Скачать презентацию Another Beautiful Mind The Fascinating Story of an Скачать презентацию Another Beautiful Mind The Fascinating Story of an

507a7084eaa2f6028a5616ac31a6d194.ppt

  • Количество слайдов: 59

Another Beautiful Mind The Fascinating Story of an Unusual Russian Mathematician and a Famous Another Beautiful Mind The Fascinating Story of an Unusual Russian Mathematician and a Famous 100 -Year-Old Unsolved Problem

Topics to be discussed • • Clay Institute of Mathematics What is mathematics? Life Topics to be discussed • • Clay Institute of Mathematics What is mathematics? Life of a mathematician More on topology Who is Grigory Perelman? Story of the Poincare Conjecture What finally happened? Questions / issues raised by this story

Clay Institute of Mathematics • The Clay Mathematics Institute (CMI) is a private, non Clay Institute of Mathematics • The Clay Mathematics Institute (CMI) is a private, non profit-foundation, based in Cambridge, MA. • The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. • The institute was founded in 1998 through the vision and generosity of Boston businessman Landon T. Clay and his wife, Lavinia D. Clay. • While the institute is best known for its Millenium Prize Problems, it carries out a wide range of activities including: ‒ Research awards and scholarships ‒ a postdoctoral program (ten Clay Research Fellows are supported each year) ‒ recognition awards and publications ‒ summer schools, conferences, workshops, public lectures, and outreach activities aimed at junior mathematicians

Millennium Prize Problems • On May 24, 2000, the Clay Mathematics Institute established seven Millennium Prize Problems • On May 24, 2000, the Clay Mathematics Institute established seven Prize Problems, representing some of the most difficult problem which mathematicians were grappling with at the turn of second Millennium ‒ These 7 problems were chosen by the founding Scientific Advisory Board of Clay Mathematics Institute, which conferred with leading experts worldwide. ‒ These classical problems were formulated between 1850 and 1970 and have been worked on by the best mathematical minds in the world but have resisted solution for many years ‒ A prize fund of $7 million was established, $1 million for each problem ‒ Rules were established for winning the prize money • The reclusive Russian mathematician Grigory Perelman (born 1966), in a series of short articles posted on the Internet beginning in late 2002, solved the Poincaré conjecture, one of the most fundamental problems in the entire field of mathematics. • March 18, 2010. The Clay Mathematics Institute (CMI) announces today that Dr. Grigory Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture

Rules for Winning Millennium Prize • • Publication of solution in a refereed mathematics Rules for Winning Millennium Prize • • Publication of solution in a refereed mathematics publication of worldwide repute, and it must also have general acceptance in the mathematics community two years after. Following two-year waiting period, the SAB (Scientific Advisory Board) will decide whether a solution merits detailed consideration. In the affirmative case, the SAB will constitute a special advisory committee who are experts in the area of the problem. • The SAB will seek advice from internationally-recognized mathematical experts in the area of the problem. Each component of proposed solution shall be verified by one or more members of this special advisory committee. • Following report of special advisory committee, the SAB will make a recommendation to the Directors. The SAB may recommend the award of a prize to one person or that a prize be divided among multiple solvers of a problem or theirs. • The SAB will pay special attention to the question of whether a prize solution depends crucially on insights published prior to the solution under consideration and may recommend recognition of such prior work in the prize citation

Thoughts about mathematics • Mathematics can call to mind many thoughts to different people. Thoughts about mathematics • Mathematics can call to mind many thoughts to different people. ‒ I can’t understand it; I hate it ‒ I was always bad at math ‒ I’m good with numbers ‒ I’m not going to understand anything about this talk; why am I here? !

Definitions of mathematics from the Web • A science (or group of related sciences) Definitions of mathematics from the Web • A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement • Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions. • An abstract representational system used in the study of numbers, shapes, structure and change and the relationships between these concepts • Mathematics deal with particular problems related to logic, space, transformations, numbers and more general ideas which encompass these concepts. • Mathematics is a study of patterns and relationships; a science and a way of thinking; an art, characterized by order and internal consistency; a language, using carefully defined terms and symbols

So, What Is Mathematics, Exactly? • Mathematics can, broadly speaking, be subdivided into the So, What Is Mathematics, Exactly? • Mathematics can, broadly speaking, be subdivided into the study of quantity (numbers), structure, space, and change

Topics of mathematics • Structure: Combinatorics, Number Theory, Group Theory, Graph Theory, Order Theory Topics of mathematics • Structure: Combinatorics, Number Theory, Group Theory, Graph Theory, Order Theory • Space: Geometry, Trigonometry, Differential Geometry, Topology, Fractal Geometry, Measure Theory • Change: Calculus, Vector Calculus, Differential Equations, Dynamical Systems, Chaos Theory, Complex Analysis • Foundations: Mathematical logic, Set theory, Category theory • Theoretical Computer Science: Theory of computation, cryptography • Applied Mathematics: Mathematical physics, Fluid dynamics, Numerical analysis, Optimization, Probability theory, Statistics, Financial Mathematics, Game Theory, Mathematical Biology, Mathematical Chemistry, Mathematical Economics

