||Ａ： Mathematics subjects currently studied at university mathematics department
are roughly classified as algebra, geometry, and analysis.
Algebra is an advanced form of polynomial calculation learned in high school mathematics.
Number theory includes studies of the properties of integers, e.g. what
kind of distribution do prime numbers take on a number line?
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,
71, 73, 79, 83, 89, 97, ・・・
A set of all the zeros of some polynomials such as the following equation
is designated as an algebraic variety.
y2 = x3 + a x + b (where a and b are constant)
Algebraic geometry is the study of geometric properties of an algebraic variety. A set with
a well-behaved product, such as all the maps from a cube to a cube, or
all the 2 × 2 invertible matrices, is designated as a group. The group
theory, which studies various properties of a group, is also an important
field of algebra.
Geometry is a discipline including the study of properties of figures and shapes.
It includes topology, in which one studies invariant properties of a figure
in continuous deformation, and differential geometry, in which one studies
invariant properties of a figure in smooth deformation
No two-dimensional, closed, and curved surface lacks a boundary,
other than a spherical surface and a curved surface some holes like a ring buoy.
That is one of the simplest results of topology.
A bubble forms a curved surface called a minimal surface.
Its study is an important theme of differential geometry.
Analysis is an advanced form of calculus studied in high school mathematics. For
example, using Euler equation: e^iz = cos z+i sin z, exponential function e^z can be extended to a complex valued function
of a complex variable. It demonstrates that the exponential function and
trigonometric functions are functions of the same kind. Analysis includes
function theory, in which one studies properties of such functions of complex
variables. In Fourier analysis, one studies properties of periodic functions,
expressing them as sums of sine and cosine functions. In the theory of
differential equations, one studies solutions of differential equation
such as y"+ a(x) y'+ b(x) = 0. In functional analysis, one studies properties of infinite dimensional
vector spaces such as the space C (R) all the continuous functions on the
real axis. Probability theory is used for studies of regular and complicated
matter like a fractal. For example, a differential equation describing
the properties of fluids has a well-behaved solution. This is a prevailing
research theme of analysis.
Other subjects of mathematics include foundations of mathematics in which one studies properties of mathematical logics and axioms.