to be in “ordinary” three-dimensional space. But if it flies far enough in a straight line in any direction, it will eventually circumnavigate the 3-sphere and find itself back where it started, just like the insect on the balloon or someone taking a trip around the world. Spheres exist for dimensions other than three as well. The 1-sphere is also familiar to you: it is just a circle (the rim of a disk, not the disk itself). The n-dimensional sphere is called an n-sphere.

After Poincare proposed his conjecture about the 3-sphere, many scientists tried to prove it, but a major step in closing the three-dimensional problem came in November 2002, when Grigory Perelman, a mathematician at the Steclov Institute of Mathematics in St.Petersburg, posted a paper on the www.arxiv.org Web server that is widely used by physicists and mathematicians as a clearinghouse of new research.

Perelman’s work extends and completes a program of research that Richard S. Hamilton of Columbia University explored in the 1990s. Perelman’s calculations and analysis blew away several roadblocks that Hamilton had run into and could not overcome. Perelman’s work withstood two years of scrutiny before he became eligible for the prize.

 

2. Find in the text equivalents to the phrases:

- 3-х мерное многообразие с наименее сложной структурой;

- никакое другое многообразие не имеет таких же свойств;

- проблема Пуанкаре гласит;

- некомпактные многообразия

- край диска;

- очень маленькая кривизна;

- своеобразный информационный центр по обсуждению нового исследования

 

3. Form phrases by matching the words  from 1-9 with the words  a-i:

 

1. funda­mental 2.starting  3. magnifying  4. a slight  5. far

6. n-dimensional                 7. crawling     8. conjecture              9. tiny

a) insect   b) glass   c) enough    d) holds                  e) sphere

f) curvature        g) questions      h) insect                    i) point.

 

 

TALKING POINT

4 . Discuss in pairs:

· questions topology concerns with;

·  the Poincare conjecture problem;

· attempts to prove the Poincare conjecture;

· difficulties withstood by Perelman before he was offered the award.

 

VOCABULARY STUDY

5. Choose the word to complete the text:

  To mathematicians, Grigori Perelman's … (1) of the Poincaré conjecture qualifies at least as the Breakthrough of the Decade. But it has taken them a good    part of that decade to … (2) themselves that it was for real. In 2006, nearly 4 years after the Russian mathematician released the first of three papers … (3) the proof, researchers finally reached a consensus that Perelman had solved one of the subject's most venerable problems. But the solution touched off a storm of controversy and drama that threatened to overshadow the brilliant work.

Perelman's proof has fundamentally … (4) two distinct branches of mathematics. First, it solved a problem that for more than a century was the indigestible seed at the core of … (5), the mathematical study of abstract shape. Most mathematicians expect that the work will lead  to a much broader result, a proof of the geometrization conjecture: essentially, a "periodic table" that brings clarity to the study of three-dimensional spaces, much as Mendeleev's table did for … (6).

   While bringing new results to topology, Perelman's work brought new … (7) to geometry. It cemented the central role of geometric evolution equations, powerful machinery for transforming hard-to-work-with spaces into more-manageable ones. Earlier studies of such equations always ran into "singularities" at which the equations break down. Perelman dynamited that roadblock.

"This is the first time that mathematicians have been able to understand the structure of singularities and the … (8) of such a complicated system," said Shing-Tung Yau of Harvard University at a lecture in Beijing this summer. "The methods developed … should shed light on many natural systems, such as the Navier-Stokes equation (of fluid dynamics) and the Einstein equation (of general relativity)."

1. a) theory                      b) proof                         c) idea

2.  a) confine                 b) concern                     c) convince

3. a) outlining               b) declaring                  c) claiming

4. a) altered                   b) diverted                    c) proved

5. a) fluid dynamics      b) mechanics                 c) topology

6. a) biology                  b) chemistry                  c) geometry

7. a) techniques             b) tools                          c) methodology

8. a) development          b) organization              c) structure

Read the text.

M. Poincaré was a mathematician, geometer, philosopher, and man of letters, who was a kind of poet of the infinite, a kind of bard of science.                                (quotation from an address at the funeral)

Henri Poincaré's was born in Nancy where his father was Professor of Medicine at the University. Henri was “... ambidextrous and was nearsighted”; during his childhood he had poor muscular coordination and was seriously ill for a time with diphtheria. He received special instruction from his gifted mother and excelled in written composition while still in elementary school.

