Proof

Maxim Kontsevich

2009-01-28T00:09:00.000-08:00

Born into the family of Lev Rafailovich Kontsevich – Soviet orientalist and author of the Kontsevich system. After ranking second in the All-Union Mathematics Olympiads, he attended Moscow State University but left without a degree in 1985 to become a researcher at the Institute for Problems of Information Transmission in Moscow. In 1992 he received his Ph.D. at the University of Bonn under Don Bernard Zagier. His thesis claims to prove a conjecture by Edward Witten that two quantum gravitational models are equivalent. Currently he is a Professor at the Institut des Hautes Etudes Scientifiques (IHÉS) in Bures-sur-Yvette, France and Distinguished Professor at University of Miami in Coral Gables, Florida, U.S.

His work concentrates on geometric aspects of mathematical physics, most notably on knot theory, quantization, and mirror symmetry. His most famous result is a formal deformation quantization that holds for any Poisson Manifolds . He also introduced knot invariants defined by complicated integrals analogous to Feymann integrals. In topological field theory, he introduced the moduli space of stable maps, which may be considered a mathematically rigorous formulation of the Feymann integral for topological string theory. These results are a part of his "contributions to four problems of geometry" for which he was awarded the Fields Medal in 1998.

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Stability structures, motivic Donaldson-Thomas invariants and cluster transformations,with Yan Soibelman.
Notes on motives in finite characteristic
On Malliavin measures, SLE and CFT, with Yuri Suhov
Automorphisms of the Weyl algebra, with Alexei Belov-Kanel
Affine structures and non-archimedean analytic spaces, with Yan Soibelman
Homological mirror symmetry and torus fibrations, with Yan Soibelman
Deformations of algebras over operads and Deligne's conjecture,with Yan Soibelman
Operads and Motives in Deformation Quantization

Roy J. Glauber

2009-01-27T23:11:00.000-08:00

Glauber was born in 1925 in New York City, a member of the 1941 graduating class of the Bronx High School of Science, and went on to do his undergraduate work at Harvard University. After his sophomore year he was recruited to work on the Manhattan Project, where (at the age of 18) he was one of the youngest scientists at Los Almos. His work involved calculating the critical mass for the atom bomb. After two years at Los Alamos, he returned to Harvard, receiving his bachelor's degree in 1946 and his PhD in 1949.

Glauber has received many honors for his research, including the A.A. Michelson Medal from the Franklin Institute in Philadelphia (1985), the Max Born Award from the Optical Society of America (1985), the Dannie Heineman Prize for Mathematical Physics the American Physical Society (1996), and the 2005 Nobel Prize in Physics. On 22nd April 2008, Professor Glauber was awarded the 'Medalla de Oro del CSIC' ('CSIC's Gold Medal') in a ceremony held in Madrid, Spain.

He currently lives in Arlington, Massachusetts and is the Mallinckrodt Professor of Physics at Harvard University, where both past and present students enthusiastically praised his teaching to Harvard Crimson reporters.

Glauber has two children, a son and a daughter, and five grandchildren.

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Title Quantum Optics and Heavy Ion Physics
Density Operators for Fermions (with Kevin E. Cahill)

Terence Tao

2009-01-17T02:10:00.000-08:00

Terence Chi-Shen Tao FRS (born July 17, 1975, Adelaide, South Australia) is an Australian mathematician working primarily on harmonic analysis, partial differential equations, combinatorics, analytic number theory and representation theory. His single most famous result is a proof, in joint work with British mathematician Ben J. Green, that there exist arbitrarily long arithmetic progressions of prime numbers (the Green–Tao theorem). Tao is currently a professor of mathematics at the University of California, Los Angeles.

In August 2006, he was awarded a Fields Medal, widely considered the top honor a mathematician can receive. Just one month later, in September 2006, he was awarded a MacArthur Fellowship. He was elected a Fellow of the Royal Society on May 18, 2007.