Life of a (research) mathematician • Mathematicians are known by their specialty, in branches Life of a (research) mathematician • Mathematicians are known by their specialty, in branches of mathematics ‒ In addition to teaching, mathematicians do research on esoteric problems that may or may not have application in the real world. ‒ They may work for years on a problem, solving just a piece of the problem but thereby providing insights to this or other problems that help a younger mathematician solve it at a later time. ‒ Most of them (not all) collaborate in their work ‒ Most research is funded by grants, usually from the Defense Department • Most mathematicians work for universities, but some large corporations employ mathematicians specifically for research ‒ Benoit Mandelbrot, a famous Polish mathematician, spent most of his career at IBM's Thomas J. Watson Research Center doing theoretical research ‒ Thousands of firms employ research mathematicians doing applied mathematics to support their commercial products (Google, Microsoft, etc. )

Recognition of Mathematicians • Mathematicians follow certain patterns to gain fame and recognition. ‒ Recognition of Mathematicians • Mathematicians follow certain patterns to gain fame and recognition. ‒ Publication is critical. ‒ Mathematical conferences to present research results ‒ Prizes: the prestigious Fields Medal, or the Abel Prize, which are comparable to the Nobel prize in other fields. ‒ Competition, friendly and otherwise. If a mathematician has been working on a problem for many years and a young upstart discovers the solution, or even a new development, where does that leave the long-time researcher? He or she has invested years of his or her career in this problem. • Mathematicians value content over form

Brief History of Geometry • Geometry is the branch of mathematics concerned with questions Brief History of Geometry • Geometry is the branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. • Geometry is one of the oldest mathematical sciences. • Geometry is the most powerful branch of mathematics for describing physical space • All of physics and chemistry is based in geometry, impossible to separate the math of geometry from the laws of physics • Newton, Euler, Gauss, Maxwell: both great mathematicians and great physicists • Geometry useful in engineering, architecture

Geometry History - continued 3 great revolutions in the history of geometry ‒ 300 Geometry History - continued 3 great revolutions in the history of geometry ‒ 300 BC: Euclid – Greek - father of Geometry (Euclidean geometry), published textbook Elements, one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19 th or early 20 th century ‒ 1600 s: Decartes – (“father” of modern philosophy) developed a coordinate system allowing geometric shapes to be expressed in algebraic equations (this led to calculus) ‒ 1900 s: Poincare – development of topology • First great topology theorems predated Poincare, in the 17 th Century • Euler formula: relationship beteeen # sides, #edges, and # points in a geometric figure S + P = E +2

What is Topology? • Definition 1: Topology is the part of geometry that does What is Topology? • Definition 1: Topology is the part of geometry that does not depend on specific measurement of distances and angles • Definition 2: Topology is the branch of mathematics concerned with basic properties of geometric figures that remain unchanged when they are stretched, shrunk, deformed, or distorted as long as they are not ripped or punctured • Topology is a relatively new field of mathematics • Issues in topology that we will discuss ‒ Connectedness ‒ Spaces being (topologically) equivalent ‒ Dimension

Mobius Strip (best known object in topology) • Consider the idea of a surface Mobius Strip (best known object in topology) • Consider the idea of a surface • Piece of paper (or the table top) has 2 surfaces, top and bottom. • Any line drawn on one surface does not cross to the other surface • Mobius strip is a 1 -surface, or 1 -sided line drawn on one side is continuous • Demonstrate, also cut in half

More on the Mobius Strip • Mobius strip discovered in 19 th century (earliest More on the Mobius Strip • Mobius strip discovered in 19 th century (earliest reference to it was as a parlor trick in 1882) • Ubiquitous symbol for recycling, where it represents the process of transforming waste materials into useful resources • Useful applications of Mobius Strip ‒ conveyor belts (to make them last longer, since "each side" gets the same amount of wear ‒ continuous-loop recording tapes (to double the playing time) ‒ In the 1960's Sandia Laboratories used Möbius Strips in the design of versatile electronic resistors. ‒ Mobius strips are common in the manufacturing of fabric computer printer and typewriter ribbons, allowing the ribbon to be twice as wide as the printhead while using both half-edges evenly.