In 1862 Henri entered the Lycée in Nancy (now renamed the Lycée Henri Poincaré in his honour). He spent eleven years at the Lycée and during this time he proved to be one of the top students in every topic he studied. Henri was described by his mathematics teacher as a "monster of mathematics" and he won first prizes in the concours général, a competition between the top pupils from all the Lycées across France.

Poincaré entered the École Polytechnique in 1873, graduating in 1875. He was well ahead of all the other students in mathematics but, perhaps not surprisingly given his poor coordination, performed no better than average in physical exercise and in art. Music was another of his interests but, although he enjoyed listening to it, his attempts to learn the piano while he was at the École Polytechnique were not successful. His memory was remarkable and he retained much from all the texts he read but not in the manner of learning by rote, rather by linking the ideas he was assimilating particularly in a visual way. His ability to visualise what he heard proved particularly useful when he attended lectures since his eyesight was so poor that he could not see the symbols properly that his lecturers were writing on the blackboard. After graduating from the École Polytechnique, Poincaré continued his studies at the École des Mines.

After completing his studies at the École des Mines Poincaré spent a short while as a mining engineer at Vesoul while completing his doctoral work.. Immediately after receiving his doctorate, Poincaré was appointed to teach mathematical analysis at the University of Caen. He was to remain there for only two years before being appointed to a chair in the Faculty of Science in Paris in 1881. In 1886 Poincaré was nominated for the chair of mathematical physics and probability at the Sorbonne and he also was appointed to a chair at the École Polytechnique. Poincaré held these chairs in Paris until his death at the early age of 58.

Poincaré was a scientist preoccupied by many aspects of mathematics, physics and philosophy, and he is often described as the last universalist in mathematics. He made contributions to numerous branches of mathematics, celestial mechanics, fluid mechanics, the special theory of relativity and the philosophy of science. Poincaré's Analysis situs, published in 1895, is an early systematic treatment of topology. He can be said to have been the originator of algebraic topology and, in 1901, he claimed that his researches in many different areas such as differential equations and multiple integrals had all led him to topology. For 40 years after Poincaré published the first of his six papers on algebraic topology in 1894, essentially all of the ideas and techniques in the subject were based on his work. Poincaré conjecture remained for many years as one of the most baffling and challenging unsolved problems in algebraic topology.

In applied mathematics he studied optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and cosmology. In the field of celestial mechanics he studied the three-body-problem, and the theories of light and of electromagnetic waves. He is acknowledged as a co-discoverer, with Albert Einstein and Hendrik Lorentz, of the special theory of relativity. Poincaré achieved the highest honours for his contributions of true genius. He was elected to the Académie des Sciences in 1887 and in 1906 was elected President of the Academy. The breadth of his research led him to being the only member elected to every one of the five sections of the Academy, namely the geometry, mechanics, physics, geography and navigation sections. He won numerous prizes, medals and awards.

 

7. Say whether the statements below are true (T) or false (F):

a) During his childhood Poincaré had poor muscular coordination.

b) He received special instruction from his gifted mother and excelled in written composition while still in elementary school.

c) Immediately after receiving his doctorate, Poincaré was appointed to teach mathematical analysis at the University of Berkely.

d) Poincaré was a scientist preoccupied by only one aspect of mathematics – topology.

e) Poincaré conjecture remained for many years as one of the most baffling and challenging unsolved problems in algebraic topology.

f) He was acknowledged as a co-discoverer, with Albert Einstein and Hendrik Lorentz, of the special theory of relativity.

 

Translate into English

Свойства двумерных многообразий были хорошо известны уже в середине девятнадцатого века. Однако оставалось неясным, справедливо ли для трех измерений то, что истинно в случае двух измерений. Пуанкаре предположил, что все замкнутые односвязные трехмерные многообразия (финитные многообразия без дырок) - являются сферами. Эта гипотеза имела особенно важное значение для ученых, исследующих самое большое трехмерное многообразие - нашу вселенную. Математическое доказательство этой гипотезы было, тем не менее, совсем не легким. Большинство попыток вело исследователей в тупик (to lead to a deadlock), но некоторые послужили источником важных математических открытий, таких как лемма (lemma) Дена, теорема сферы и теорема о петле, ставших базовыми теоремами современной топологии. 22 августа 2006 г. Григорию Перельману присуждена международная премия «Медаль Филдса» за решение гипотезы Пуанкаре. Однако российский учёный отказался от присутствия на церемонии вручения премии.

 

GRAMMAR PRACTICE: Conditionals –I (GR-23 p.209)

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