These are topics in mathematics
at the current cutting edge of superstring research.

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.

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.
The current interest in NCG among physicists
of the 21st century has been stimulated by work by French mathematician
Alain Connes.