Concepts in Topology Connectedness: A geometrical figure is “simply connected” if any simple closed Concepts in Topology Connectedness: A geometrical figure is “simply connected” if any simple closed curve can be shrunk to a point continuously in the set. A sphere is simply connected because every loop can be contracted (on the surface) to a point. • • Examples of figures that are simply connected ‒ Sphere ‒ Football ‒ Dice Examples of figures that are not simply connected ‒ Bagel ‒ Coffee Cup

Topologically equivalent Two objects are “topologically equivalent” when one can be shaped into the Topologically equivalent Two objects are “topologically equivalent” when one can be shaped into the other • Examples that are “topologically equivalent” ‒ Melon = dice = baseball bat ‒ Donut = bagel = coffee cup ‒ BUT melon ≠ donut

Working in dimensions • Dimension of an object or space: minimum number of coordinates Working in dimensions • Dimension of an object or space: minimum number of coordinates needed to specify a point, from a reference point, within it – No dimensions: a point (because there is no reference point) – One dimension: a line ‒ Locate a point on the line by one number, or coordinate (starting with a reference point) – Two dimensional space is a plane ‒ Locate a point on the plane by 2 numbers, from a reference point ‒ Three dimensional space : locate a spot by 3 numbers, from a reference point • Next: 4 dimensional space, 4 th dimension is time • Mathematicians work in “n” dimensions ‒ Location can be expressed by a vector (1, 2, 3, . . . n)

Another concept: Topological Surface • On a “closed” 2 -dimensional surface (not necessarily flat), Another concept: Topological Surface • On a “closed” 2 -dimensional surface (not necessarily flat), that is one with boundaries, any two points can be connected by one line that lies entirely within the surface • Now, consider the surface of a 3 -dimensional sphere ‒ Any two points can be connected by one line (not necessarily straight) • Thus, the topological surface of a 3 -dimensional sphere is a 2 -dimensional space • In topology terms, the sphere is called a ball, and the surface of the ball is called a sphere, so we would say ‒ “the topological surface is a 3 -dimensional ball is a 2 -dimensional sphere” ‒ “the topological surface of a 4 -dimensional ball is a 3 -dimensional sphere” ‒ “the topological surface of a 5 -dimensional ball is a 4 -dimensional sphere”

Bringing it together • We will go back and forth between the story of Bringing it together • We will go back and forth between the story of Grigory Perelman, Russian mathematician, and attempts by him and by others to solve the Poincare Conjecture, one of the 7 Clay problems in the field of topology • Story goes from the time Poincare first proposed the question, in 1904 until March 18, 2010 when the Clay Institute announced the prize • Story of both the problem and the man (Perelman) involves mathematicians, egos, careers, difficult personalities, etc.

Grigory Perelman, Russian mathematician SUMMARY • Born in 1966 in St. Petersburg, Jewish, mother Grigory Perelman, Russian mathematician SUMMARY • Born in 1966 in St. Petersburg, Jewish, mother gave ujp graduate studies to raise him, parents separated (father now lives in Israel) • Great difficulty communicating, poor social skills, needed help with minutae of everyday living (probably has Asperger’s) • Joined math club, small group of very bright boys • • ‒ influence of math coach Sergei Rukshin Perelman went for the Olympiad path ‒ “Never encountered a problem he couldn’t solve, ” ‒ Won gold metal in 1982, gained admission to Leningrad University Late 1980 s: came to US, studied at NYU, SUNY Stonybrook, UC Berkeley Always a recluse, returned to Russia in 90 s, worked alone Famous for solution to Poincare Conjecture

The World of Math Clubs in Soviet Russia • • • Competitive mathematics like The World of Math Clubs in Soviet Russia • • • Competitive mathematics like a sport. Children enter while in grade school. For Russian child, math club was a sort of miracle, given the great degree of uniformity to Soviet life (education, living arrangements, etc. ) Math clubs involved coaches, practice sessions, competitions Children given math problems, work on solving during sessions Some math clubs involved total immersion by students ‒ Andrei Kolmogorov : good mathematical education required boys’ club environment: hiking, sports, swimming, cross-country skiing, listening to music, discussing mathematical projects Limited opportunities for Jews in the 1960 s: ‒ Only 2 Jews accepted at Leningrad University ‒ Be on of Soviet team for International Mathematical Olympiad, guaranteed admission to Leningrad University ‒ Attend lesser university

Mathematical Olympiad (IMO) • The International Mathematical Olympiad (IMO) is an annual six-problem, 42 Mathematical Olympiad (IMO) • The International Mathematical Olympiad (IMO) is an annual six-problem, 42 point mathematical olympiad for pre-collegiate students. • First IMO was held in Romania in 1959, has since been held annually (not 1980) • About 100 countries send teams of up to six students, plus one team leader, one deputy leader, and observers. • Content ranges from extremely difficult precalculus problems to problems on branches of mathematics not conventionally covered at school ‒ Anyone with a basic understanding of mathematics should understand the problems, even if the solutions require a great deal more knowledge ‒ This creates an incentive to find elegant, deceptively simple-looking problems but which require a certain level of ingenuity. • Awards are given to a top percentage of the individual contestants. Teams not officially recognized—all scores are given only to individual contestants

