Perelman

         Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology

         In 1994, Perelman proved the soul conjecture. In 2003, he proved Thurston's geometrization conjecture. This consequently solved in the affirmative the Poincare conjecture, posed in 1904, which before its solution was viewed as one of the most important and difficult open problems in topology.

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 congress, stating: "I'm not interested in money or fame, I don't want to be on display like an animal in a zoo."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.

On 18 March 2010, it was announced that he had met the criteria to receive the first Clay Millennium prize for resolution of the Poincaré conjecture. On 1 July 2010, he turned down the prize, saying that he considers his contribution to proving the Poincaré conjecture to be no greater than that of Richard Hamilton, who introduced the theory of Ricci flow with the aim of attacking the geometrization conjecture.

         Perelman is really a person who doesn't care for fame.This kind of people are rather great.It's this kind of people that really can do great jobs and beautiful math.We need more people like him in the world of mathematics.I really admire his spirit and behavior.

Geometric Analysis

Geometric analysis is a mathematical discipline at the interface of differential geometry and differential equations. It includes both the use of geometrical methods in the study of partial differential equations (when it is also known as "geometric PDE"), and the application of the theory of partial differential equations to geometry. It incorporates problems involving curves and surfaces, or domains with curved boundaries, but also the study of Riemannian manifolds in arbitrary dimension. The calculus of variations is sometimes regarded as part of geometric analysis, because differential equations arising from variational principles have a strong geometric content. Geometric analysis also includes global analysis, which concerns the study of differential equations on manifolds, and the relationship between differential equations and topology
Geometric analysis is a very typical and effective way to study geometry and topology,and it has become one of the most popular branch in mathematics especially in geometry.The tool which Perelman use to solve Poincare conjecture is Ricci flow,which is just from geometric analysis.Geometric analysis really has a high position in geometry and more and more people study it to get to know geometry better.

Shing_Tung Yau

     Today I want to introduce another great mathematician who is a Chinese American to you. His name is ShingTung Yau.
     Shing-Tung Yau , (born April 4, 1949) is a Chinese-born American mathmatician. He won the Fields Medal in 1982.Yau's work is mainly in differential geometry, especially in geometric analysis. His contributions have had an influence on both physics and mathematics and he has been active at the interface between geometry and theoretical physics. His proof of the positive energy theorem in general relativity demonstrated—sixty years after its discovery—that Einstein's theory is consistent and stable. His proof of the Calabi conjecture allowed physicists—using Calabi–Yau compactification—to show that string theory is a viable candidate for a unified theory of nature. Calabi–Yau manifolds are among the ‘standard toolkit’ for string theorists today.
      Yau is one of the world's greatest mathematician in differential geometry nowadays,and he changed the whole world of geometry by developing the geometric analysis-using pde as a tool to solve geometric problems.He is one of the few people who won Fields Prize and Wolf Prize.His job is really great.What's more,he also did a lot of things for Chinese mathematics.Although some people have some different opinions.But I think he is a good mathematician.

Shiing-Shen Chern

     I have talked about several mathematicians I like in my blog,so next two blogs I will talk about several great mathematicians from China.They are really good examples for us to follow.And their experience and successful show we Chinese also have good talents for mathematics.The first Chinese mathematician I want to talk about is Shiing-Shen Chern.

      Shiing-Shen Chern (October 26, 1911 – December 3, 2004) was a Chinese-born American mathematician. He was regarded as one of the leaders in differential geometry of the twentieth century. Chern's work extends over all the classic fields of differential geometry. It includes areas currently fashionable (the Chern-Simons theory arising from a 1974 paper written jointly with Jim-Simons), perennial (the Chern-Weil theory linking curvature invariants to characteristic classes from 1944, after the Weil paper of 1943 on the Gauss-Bonnet theorem), the foundational (Chern classes), and some areas such as projective differential geometry. He published results in integral geometry,value distribution theory of holomophic functions and minimal submanifolds.

He was a follower of Cartan, working on the 'theory of  equivalence' in his time in China from 1937 to 1943, in relative isolation. In 1954 he published his own treatment of the pseudo group problem that is in effect the touchstone of Cartan's geometric theory. He used the moving frame method with success only matched by its inventor; he preferred in complex manifold theory to stay with the geometry, rather than follow the potential theory . Indeed, one of his books is entitled, "Complex Manifolds without Potential Theory". In the last years of his life, he advocated the study of Finsler Geometry, writing several books and articles on the subject.

       Chern is one of the leaders in differential geometry and he really did great jobs in this field.In fact he uses his idea and method to affect the whole world of geometry.He is not only a great mathematician,but also a great educator.He really did a lot of things for Chinese mathematics.He tried his best to select students who were good  at mathematics to send them abroad for further study and his students have become great mathematicians in all field of math.He is really great,and I really admire him.

Emmy Noether.

