<s>
Noncommutative	B-Algorithm
algebraic	I-Algorithm
geometry	I-Algorithm
is	O
a	O
branch	O
of	O
mathematics	O
,	O
and	O
more	O
specifically	O
a	O
direction	O
in	O
noncommutative	O
geometry	O
,	O
that	O
studies	O
the	O
geometric	O
properties	O
of	O
formal	O
duals	O
of	O
non-commutative	O
algebraic	O
objects	O
such	O
as	O
rings	O
as	O
well	O
as	O
geometric	O
objects	O
derived	O
from	O
them	O
(	O
e.g.	O
</s>
<s>
For	O
example	O
,	O
noncommutative	B-Algorithm
algebraic	I-Algorithm
geometry	I-Algorithm
is	O
supposed	O
to	O
extend	O
a	O
notion	O
of	O
an	O
algebraic	O
scheme	O
by	O
suitable	O
gluing	O
of	O
spectra	O
of	O
noncommutative	O
rings	O
;	O
depending	O
on	O
how	O
literally	O
and	O
how	O
generally	O
this	O
aim	O
(	O
and	O
a	O
notion	O
of	O
spectrum	O
)	O
is	O
understood	O
in	O
noncommutative	O
setting	O
,	O
this	O
has	O
been	O
achieved	O
in	O
various	O
level	O
of	O
success	O
.	O
</s>
<s>
Much	O
of	O
the	O
motivation	O
for	O
noncommutative	O
geometry	O
,	O
and	O
in	O
particular	O
for	O
the	O
noncommutative	B-Algorithm
algebraic	I-Algorithm
geometry	I-Algorithm
,	O
is	O
from	O
physics	O
;	O
especially	O
from	O
quantum	O
physics	O
,	O
where	O
the	O
algebras	O
of	O
observables	O
are	O
indeed	O
viewed	O
as	O
noncommutative	O
analogues	O
of	O
functions	O
,	O
hence	O
having	O
the	O
ability	O
to	O
observe	O
their	O
geometric	O
aspects	O
is	O
desirable	O
.	O
</s>
<s>
The	O
methods	O
of	O
noncommutative	B-Algorithm
algebraic	I-Algorithm
geometry	I-Algorithm
are	O
analogs	O
of	O
the	O
methods	O
of	O
commutative	O
algebraic	O
geometry	O
,	O
but	O
frequently	O
the	O
foundations	O
are	O
different	O
.	O
</s>
<s>
There	O
are	O
,	O
a	O
bit	O
weaker	O
,	O
reconstruction	O
theorems	O
from	O
the	O
derived	O
categories	O
of	O
(	O
quasi	O
)	O
coherent	O
sheaves	O
motivating	O
the	O
derived	B-Algorithm
noncommutative	I-Algorithm
algebraic	I-Algorithm
geometry	I-Algorithm
(	O
see	O
just	O
below	O
)	O
.	O
</s>