Sergei Rukshin, master coach • Troubled teen, minor juvenile offenses by age 15 ‒ Sergei Rukshin, master coach • Troubled teen, minor juvenile offenses by age 15 ‒ Expected path: trade school, military, short life of drink and violence ‒ Parents pleaded for him to get into mathematical high school • Rukshin fell in love with mathematics, began organizing children into math teams, trained them to do better than he did, became a teaching assistant at the Palace of Pioneers, then became math coach • Coach for >25 years, one of best in Russia ‒ His students have won >70 Mathematical Olympiad medals, >40 gold ones ‒ Half of Russian competitors come from Rukshin’s club ‒ Rukshin still runs Mathematics Education Center, >200 children, 11 and older

More on Rukshin • • • His method: engage every child in separate conversation More on Rukshin • • • His method: engage every child in separate conversation about child’s particular successes, difficulties, and mistakes ‒ No child could coast at any time ‒ Every child had to present solution to entire group ‒ Mathematics became the most important thing in children’s lives Rukshin’s life work: create world-class mathematics competitors Developed special relationship with Perelman that continued outside the activity of the math club ‒ Traveled home on same train after club meetings ‒ Influenced Perelman to take interest in music ‒ Took care of Perelman’s day-to-day needs (laundry, cleaning) at summer camp ‒ Spent one summer teaching him English (Perelman was age 14) so P could enter specialized math high school

Perelman in High School • • • Soviet system based on uniformity – everyone Perelman in High School • • • Soviet system based on uniformity – everyone taught the same thing Initially, two types of specialized high schools, those devoted to math and physics, and those devoted to foreign language. Now, specialized schools in sports and music Perelman attended Leningrad’s Specialized Mathematics School Number 239 ‒ This is the alma mater of numerous winners of International Math Olympiads ‒ No quota for Jews, unlike in many prestigious universities and research institutions Teachers at School 239 tried to shape students to be well-rounded by exposing them to hikes, music, literature Valery Ryzhik – teacher at School 239 – commented about Perelman ‒ Perelman was an “awe-inspiring student” ‒ Perelman never went on outings with other students ‒ Perelman never broached any personal subject; he only wanted to study math and to be left alone!

University Studies • • Graduated from Leningrad State University with a Candidate of Science University Studies • • Graduated from Leningrad State University with a Candidate of Science degree (the Soviet equivalent to the Ph. D After graduation, Perelman began work in Leningrad at the renowned Steklov Institute of Mathematics

Russia in late 1980 s • • • Reforms under Gorbachev ‒ 1985: perestroika Russia in late 1980 s • • • Reforms under Gorbachev ‒ 1985: perestroika (restructuring, political and economic reforms) ‒ 1988: golden age of glasnost (openness), ‒ 1989: first semi-democratic elections, 1 st parliamentary debate More intellectual opportunity ‒ Soviet scholars began to travel abroad ‒ For Perelman, now possible to be on a path for membership in the world of the mathematical elite Mikhail Gromov: one of world’s leading geometers ‒ Brought Perelman to international conferences ‒ Co-authored papers ‒ Recommended Perelman for positions ‒ Totally believed in Perelman, compared him to Newton

Perelman Comes to United States • • • Arrived in Us in 1992, age Perelman Comes to United States • • • Arrived in Us in 1992, age 26, for graduate studies at Courant Institute at NYU ‒ Long hair, long fingernails, same clothes every day, obtained black bread and fermented milk from Russian store in Brooklyn (he walked from Manhattan) ‒ Mother came with him, lived in Brooklyn ‒ Formed friendship with Gang Tian, interested in little outside mathematics ‒ Tian and Perelman went to lectures at Institute for Advanced Studies Spring 1993: went to SUNY Stonybrook as a postdoc ‒ Established relationship with Mike Anderson ‒ Invited to speak at International Congress of Mathematics ‒ Becoming a star because he proved important theorems Next: another postdoc at Univ of California Berkeley, to do research ‒ Engaged in discussions with Bruce Kleiner about Geometrization Conjecture (related to Poincare Conjecture) ‒ Courted by several universities, but seemed “stuck” in America, so returned to Russia, went back to lab at Steklov Institute, and moved in with his mother

Perelman Back in St. Petersburg • Perelman now worked in lab, little interaction with Perelman Back in St. Petersburg • Perelman now worked in lab, little interaction with anybody • Grew increasingly resistant to questions about his projects • Turned down prize awarded by European Mathematical Society at their 2 nd Congress ‒ “I don’t want it and I won’t accept it. ” • Did not answer emails from his American colleagues (esp Mike Anderson, of Stony Brook) • Growing isolation

Back to Topology • Topological ideas are present in almost all areas of today's Back to Topology • Topological ideas are present in almost all areas of today's mathematics. ‒ The subject of topology itself consists of several different branches, such as point set topology, algebraic topology and differential topology, which have relatively little in common • Topological ideas were expressed as early as 1736 in St. Petersburg ‒ Swiss mathematician Leonhard Euler, teaching there, published a paper on the solution to the Konigsberg bridge problem ‒ Mayor wanted Euler to devise a walking tour that would pass through each of Konigsburg’s 7 bridges exactly once. Euler proved it couldn’t be done • The term "Topologie" was introduced in German in 1847 • "Topology, " its English form, was first used in 1883 • The term topologist in the sense of a specialist in topology was used in 1905 • Modern topology depends strongly on the ideas of set theory, in the later part of the 19 th century

