Title: Sequences of Trees and Higher-Order Renormalization Group Equations
| Speaker: | William Dugan |
| Affiliation: | University of Waterloo |
| Room: | MC 6483 |
Abstract:
In
1998,
Connes
and
Kreimer
introduced
a
combinatorial
Hopf
algebra HCK
on
the
vector
space
of
forests
of
rooted
trees
that
precisely
explains
the
phenomenon
of
renormalization
in
quantum
eld
theory.
This
Hopf
algebra
has
been
of
great
interest
since
its
inception,
as
it
connects
the
disciplines
of
algebra,
combinatorics,
and
physics,
providing
interesting
questions
in
each.
In
this
thesis
we
introduce
the
notion
of
higher-order
renormalization
group
equations,
which
generalize
the
usual
renormalization
group
equation
of
quantum
eld
theory,
and
further
dene
a
corresponding
notion
of
order
on
certain
sequences
of
trees
constituting
elements
of
the
completion
of
HCK.
We
also
give
an
explication
of
a
result,
due
to
Foissy,
that
characterizes
which
sequences
of
linear
combinations
of
trees
with
one
generator
in
each
degree
generate
Hopf
subalgebras
of
HCK.
We
conclude
with
some
results
towards
classifying
these
sequences
by
their
order
(when
such
an
order
is
admitted),
and
by
presenting
a
new
family
of
second-order
sequences
of
which
the
sequence
of
generators
of
the
Connes-Moscovici
subalgebra
is
a
member.