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Tom1992

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http://superstringtheory.com/math/index.html

Real analysis

In real analysis, students learn abstract properties of real functions as mappings, isomorphism, fixed points, and basic topology such as sets, neighborhoods, invariants and homeomorphisms.

Complex analysis

Complex analysis is an important foundation for learning string theory. Functions of a complex variable, complex manifolds, holomorphic functions, harmonic forms, Kähler manifolds, Riemann surfaces and Teichmuller spaces are topics one needs to become familiar with in order to study string theory.

Group theory

Modern particle physics could not have progressed without an understanding of symmetries and group transformations. Group theory usually begins with the group of permutations on N objects, and other finite groups. Concepts such as representations, irreducibility, classes and characters.

Differential geometry

Einstein's General Theory of Relativity turned non-Euclidean geometry from a controversial advance in mathematics into a component of graduate physics education. Differential geometry begins with the study of differentiable manifolds, coordinate systems, vectors and tensors. Students should learn about metrics and covariant derivatives, and how to calculate curvature in coordinate and non-coordinate bases.

Lie groups

A Lie group is a group defined as a set of mappings on a differentiable manifold. Lie groups have been especially important in modern physics. The study of Lie groups combines techniques from group theory and basic differential geometry to develop the concepts of Lie derivatives, Killing vectors, Lie algebras and matrix representations.

Differential forms

The mathematics of differential forms, developed by Elie Cartan at the beginning of the 20th century, has been powerful technology for understanding Hamiltonian dynamics, relativity and gauge field theory. Students begin with antisymmetric tensors, then develop the concepts of exterior product, exterior derivative, orientability, volume elements, and integrability conditions.

Homology

Homology concerns regions and boundaries of spaces. For example, the boundary of a two-dimensional circular disk is a one-dimensional circle. But a one-dimensional circle has no edges, and hence no boundary. In homology this case is generalized to "The boundary of a boundary is zero." Students learn about simplexes, complexes, chains, and homology groups.

Cohomology

Cohomology and homology are related, as one might suspect from the names. Cohomology is the study of the relationship between closed and exact differential forms defined on some manifold M. Students explore the generalization of Stokes' theorem, de Rham cohomology, the de Rahm complex, de Rahm's theorem and cohomology groups.

Homotopy

Lightly speaking, homotopy is the study of the hole in the donut. Homotopy is important in string theory because closed strings can wind around donut holes and get stuck, with physical consequences. Students learn about paths and loops, homotopic maps of loops, contractibility, the fundamental group, higher homotopy groups, and the Bott periodicity theorem.

Fiber bundles

Fiber bundles comprise an area of mathematics that studies spaces defined on other spaces through the use of a projection map of some kind. For example, in electromagnetism there is a U(1) vector potential associated with every point of the spacetime manifold. Therefore one could study electromagnetism abstractly as a U(1) fiber bundle over some spacetime manifold M. Concepts developed include tangent bundles, principal bundles, Hopf maps, covariant derivatives, curvature, and the connection to gauge field theories in physics.

Characteristic classes

The subject of characteristic classes applies cohomology to fiber bundles to understand the barriers to untwisting a fiber bundle into what is known as a trivial bundle. This is useful because it can reduce complex physical problems to math problems that are already solved. The Chern class is particularly relevant to string theory.

Index theorems

In physics we are often interested in knowing about the space of zero eigenvalues of a differential operator. The index of such an operator is related to the dimension of that space of zero eigenvalues. The subject of index theorems and characteristic classes is concerned with

Supersymmetry and supergravity

The mathematics behind supersymmetry starts with two concepts: graded Lie algebras, and Grassmann numbers. A graded algebra is one that uses both commutation and anti-commutation relations. Grassmann numbers are anti-commuting numbers, so that x times y = –y times x. The mathematical technology needed to work in supersymmetry includes an understanding of graded Lie algebras, spinors in arbitrary spacetime dimensions, covariant derivatives of spinors, torsion, Killing spinors, and Grassmann multiplication, derivation and integration, and Kähler potentials.

K-theory

Cohomology is a powerful mathematical technology for classifying differential forms. In the 1960s, work by Sir Michael Atiyah, Isadore Singer, Alexandre Grothendieck, and Friedrich Hirzebruch generalized coholomogy from differential forms to vector bundles, a subject that is now known as K-theory.

Witten has argued that K-theory is relevant to string theory for classifying D-brane charges. D-brane objects in string theory carry a type of charge called Ramond-Ramond charge. Ramond-Ramond fields are differential forms, and their charges should be classifed by ordinary cohomology. But gauge fields propagate on D-branes, and gauge fields give rise to vector bundles. This suggests that D-brane charge classification requires a generalization of cohomology to vector bundles -- hence K-theory.

Noncommutative geometry (NCG for short)

Geometry was originally developed to describe physical space that we can see and measure. After modern mathematics was freed from Euclid's Fifth Axiom by Gauss and Bolyai, Riemann added to modern geometry the abstract notion of a manifold M with points that are labeled by local coordinates that are real numbers, with some metric tensor that determines an extremal length between two points on the manifold.

Much of the progress in 20th century physics was in applying this modern notion of geometry to spacetime, or to quantum gauge field theory.

In the quest to develop a notion of quantum geometry, as far back as 1947, people were trying to quantize spacetime so that the coordinates would not be ordinary real numbers, but somehow elevated to quantum operators obeying some nontrivial quantum commutation relations. Hence the term "noncommutative geometry," or NCG for short.

I was wondering if this list is the same for loop quantum gravity. If not, what can be removed and what should be added?