 • Definitions needed for understanding Poincare Conjecture Balls and spheres ‒ We think • Definitions needed for understanding Poincare Conjecture Balls and spheres ‒ We think of earth as a sphere, but in mathematical terms, it is a 3 -dimensional ball ‒ The surface of a 3 -dimensional ball is a 2 -dimensional sphere ‒ The surface of a 2 -dimensional ball is a 1 -dimensional sphere (a circle) ‒ The surface of a 4 -dimensional ball is a 3 -dimensional sphere • When each neighborhood of the earth is mapped onto a piece of paper, we have parallel gridlines not crossing, triangles having sum of angles = 180 degrees, etc. • Parallel streets do not cross (i. e. , 1 st and 2 nd Avenue in NYC) • But when all atlas pages are put together to be the earth, we would need to smooth out edges and ultimately get a globe, which would reflect the Earth’s curvature ‒ In this case, parallel lines would cross; sum of angles in a triangle ≠ 180°

Definitions • Euclidean geometry – flat space, parallel lines don’t cross ‒ Example: a Definitions • Euclidean geometry – flat space, parallel lines don’t cross ‒ Example: a map on paper • Non-Euclidean geometry – curved space, parallel lines do cross ‒ Example: a globe • Manifold: a space that can be divided into many neighborhoods ‒ Each neighborhood has a basic Euclidean geometry, but when put together, it becomes a curved space ‒ Consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round. On small scales that we see, the Earth does indeed look flat. ‒ In general, any object that is nearly "flat" on small scales is a manifold

So what is the story of the Poincare Conjecture? • Henri Poincare (1854 – So what is the story of the Poincare Conjecture? • Henri Poincare (1854 – 1912): French mathematician and physicist ‒ He made many contributions to different fields of pure and applied mathematics ‒ In a 1904 paper, he formulated one of the most famous problems in mathematics, known as the Poincare conjecture: “Every simply connected, closed 3 -manifold is homeomorphic to the 3 -sphere” • Why is Poincare such a famous problem? ? Because its solution has eluded mathematicians ever since the problem was proposed!!!

The Poincare Conjecture Poincaré knew that a two dimensional sphere (the surface of a The Poincare Conjecture Poincaré knew that a two dimensional sphere (the surface of a 3 -dimensional ball) is essentially characterized by the property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This Poincare conjecture states that the only compact three dimensional simply connected manifold is a three dimensional sphere. (Remember: a manifold is a curved space that is nearly flat in small neighborhoods)

What is the Importance of the Poincare Conjecture? • “It's pure knowledge. Not all What is the Importance of the Poincare Conjecture? • “It's pure knowledge. Not all knowledge brings direct benefits to mankind. But to mathematicians, it's a great achievement that closes a major hole in their understanding. ” • “Indirect benefits. . the primary benefit of Perelman's work is not the answer to the a big question, but the techniques he developed in the process of tackling it. • That's the practical benefit of big, "pure" problems like Poincare and most of the Clay Institute problems. The answer to the question itself isn't as important as the machinery that has to be built to get there. Once an engine is built, industrious people will always find a way to drive it interesting places, even if the guy who invented it regarded it as an end in its own right and doesn't care where it goes. • Its real significance is that it's a basic structural statement about how the world is organized. (If you prefer order to chaos, that's something worth caring about. )

Attempts to Solve Poincare Conjecture • John H. C. Whitehead (Englishman), 1934 ‒ Submitted Attempts to Solve Poincare Conjecture • John H. C. Whitehead (Englishman), 1934 ‒ Submitted proof to Quarterly Journal of Mathematics, but made an error so on Feb 8, 1935, he submitted a recantation • Christos Papakyriakopoulos (Papa), b. 1914, moved to Princeton in 1948. Devoted last 15 years of his life to one single aim – proving the Poincare Conjecture (1960 s until his death in 1976). ‒ His approach: prove something more manageable. First attempt was 2 conjectures, but NYU mathematician found holes in this theory ‒ In all, Papa came up with 13 new conjectures that, if proved, would prove the PC. But he failed to do this. ‒ Great secrecy in his work ‒ Last 12 years of his life, published only 1 paper

Other attempts • Elvira Strasser-Rapaport ‒ Came to mathematics at age 43 after raising Other attempts • Elvira Strasser-Rapaport ‒ Came to mathematics at age 43 after raising 2 daughters ‒ Proved one of Papa’s conjectures, but then stalled • RH Bing (Texan) ‒ Fascinated by PC for most of his adult life ‒ Published related paper in 1958, but only a partial solution • Ken’iti Koseki (Japanese) ‒ Published 107 page paper in 1958 in Japanese journal, but the mathematics community discounted it • Edwin Moise, Princeton ‒ Proved in 1952 that PC was correct for dimension 3 ‒ In competition with Papa to see who could solve it first ‒ Could never find solution, eventually turned away from mathematics altogether ‒ Wrote textbook in geometry and then moved on to literary criticism ‒ BUT, paper he wrote in 1952 would prove to be an essential ingredient in the ultimate proof of PC

