In
the
spring
semester
of
2014,
I
conducted
research
in
the
field
of
noncommutative
algebra
and
algebraic
geometry
under
the
supervision
of
Professor
Jason
Bell.
To
describe
my
research,
I
will
begin
by
discussing
related
questions
arising
in
geometry
and
then
show
how
these
questions
provide
motivation
for
the
problem
I
ended
up
solving.
This
solution
ended
up
being
part
of
a
paper,
written
with
Jason
Bell
and
Colin
Ingalls,
which
has
been
submitted
to
the
Bulletin
of
the
London
Math
Society.
An
algebraic
variety
X
is
the
solution
set
of
a
system
of
polynomial
equations
in
n
variables.
An
algebraic
variety
X
is
said
to
be
projective
if
it
a
closed
subset
of
a
projective
space.
Given
two
projective
algebraic
varieties
X
and
Y
over
an
algebraically
closed
field
k,
a
fundamental
question
in
algebraic
geometry
is
to
determine
whether
or
not
they
are
birationally
equivalent
(this
is
a
useful
notion
of
being
“the
same”
for
varieties).
This
birational
classification
was
completely
achieved
in
the
case
of
surfaces
and
can
be
described
in
terms
of
a
short
list
of
invariants.
But
to
answer
these
types
of
questions
in
general,
mathematicians
have
created
many
“birational
invariants”
that
control
the
maps
between
X
and
Y.
A
famous
result
in
this
direction
is
the
Riemann-Hurwitz
theorem,
which
implies
that
if
X
and
Y
are
smooth
irreducible
projective
complex
curves
and
there
is
a
surjective
morphism
from
X
to
Y
then
the
genus
of
X
is
greater
than
or
equal
to
the
genus
of
Y.
In
algebraic
geometry,
there
is
a
dictionary
between
geometric
concepts
and
algebraic
concepts
and
this
dictionary
allows
one
to
translate
the
notion
of
birational
isomorphism
of
varieties
to
isomorphisms
of
their
function
fields
and
in
particular
it
makes
sense
to
talk
about
invariants
for
fields.
As
an
example,
the
Riemann-Hurwitz
theorem
can
be
formulated
by
saying
that
if
X
and
Y
are
smooth
irreducible
projective
complex
curves
and
the
function
field
of
Y
embeds
in
the
function
field
of
X
then
the
genus
of
X
is
greater
than
or
equal
to
the
genus
of
Y.
Taking
this
field-theoretic
point
of
view,
it
is
often
possible
to
find
noncommutative
analogues
of
invariants
that
extend
geometric
invariants
to
the
setting
of
division
rings.
This
is
the
starting
point
for
noncommutative
algebraic
geometry.
To
further
understand
noncommutative
geometry
and
deepen
the
analogy
with
classical
geometry,
Artin
gives
a
conjectured
classification
of
“noncommutative
surfaces”
akin
to
the
birational
classification
of
surfaces.
With
Artin’s
conjecture
as
motivation,
we
proved
a
noncommutative
analogue
of
the
Riemann-Hurwitz
theorem
mentioned
above:
Let
X
and
Y
be
smooth
irreducible
algebraic
curves
over
the
complex
numbers,
and
let
D
(X)
and
D
(Y)
be
the
respective
quotient
division
rings
of
the
rings
of
differential
operators
of
X
and
Y.
If
there
is
a
C-algebra
embedding
of
D
(X)
into
D
(Y)
then
the
genus
of
X
is
less
than
or
equal
to
the
genus
of
Y.
Here
the
quotient
division
ring
of
the
ring
of
differential
operators
on
X
plays
the
noncommutative
analogue
of
the
function
field
of
a
surface.
To
prove
this,
one
first
shows
that
ring
of
differential
operators
on
X
can
be
described
in
terms
of
the
ordinary
function
field
of
X
along
with
some
derivation
of
the
function
field.
The
quotient
ring
D
(X)
is
then
obtained
by
inverting
the
non-zero
elements.
If
one
chooses
a
trivial
derivation,
our
statement
is
just
the
Riemann-Hurwitz
theorem,
and
this
justifies
calling
our
result
a
“noncommutative”
Riemann-Hurwitz
theorem.
Unfortunately,
understanding
the
quotient
division
ring
D
(X)
of
a
curve
X
is
not
easy
as
understanding
its
function
field,
as
it
is
a
noncommutative
localization,
which,
in
general,
is
horribly
behaved.
To
get
around
this,
we
reduced
to
working
with
models
of
our
curves
over
finitely
generated
rings
and
then
showed
that
one
could
reduce
modulo
various
prime
ideals
of
our
finitely
generated
rings
while
still
preserving
the
relevant
geometric
information.
Finally,
we
showed
that
modulo
these
primes
one
could
reduce
to
the
commutative
case,
albeit
in
positive
characteristic.
After
reducing
to
a
commutative
problem,
we
used
known
results
from
algebraic
geometry
to
obtain
our
genera
inequality.
It
is
worth
noting
that
this
result
does
not
directly
prove
the
conjecture
of
Artin.
Much
more
robust
invariants
are
needed
to
accomplish
this.
The
first
half
of
my
research
term
was
spent
trying
to
find
a
noncommutative
analogue
of
the
genus
for
the
ring
of
differential
operators.
This
noncommutative
genus,
if
it
exists,
should
equal
the
classical
genus
when
the
derivation
is
trivial.
We
obtained
some
partial
characterizations,
but
it
didn’t
behave
as
well
as
we
would
have
liked.
However,
I
was
able
to
fix
enough
of
these
problems
to
obtain
our
main
result.
I
ultimately
found
a
suitable
noncommutative
invariant
by
reducing
modulo
primes
and
then
borrowing
commutative
invariants,
and
this
was
sufficient
to
get
our
theorem.
Our
result
can
be
regarded
as
the
first
step
in
the
search
for
a
noncommutative
genus,
which,
one
hopes,
will
eventually
lead
to
a
complete
list
of
noncommutative
invariants
necessary
to
obtain
a
proof
of
Artin’s
conjecture.