He received the Salem Prize in 2000, the Bôcher Prize in 2002, and the Clay Research Award in 2003, for his contributions to analysis including work on the Kakeya conjecture and wave maps. In 2005 he received the American Mathematical Society's Levi L. Conant Prize with Allen Knutson, and in 2006 he was awarded the SASTRA Ramanujan Prize.

In 2004, Ben Green and Tao released a preprint proving what is now known as the Green-Tao theorem. This theorem states that there are arbitrarily long arithmetic progressions of prime numbers. The New York Times described it this way:

“

In 2004, Dr. Tao, along with Ben Green, a mathematician now at the University of Cambridge in England, solved a problem related to the Twin Prime Conjecture by looking at prime number progressions—series of numbers equally spaced. (For example, 3, 7 and 11 constitute a progression of prime numbers with a spacing of 4; the next number in the sequence, 15, is not prime.) Dr. Tao and Dr. Green proved that it is always possible to find, somewhere in the infinity of integers, a progression of prime numbers of equal spacing and any length.

”

For this and other work, he was awarded the Australian Mathematical Society Medal in 2005.

In 2006, at the 25th Intenational Congress of Mathematicians in Madrid, he became one of the youngest, the first Australian, and the first UCLA faculty member ever to be awarded a Fields Medal. An article by New Scientist writes of his ability:

“

Such is Tao’s reputation that mathematicians now compete to interest him in their problems, and he is becoming a kind of Mr Fix-it for frustrated researchers. “If you're stuck on a problem, then one way out is to interest Terence Tao,” says Fefferman.

”

Tao was a finalist to become Australian of the Year in 2007.

In April 2008 Tao received the Alan T. Waterman Award, which recognizes an early career scientist for outstanding contributions in their field. In addition to a medal, Waterman awardees also receive a $500,000 grant for advanced research.

In December 2008 he was named The Lars Onsager lecturer of 2008, for “his combination of mathematical depth, width and volume in a manner unprecedented in contemporary mathematics”. He was presented the Onsager Medal, and held his Lars Onsager lecture entitled “Structure and randomness in the prime numbers” at NTNU, Norway.

Click to Download : Global regularity of wave maps V. Large data local wellposedness and perturbation theory in the energy clas.

Click to Download : Global existence and uniqueness results for weak solutions of the focusing mass-critical non-linear Schrödinger equatio.

Click to Download : Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy clas.

Click to Download : Global regularity of wave maps III. Large energy from $\R^{1+2}$ to hyperbolic space.

Click to Download : A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential.

Click to Download : On the testability and repair of hereditary hypergraph properties (with Tim Austin).

Click to Download : Random Matrices: The circular Law (with Van Vu).

Click to Download : Multi-linear multipliers associated to simplexes of arbitrary length (with Camil Muscalu and Christoph Thiele).

Click to Download : Linear Equations in Primes (with Ben Green).

Click to Download : Quadratic Uniformity of the Mobius Function (with Ben Green).

Click to Download : Global behaviour of nonlinear dispersive and wave equations.

Click to Download : A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equation.

Click to Download : New bounds for Szemeredi's Theorem, I: Progressions of length 4 in finite field geometries (with Ben Green)

Click to Download : An inverse theorem for the Gowers U^3 norm (with Ben Green)

Click to Download : The Gaussian primes contain arbitrarily shaped constellations

Andrei Okounkov

2009-01-16T01:17:00.000-08:00

Andrei Yuryevich Okounkov (Russian: Андрей Юрьевич Окуньков, Andrej Okun'kov) (born 1969) is a Russian mathematician who works on representation theory and its applications to algebraic geometry, mathematical physics, probability theory and special functions.

He has worked on the representation theory of infinite symmetric groups, the statistics of plane partitions, and the quantum cohomology of the Hilbert scheme of points in the complex plane. Much of his work on Hilbert schemes was joint with Rahul Pandharipande.

Okounkov along with Pandharipande, Nikita Nekrasov, and Davesh Maulik, has formulated well-known conjectures relating the Gromov-Witten invariants and Donaldson-Thomas invariants of threefolds.