More attempts, continued • Wolfgang Haker (1928), German ‒ Best known for solution of More attempts, continued • Wolfgang Haker (1928), German ‒ Best known for solution of 4 -color problem in topology, gained him worldwide recognition ‒ Became obsessed with Poincare Conjecture ‒ Coined Poincaritis: getting so wrapped up in trying to prove PC that sufferer would do little useful work for tens of years • Valentin Poenaru, Budapest ‒ Became interested in PC while studying at Princeton in 1960 s ‒ In competition with Hakem for over 10 years ‒ While at Penn State in 1994, outlined a program for proof of PC, but never managed to prove all steps ‒ Still working on PC in 2004!! • There are more!!!

Solving Poincare Conjecture for higher dimensions • Surprisingly, this was easier • John Stallings: Solving Poincare Conjecture for higher dimensions • Surprisingly, this was easier • John Stallings: in 1960, submitted proof for dimensions 7 and higher • Steven Smale: proved PC for dimensions 5 and higher • 1982: Michael Friedman did a proof for dimension 4, this was hailed as a breakthrough, and he received a Fields Medal ‒ Fields Medal is like the Nobel Prize for mathematicians under age 40 • Richard Hamilton (Berkeley), 1980 s, worked on PC, created a blueprint – another way for approaching the PC, but then got stuck at age 40! • BUT NO PROOF FOR DIMENSION 3!!!

Path to solution • In the world of top mathematicians, the intellectual elite are Path to solution • In the world of top mathematicians, the intellectual elite are the people who open new horizons by posing questions no one else has though to ask (Poincare) • Next are the mathematicians who devise ways to answer those questions. They come up with techniques, solve related problems that give new insights • Finally, the persistent, exacting mathematician that finishes the job!

Other players in development of Perelman’s career • Mikhail Gromov: instrumental in introducing Perelman Other players in development of Perelman’s career • Mikhail Gromov: instrumental in introducing Perelman to the international community • Jeff Cheeger (geometer, NYU): first met in 1991, advised and interacted with him during 90 s and after PC had been printed, in 2003 • Michael Anderson (Stonybrook): helped Perelman with personal affairs, reached out by email when P became reclusive • Gang Tian (NYU, later MIT): friend of Perelman at Courant, both postdocs. Perelman, in April 2003, presented his proof at MIT, at invitation of Tian • John Morgan (Columbia): collaborated with Gang Tian in verifying Perelman's proof of the Poincaré conjecture • Bruce Kleiner (NYU): friend from Berkeley, conversations about Geometrization Conjecture (a long-unsolved problem that included Poincare Conjecture) • John Lott (Univ of California, Berkeley): worked with Kleiner on verifying proof

Back to Perelman • Perelman began to work on the Poincare Conjecture in the Back to Perelman • Perelman began to work on the Poincare Conjecture in the late 1990 s, when he began to disappear • No contact with his “friends” • Refused to answer questions about his work • Left his job at Steklov lab

The Proof Emerges (Tuesday, November 12, 2002) • Perelman sent email to almost 1 The Proof Emerges (Tuesday, November 12, 2002) • Perelman sent email to almost 1 dozen US mathematicians drawing attention to his website • He had posted a paper on website hosted by Cornell library for purposes of facilitating electronic communication between mathematicians • This was the first of 3 papers (preprints) with the results of Perelman’s work on the Poincare Conjecture (his proof) • Every recipient of P’s email had worked on PC for years, so conflicted response • For 100 years, proofs appeared, then were debunked. How to determine if this proof was correct? ?

Reaction to Perelman’s Email • Flurry of emails followed • Mike Anderson sent mails Reaction to Perelman’s Email • Flurry of emails followed • Mike Anderson sent mails to colleagues • Within hours, he began getting responses from geometers from all over the world • Many mathematicians set aside their own professional ambitions to devote themselves to deciphering and interpreting Perelman’s work. Everyone knew how seriously Perelman should be taken • Richard Hamilton silent ‒ Richard Hamilton, professor at Columbia in field of differential geometry, best known for having discovered Ricci flows ‒ Ricci flow community centered around Hamilton ‒ Ricci flow played important role in Perelman’s solution of Poincare Conjecture

Perelman Emerges from Hermithood • Perelman received invitations to speak. He agreed to come Perelman Emerges from Hermithood • Perelman received invitations to speak. He agreed to come to the US for 1 month ‒ He writes up 2 nd preprint, 22 pages, 8 pages shorter than the 1 st preprint ‒ Arrived at MIT in April, 2003. Lecture hall packed ‒ Perelman’s rules: no publicity, no videotaping ‒ Presentation lucid, organized, even playful • For 2 weeks, Perelman gave talks, held discussions with Tian, was relaxed, friendly • Received offers of professorship at MIT (not interested) • April 15 New York Times headline: Russian Reports He Has Solved Celebrated Math Problem

Perelman’s reaction to attention • Perelman incensed over NY Times article ‒ He “reported” Perelman’s reaction to attention • Perelman incensed over NY Times article ‒ He “reported” nothing ‒ Poincare Conjecture “celebrated” in newspaper was “vulgar” • Perelman bristly on the subject of the $1 million prize, he was not interested in money. • Perelman had no interest in anyone “vetting” his proof. • Perelman revolted against scientific journals. • Perelman never wanted publicity.

