the role of Chern Simons theory in solving the fractional quantum Hall effect mystery

http://www.nim.nankai.edu.cn/activites/conferences/dgmtp/paper/Jainendra%20Jain/chern.pdf

I found this website , I think is from Penn State and NSF. it a project that describe the Chern Simons theory's role.

Chern-Simons theory:prediction of composite fermion, a new class of topological fermions in physics. what is special about them? is the question in the outline.

The Chern-Simons theory unifies the fractional quantum Hall with the well understood integral quantum Hall effect, in the process leading to the prediction and the subsequent discovery of a new class of topological fermions in nature.

chern classes

11/21/08

http://arxiv.org/abs/math.AG/0512486

it write by Constantin Teleman and Christopher T. Woodward. submitted on 21 Dec 2005

this website is describe the chern classes and The Newstead-Ramanan conjecture for Chern classes.

The Chern classes of a complex manifold are the Chern classes of its tangent bundle. The th Chern class is anobstruction to the existence of  everywhere complex linearly independent vector fields on that vector bundle. The th Chern class is in the th cohomology group of the base space.in topology, differential geometry, and algebraic geometry, it is often of vital importance to know how many linearly independent sections a vector bundle has. The Chern classes can be used to provide a great deal of information in answering this question. 

Sixth Annotation: Chern-Simons Theory and Topological Strings

Chern-Simons Theory and Topological Strings

http://arxiv.org/abs/hep-th/0406005

wrote by Marcos Marino sumitted on 1 Jun 2004, last revised 9 may 2005.

It says that they review the relation between Chern-Simons gauge theory and topological string theory. 

This relation has made possible to give an exact solution of string theory on these spaces to all orders in the string coupling

constant.

also,I found up a book name Chern-Simon theory, Matrix Models, and Topological strings.

According the overview is In recent years, the old idea that gauge theories and string theories are equivalent has been implemented and developed in various ways, and there are by now various models where the string theory / gauge theory correspondence is at work. One of the most important examples of this correspondence relates Chern-Simons theory, a topological gauge theory in three dimensions which describes knot and three-manifold invariants, to topological string theory, which is deeply related to Gromov-Witten invariants.

New Thesis Statement and Introductions

     Chern is one of the leaders in differential geometry of the twentieth century. he use his math to define a lot theory that we don't know. he is a chinese but why he become a famous math- ematicians because of the string theory. Chern's work spreads over all the classic fields of differential geometry. It includes areas currently fashionable, perennial , the foundational, and some areas such as projective differential geometry and webs that have a lower profile. He published results in integral geometry, value distribution theory of holomorphic functions, and minimal submanifolds.

        chern use his math to show the scientist that we have this theory, but the problem is we still can not make it become true, we only can know string theory is to be a theory of quantum gravity, then the average size of a string should be somewhere near the length scale of quantum gravity, called the Planck length, which is about 10^-33 centimeters, or about a millionth of a billionth of a billionth of a billionth of a centimeter. 

       A important proof was the forerunner of other invariants which bear his name, Chern classes, Chern-Weil homomorphism and Chern-Simons invariants, which have become essential tools not only in differential geometry but in other areas of mathematics such as topology and algebraic geometry and also mathematical physics. A large part of modern algebraic geometry would not exist without Chern classes. 

His area of research was differential geometry where he studied the Chern characteristic classes in fibre spaces. These are important not only in mathematics but also in mathematical physics. He worked on characteristic classes during his 1943-45 visit to Princeton and, also at this time, he gave a now famous proof of theGauss-Bonnet formula.

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Introduction

      every scientist must is a mathematician. mathematician can measure space and maybe anything. chern is the famous Chinese American mathematician, he use his math to define the string theory(a very important theory), if  you can really understand it and use it. maybe we can pass the time go back 100 years ago, you can think about how special mathematician can do. also, he define and improve the differential geometry.