Okounkov has an Erdős number of at most three, via Anatoly Vershik and Gregory A. Freiman.

In 2006, at the 25th International Congress of Mathematicians in Madrid, Spain he received the Fields Medal "for his contributions to bridging probability, representation theory and algebraic geometry."

Click to Download : Random partitions and instanton counting
Click to Download : Random surfaces enumerating algebraic curves
Click to Download : Random trees and moduli of curves
Click to Download : Symmetric functions and random partitions
Click to Download : Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram
Click to Download : Quantum Calabi-Yau and Classical Crystals (with Nikolai Reshetikhin and Cumrun Vafa)

to be continued in part 2 . .

Parelman's Proof

2009-01-15T23:23:00.000-08:00

Grigori Yakovlevich Perelman (Russian : Григорий Яковлевич Перельман), born 13 June 1966 in Leningrad, USSR (now St. Petersburg, Russia), sometimes known as Grisha Perelman, is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology. In particular, he proved Thurston's geometrization conjecture. This solves in the affirmative the famous Poincare conjecture, posed in 1904 and regarded as one of the most important and difficult open problems in mathematics until it was solved.

In August 2006, Perelman was awarded the Fields Medal for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow". Perelman declined to accept the award or to appear at the congres.

On 22 December 2006, the journal Science recognized Perelman's proof of the Poincaré Conjecture as the scientific "Breakthrough of the Year," the first such recognition in the area of mathematics.

Grigori Perelman was born in Leningrad (now Saint Petersburg) to a Jewish family on 13 June 1966. His early mathematical education occurred at the Leningrad Secondary School#239, a specialized school with advanced mathematics and physics programs. In 1982, as a member of the USSR team competing in the International Mathematical Olympiad, an international competition for high school students, he won a gold medal, achieving a perfect score. In the late 1980s, Perelman went on to earn a Candidate of Science degree (the Soviet equivalent to the Ph.D.) at the Mathematics and Mechanics Faculty of the Leningrad State University, one of the leading universities in the former Soviet Union. His dissertation was entitled "Saddle surfaces in Euclidean spaces".

After graduation, Perelman began work at the renowned Leningrad Department of Steklov Institute of Mathematics of the USSR Academy of Sciences, where his advisors were Aleksandr Danilovich Aleksandrov and Yuri Dmitrievich Burago. In the late 80s and early 90s, Perelman held posts at several universities in the United States. In 1992, he was invited to spend a semester each at the Courant Institute in New York University and Stony Brook University. From there, he accepted a two-year Miller Research Fellowship at the University of California, Bekerley in 1993. He was offered jobs at several top universities in the US, including Princeton and Stanford, but he rejected them all and returned to the Steklov Institute in the summer of 1995.

He has a younger sister, Elena, who is also a mathematician. She received a PhD from Weizmann Institute and is a biostatician at Karolinska Institute.

Perelman is a talented violinist and also plays table tennis.

Here his proof about Ricci Flow and Geometrization of Three-Manifolds :

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Andrew Wiles

2009-01-15T23:01:00.000-08:00

Sir Andrew John Wiles KBE FRS (born 11 April 1953) is a british mathematician and a professor at Princeton University, specialising in number theory. He is most famous for proving Fermat's Last Theorem.

Andrew Wiles was born in Cambridge, England in 1953 and attended King's College School, Cambridge (where his mathematics teacher, David Higginbottom first introduced Fermat's Last Theorem to him) and The Leys School, Cambridge; and earned his BA degree in 1974 after study at Merton College, Oxford, and a Ph.D. in 1980 after research at Clare College, Cambridge. His graduate research was guided by John Coates beginning in the summer of 1975. Together they worked on the arithmetic of elliptical curves with complex multiplication by the methods of Iwasawa theory.He further worked with Barry Mazur on the main conjecture of Iwasawa theory over Q and soon afterwards generalized this result to totally real fields. Taking approximately seven years to complete the work, Wiles was the first person to prove Fermat's Last Theorem, in 1995, earning him a place in history.

Here his proof about Fermat's Last Theorem :

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