Title: The Hepp bound of a matroid: flags, volumes and integrals
| Speaker: | Erik Panzer |
| Affiliation: | University of Oxford |
| Zoom: | Contact Rose McCarty |
Abstract:
Invariants
of
combinatorial
structures
can
be
very
useful
tools
that
capture
some
specific
characteristics,
and
repackage
them
in
a
meaningful
way.
For
example,
the
famous
Tutte
polynomial
of
a
matroid
or
graph
tracks
the
rank
statistics
of
its
submatroids,
which
has
many
applications,
and
relations
like
contraction-deletion
establish
a
very
close
connection
between
the
algebraic
structure
of
the
invariant
(e.g.
Tutte
polynomials)
and
the
actual
matroid
itself.
I
will
present
an
invariant,
called
the
Hepp
bound,
that
associates
to
a
matroid
a
rational
function
in
many
variables
(one
variable
for
each
element
of
the
matroid).
This
invariant
behaves
nicely
with
respect
to
duality
and
2-sums,
and
the
residues
at
its
poles
factorize
into
the
Hepp
bounds
of
sub-
and
quotient
matroids.
It
can
be
specialized
to
Crapo's
beta
invariant
and
it
is
also
related
to
Derksen's
invariant.
The
construction
is
motivated
by
the
tropicalization
of
Feynman
integrals
from
the
quantum
field
theory
of
elementary
particles
physics.
In
the
case
of
graphs,
the
Hepp
bound
therefore
obeys
further
interesting
relations
that
are
known
for
Feynman
integrals.
Due
to
this
rich
structure,
the
Hepp
bound
can
be
viewed
from
several
distinct
perspectives,
each
making
certain
properties
emerge
more
directly
than
others.
I
will
sketch
3
definitions:
1)
enumerative
-
as
a
certain
sum
over
flags
of
submatroids,
2)
analytic
-
as
an
integral,
3)
geometric
-
as
a
volume
of
a
polytope.