

Here are some of the topics
in mathematics that a person who wants to learn advanced topics in theoretical
physics, especially string theory, should become familiar with. 
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 nonEuclidean 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 noncoordinate
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 twodimensional circular disk
is a onedimensional circle. But a onedimensional 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 anticommutation
relations. Grassmann numbers are anticommuting 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. 

