Thursday, October 10, 2013 3:30 pm
-
4:30 pm
EDT (GMT -04:00)
A Proof of the Manickam-Mikl\'{o}s-Singhi Conjecture for Vector Spaces
Speaker: | Ameera Chowdhury |
---|---|
Affiliation: | Carnegie Melon University |
Room: | Mathematics and Computer Building (MC) 5158 |
Abstract:
Let
$V$
be
an
$n$-dimensional
vector
space
over
a
finite
field.
Assign
a
real-valued
weight
to
each
$1$-dimensional
subspace
in
$V$
so
that
the
sum
of
all
weights
is
zero.
Define
the
weight
of
a
subspace
$S
\subset
V$
to
be
the
sum
of
the
weights
of
all
the
$1$-dimensional
subspaces
it
contains.
We
prove
that
if
$n
\geq
3k$,
then
the
number
of
$k$-dimensional
subspaces
in
$V$
with
nonnegative
weight
is
at
least
the
number
of
$k$-dimensional
subspaces
in
$V$
that
contain
a
fixed
$1$-dimensional
subspace.
This
result
verifies
a
conjecture
of
Manickam
and
Singhi
from
1988.