     Today I will talk something about a very famous female mathematician-Emmy Noether

     Amalie Emmy Noether ,sometimes referred to as Emily or Emmy, was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical phisics. Described by Pavel Alexanderov, Albert Einstein, Jean Dieudonne, Hermann Weil, Norbert Wiener and others as the most important woman in the history of mathematics,she revolutionized the theories of rings ,fields and algebra. In physics, Noether's theorem  explains the fundamental connection between symmetry and conservation laws.

     Noether's mathematical work has been divided into three "epochs".

     In the first (1908–1919), she made significant contributions to the theories of algebraic invariants and number fields Her work on differential invariants in the calculus of variations and Noether's theorem, has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics".

     In the second epoch, (1920–1926), she began work that "changed the face of [abstract] algebra". In her classic paper Idealtheorie in Ringbereichen (Theory of Ideals in Ring Domains, 1921) Noether developed the theory of ideals on commutative rings into a powerful tool with wide-ranging applications. She made elegant use of the ascanding chain in condition, and objects satisfying it are named Noetherian in her honor. 

     In the third epoch, (1927–1935), she published major works on noncommutative algebras and hypercomplex numbers and united representation theory of groups with the theory of modules and ideals. In addition to her own publications, Noether was generous with her ideas and is credited with several lines of research published by other mathematicians, even in fields far removed from her main work, such as algebraic topology.
     I first got to know Noether also in my second year in university.When I was learning abstract algebra,I learned a very special kind of ring,which satisfies the ascending condition,and this special kind of ring is named after Noether. From that time I began to know about this great female mathematician.She really made a lot of progress in abstract and theoretical physics as a female.She wasn't only a good researcher,but also taught a lot of great students.She is so great that I have great respect for her.

Michael Atiyah

     Today I want to write something about Michael Atiyah,who is also one of my favorite mathematicians.

     Michael Francis Atiyah is a British mathematician specialising in geometry. He grew up in Sudan and Egypt and spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study. His mathematical collaborators include Raoul Bott, Friedrich Hirsebruch and Isadore Singer. Together with Hirzebruch, he laid the foundations for topological K-theory, an important tool in algebraic topology, which, informally speaking, describes ways in which spaces can be twisted. His best known result, the Atiyah-Singer index theorem, was proved with Singer in 1963 and is widely used in counting the number of independent solutions to differential equations. Some of his more recent work was inspired by theoretical physics, in particular instantons and monopoles, which are responsible for some subtle corrections in quantum field theory. He was awarded the Fields Medal in 1966, the Copley Medal  in 1988, and the Abel Prize in 2004.

     I got to know Micheal Atiyah in my second year in university.At that time I was reading a book on commutative algebra written by Atiyah. His book is really a good book ,which is brief but really contains all the useful ideas and techniques in commutative algebra.From that time,I began to know this great mathematician.I heard a lot of  stories about him.He is really famous for his omnipotence in all branches in mathematics.He has a good point of view over the whole mathematics.He also made a prediction on the development of modern mathematics.

    This November he went to Nankai University in China.It's really a pity that I was not in China and could not go and see him.I hope I can see the real Atiyah someday~~~

Galois

    
 Évariste Galois was a French mathematician born in Borg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections. He was the first to use the word "group" (French: groupe) as a technical term in mathematics to represent a group of permutations.He died from wounds suffered in a duel under questionable circumstances at the age of twenty.
      When talking about Galois,the first impression is that this guy really has talents for mathematics.He used his unique way to change the whole world mathematics.The idea of group and his Galois theory really had a huge influence in mathematics.His theory made one of the most important branches of mathematics-abstract algebra have a big development.He was the first person to find the connections between the category of field and the category of group.What  he did is really of great use in modern mathematics.Unfortunately,this man died in a duel for a girl.His death is really a big pity for mathematics.

Grothendieck

Who is Grothendieck?

Answer:He is the god in algebraic geometry

Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra,sheaf theory, and category thei into its foundations. This new perspective led to revolutionary advances across many areas of pure mathematics.

Within algebraic geometry itself, his theory of schemes has become the universally accepted language for all further technical work. His generalization of the classical Riemann-Roch theorem launched the study of algebraic and topological K theory. His construction of new cohomology theories has left deep consequences for algebraic number theory,algebraic topology, and representation theory. 

One of his most celebrated achievements is the discovery of the first arithmetic Weil cohomology theory. This key result opened the way for a proof of the Weil conjectures, ultimately completed by his student Pierre. To this day, l-adic cohomology remains a fundamental tool for number theorists, with important applications to the Langlands programme.

Grothendieck’s way of thinking has influenced generations of mathematicians long after his departure from mathematics. His emphasis on the role of universal properties brought category into the mainstream as an important organizing principle. His notion of abelian category is now the basic object of study in homological algebra.

His work really has created a new page for algebraic geometry.That's why he is called the god~~

http://en.wikipedia.org/wiki/Alexander_Grothendieck