Exposure and offers after MIT • Stony Brook (invitation of friend Michael Anderson) for Exposure and offers after MIT • Stony Brook (invitation of friend Michael Anderson) for 2 weeks: fantastic clarity in his lectures • Columbia, April 26 (invitation of John Morgan) Sat April 26 • Princeton (received offers of professorship, but annoyed over this courtship) • Meeting with Jim Simons, highly successful and influential hedge fund manager who had built math department at Stony Brook to be world class ‒ Perelman could come to US on any terms he chose (money, time, etc. ) ‒ Perelman said no. He abhorred the idea of being some department’s prized possession.

Reaction of Richard Hamilton • Hamilton was person who could speak authoritatively on the Reaction of Richard Hamilton • Hamilton was person who could speak authoritatively on the proof ‒ Perelman did not belong to Ricci flow community surrounding Hamilton. ‒ Perelman had approached Hamilton twice, asking for clarification on a problem. 2 nd time, Hamilton failed to respond. ‒ Hamilton was keeping silent on Perelman’s proof. • Hamilton had not attended Perelman’s lectures at MIT or Stonybrook, but did attend Columbia lecture ‒ In small group setting (Hamilton, Perelman, Morgan, and Gromov) Hamilton gave endorsement on 1 st part of paper but withheld judgment on 2 nd part ‒ Hamilton: “proper but distant. ” ‒ Level of questions indicated Hamilton hadn’t delved into proof

Follow-up to U. S. Trip • Perelman: back to Russia at end of April, Follow-up to U. S. Trip • Perelman: back to Russia at end of April, 2003, but showed no inclination to be involved with continuing work on his proof • Kleiner, Lott, Tian, Morgan: held workshop on 1 st preprint. These 4 were involved closely reading Perelman’s paper. ‒ Perelman refused to look at notes of Kleiner and Lott, so Kleiner and Lott went to work on Perelman’s papers without him. (Kleiner and Lott notes were published in November 2008) ‒ Sept 2004: Tian emailed Perelman saying that they now understood proof (after 1 ½ years). Told him about a book idea. No response. ‒ Morgan and Tian completed manuscript in 2006, mailed it to Perelman. Package returned, service refused. • Summer 2004 – workshop in Princeton sponsored by Clay Institute • Summer 2005: Clay Institute sponsored month-long workshop on proof

Understanding Perelman • • • Asperger’s Syndrome – defined in 1940 s, Austrian pediatrician Understanding Perelman • • • Asperger’s Syndrome – defined in 1940 s, Austrian pediatrician Hans Asperger. Part of the autism spectrum ‒ High IQ but social maturity and social reasoning delayed ‒ Difficulty making friends, trouble communicating ‒ Difficulty understanding and controlling emotions ‒ Needs assistance in organizing their lives (dependent on mother) Simon Baren-Cohen, a British psychologist, did a study at Cambridge and found that mathematicians are 3 -7 times more likely than other students to have the diagnosis of autistic condition. Baren-Cohen developed a statistic called the AS quotient (Autism Spectrum Quotient). He gave his test to Cambridge students and winners of the British Mathematical Olympiad. Findings: ‒ mathematicians scored higher than other scientists ‒ Scientists scored higher than students in the humanities

Russian math culture • Asperger culture build into larger Russian culture of mathematics. • Russian math culture • Asperger culture build into larger Russian culture of mathematics. • Russian math prodigies are grouped with others of their kind, tradition of forgiving mathematicians their autistic rudeness ‒ Walking away from conversation ‒ Utter disregard for social convention ‒ No exchange of pleasantries • Other traits of Asperger people: ‒ Believer in single truth, unable to adjust for human limitations ‒ They don’t take into account what the listener needs to know ‒ Another theory: everything they hear taken literally ‒ autistic brain cannot empathize ‒ Rushkin’s math club had been a refuge. He sheltered the “black sheep. ”

Perelman’s Life Back in Russia (after April 2003) • • • Perelman returned to Perelman’s Life Back in Russia (after April 2003) • • • Perelman returned to Steklov Institute in St. Petersburg. He gave seminars, received promotion to “lead researcher” Human interaction dwindling over 8 years since he got his post doc. His only “close” relationship was with Ladyzhenskaya, head of the lab, but she died in Jan 2004 at age 82. After that, he rarely talked to anyone. Russian Academy of Sciences was putting its house in order since the chaos of the 90 s, so Steklov Institute becoming increasingly uncomfortable for Perelman ‒ Demanded paperwork, reports on research, publishing activity Perelman was becoming strange over money. ‒ He refused accept payment for expenses for trip to Moscow ‒ He refused to accept a distribution of funds left over from leftover grant money ‒ Perelman asked the exact amount and came back later with 8000 rubles in cash to pay back. He finally prevailed. Finally, Perelman quit his job at the lab in December 2005. “I have been disappointed in mathematics and I want to try something else. I quit. ” He canceled his email account and left. Something just snapped.

