What kinds of math do string theorists use and why? And how has string theory changed mathematics?

Answer 1

The language of physics is mathematics. In order to study physics seriously, one needs to learn mathematics that took generations of brilliant people centuries to work out. Algebra, for example, was cutting-edge mathematics when it was being developed in Baghdad in the 9th century. But today it's just the first step along the journey.

Algebra

Algebra provides the first exposure to the use of variables and constants, and experience manipulating and solving linear equations of the form y = ax + b and quadratic equations of the form y = ax2+bx+c.
Geometry

Geometry at this level is two-dimensional Euclidean geometry, Courses focus on learning to reason geometrically, to use concepts like symmetry, similarity and congruence, to understand the properties of geometric shapes in a flat, two-dimensional space.

Trigonometry

Trigonometry begins with the study of right triangles and the Pythagorean theorem. The trigonometric functions sin, cos, tan and their inverses are introduced and clever identities between them are explored.

Calculus (single variable)

Calculus begins with the definition of an abstract functions of a single variable, and introduces the ordinary derivative of that function as the tangent to that curve at a given point along the curve. Integration is derived from looking at the area under a curve,which is then shown to be the inverse of differentiation.

Calculus (multivariable)

Multivariable calculus introduces functions of several variables f(x,y,z...), and students learn to take partial and total derivatives. The ideas of directional derivative, integration along a path and integration over a surface are developed in two and three dimensional Euclidean space.

Analytic Geometry

Analytic geometry is the marriage of algebra with geometry. Geometric objects such as conic sections, planes and spheres are studied by the means of algebraic equations. Vectors in Cartesian, polar and spherical coordinates are introduced.

Linear Algebra

In linear algebra, students learn to solve systems of linear equations of the form ai1 x1 + ai2 x2 + ... + ain xn = ci and express them in terms of matrices and vectors. The properties of abstract matrices, such as inverse, determinant, characteristic equation, and of certain types of matrices, such as symmetric, antisymmetric, unitary or Hermitian, are explored.

Ordinary Differential Equations

This is where the physics begins! Much of physics is about deriving and solving differential equations. The most important differential equation to learn, and the one most studied in undergraduate physics, is the harmonic oscillator equation, ax'' + bx' + cx = f(t), where x' means the time derivative of x(t).

Partial Differential Equations

For doing physics in more than one dimension, it becomes necessary to use partial derivatives and hence partial differential equations. The first partial differential equations students learn are the linear, separable ones that were derived and solved in the 18th and 19th centuries by people like Laplace, Green, Fourier, Legendre, and Bessel.

Methods of approximation

Most of the problems in physics can't be solved exactly in closed form. Therefore we have to learn technology for making clever approximations, such as power series expansions, saddle point integration, and small (or large) perturbations.

Probability and statistics

Probability became of major importance in physics when quantum mechanics entered the scene. A course on probability begins by studying coin flips, and the counting of distinguishable vs. indistinguishable objects. The concepts of mean and variance are developed and applied in the cases of Poisson and Gaussian statistics.

Answer 2

non-Euclidean geometry (especially Minkowskian), Riemann surfaces, complex and hypercomplex numbers, n-dimensional manifolds, symmetry groups, number theory (for quantum aspects), and of course calculus and tensors.

As is often the case, most of the math was developed by mathematicians before anyone figured out there was a need for it.

String theory raised the bar for the degree of mathematical knowledge and skill needed to practice theoretical physics. See the reference to get an idea of how overwhelming it really is.