One of the most important objects in analysis for
algebraic topologists is
C
^{
*
}
algebras
. For example, they play an important role in
K
theory
. Algebraic analysis has a lot in common with
algebraic topology in the sense that we use
homological algebra
as a fundamental tool.
On the other hand, probability seems to be far from
algebraic topology. According to
this blog post by Tao
, algebra and geometry are suitable for handling
structured objects, while analysis and probability are
suitable for studying pseudorandom objects. Of course
there are intermediate objects (called hybrid objects
by Tao) and we need to use all possible tools to study
such objects.
Are
the homotopy groups of spheres
hybrid objects? There are interesting attempts by
Tsui and Wang
[
TW
]
and Guth
[
Gut
]
in which they studied the homotopy groups of spheres
by analytic tools.
Some people are studying applications of topology to
PDE
[
Pra
]
.