The Story Continues • • June 2006, issue of Asian Journal of Mathematics, 300 The Story Continues • • June 2006, issue of Asian Journal of Mathematics, 300 pages devoted to an article by 2 Chinese mathematicians Huai-dong Cao and Xi-Ping Zhu, entitled “A Complete Proof of the Poincare and Geometrization Conjectures – Application of the Hamilton-Perelman Theory of the Ricci Flow. ” ‒ These mathematicians had worked under Shing-Tung Yau, Harvard professor and Fields Medalist, close friend to Hamilton. ‒ Authors claim: Hamilton and Perelman laid the groundwork for proofs of these conjectures but that they themselves made the breakthrough, so they should get the $1 million and credit for the proof. ‒ Yau, at a pres conference said: Hamilton contributed 50%, Perelman 25%, Cao, Zhu, and Yau 30%. Week later, Yau gave credit to Chinese mathematicians for “completely solving the puzzle. ” ‒ New Yorker article on August 29 by Sylvia Nasar tracing story. Anderson was quoted: “Yau wants to be the king of geometry. He believes everything should issue from him, that he should have oversight. He doesn’t like people encroaching on his territory. ” Mathematical scandal, mathematicians not accustomed to this much controversy.

More drama surrounding Perelman • • International Congress of Mathematicians (ICM) was to hold More drama surrounding Perelman • • International Congress of Mathematicians (ICM) was to hold annual meeting August 22, 2006 in Madrid Topics to be discussed: Perelman’s proof, Kleiner and Lott notes, Cao-Zhu’s paper Perelman had been invited to speak at ICM meeting a year before but he refused. He was also going to be awarded the Fields Medal. But would he accept? ‒ Fields Medal committee dispatched president of the International Mathematical Union, Sir John Ball, Oxford Professor, to St. Petersburg. ‒ But Perelman would not accept medal. The prospect of the Fields Medal award was making him too conspicuous, getting roped into limelight. Aug 22: John Ball announced names of Fields Medal recipients at ICM meeting in Madrid. Perelman was one. By fall 2006, the idea of credit given to Cao and Zhu was put to rest. A pdf file began circulating among mathematicians that showed sizeable portions of their proof matching verbatim excerpts fro Kleiner and Lott’s notes on the 1 st Perelman preprint, posted on the web in 2003. December 2006: Cao and Zhu published a revised article giving credit for the proof to Perelman and Hamilton.

Russian Media Frenzy • Journalists, including from tabloids, began calling Perelman • Channel 1 Russian Media Frenzy • Journalists, including from tabloids, began calling Perelman • Channel 1 (TV) reaching 98% of households, reported that Perelman had turned down the $1 million prize. • School 239, his former teachers began offering opinions on his sanity • People began snapping pictures of Perelman on their mobile phones • Word circulated that the reason he didn’t go to the ICM conference in Madrid was because he couldn’t afford a ticket. Everyone wanted to give him money ‒ Cabinet minister asked to talk to him. ‒ A private Moscow foundation, in cooperation with Rushkin, cooked up scheme to give money to his mother. When he heard her on the phone, he ripped it out of her hand, screaming. ‒ Perelman”s response: he was no longer doing math

What About the $1 Million Prize? • By Fall 2006, the issue of authorship What About the $1 Million Prize? • By Fall 2006, the issue of authorship of proof was settled. The Millenium prize rules required a 2 -year waiting period. ‒ The committee would make recommendations in Fall 2009. ‒ Expectation: the million dollar prize would go to Perelman, but what would he do? ‒ Perelman turned down all offers of money as well as friendship. ‒ Perelman even ended his last remaining close personal relationship, with Rushkin, his former math coach. • Gromov: was the only one to understand Perelman’s refusal to accept prize. Reason: “because Clay is nothing, from his point of view. ”

Discussion Questions • Should there be large awards and prize money given for something Discussion Questions • Should there be large awards and prize money given for something like Poincare Conjecture that has no direct application to anything? ‒ Note: private vs. public money • What are your ideas on government giving money for research that directly supports the public good (ie. cancer research) vs basic research (ie, grants from the Defense Dept to mathematicians)? • Does the prize money play a role in stimulating the work ? • Why does math not play the same kind of role of importance with US students that it does with Russian students?