Julia Brandes, Department of Pure Mathematics, University of Waterloo
"Optimal mean value estimates beyond Vinogradov's mean value theorem"
Mean
values
for
exponential
sums
play
a
central
role
in
the
study
of
diophantine
equations.
In
particular,
strong
upper
bounds
for such
mean
values
control
the
number
of
integer
solutions
of
the corresponding
systems
of
diagonal
equations.
Since
the groundbreaking
resolution
of
Vinogradov's
mean
value
theorem
by Wooley
and
Bourgain,
Demeter
and
Guth,
we
can
now
prove
optimal upper
bounds
for
mean
values
connected
to
translation-dilation-invariant
systems.
This
has
inspired
Wooley's call
for
a
"Big
Theory
of
Everything",
a
challenge
to
establish optimal
mean
value
estimates
for
any
mean
values
associated
with systems
of
diagonal
equations.
We
establish
optimal
bounds
for
a
family
of
mean
values
that
are not
of
Vinogradov
type.
This
is
the
first
time
bounds
of
this quality
have
been
obtained
for
non-translation-dilation-invariant systems.
As
a
consequence,
we
establish
the
analytic
Hasse principle
for
the
number
of
solutions
of
certain
systems
of quadratic
and
cubic
equations
in
fewer
variables
than
hitherto thought
necessary.
This
is
joint
work
with
Trevor
Wooley.
MC 5501