**
Jacob
Campbell,
Department
of
Pure
Mathematics,
University
of
Waterloo**

"Finite free convolutions"

In
a
remarkable
series
of
papers
from
2013-15,
Marcus,
Spielman,
and
Srivastava
solved
some
important
problems
in
graph
theory,
as
well
as
the
Kadison-Singer
problem
in
operator
algebras.
A
central
role
is
played
by
expected
characteristic
polynomials
of
sums
and
products
of
randomly
rotated
matrices.

These
random
matrices
are
perennial
objects
of
interest
in
free
probability,
since
they
asymptotically
approximate
free
convolution.
It
turns
out
that
looking
at
their
expected
characteristic
polynomials,
non-asymptotically,
yields
well-defined
"finite
free
convolution"
operations
on
polynomials,
and
in
turn
the
beginnings
of
a
"finite
free
probability"
theory.
One
particular
feature,
which
forms
the
connection
with
graph
theory
but
is
interesting
in
its
own
right,
is
what
one
might
call
a
"quadrature"
phenomenon:
the
continuous
groups
of
rotations
can
be
replaced
with
certain
finite
subgroups
of
reflections
without
changing
the
convolutions.

In
joint
work
with
Zhi
Yin,
we
approach
these
finite
free
convolutions
and
quadrature
results
using
techniques
from
combinatorial
representation
theory,
namely
Weingarten
calculus
and
Schur/zonal
functions.
I
will
explain
our
approach,
without
assuming
any
particular
knowledge
of
random
matrices
or
combinatorics.
Time
permitting,
I
will
mention
some
of
the
ways
these
ideas
parallel
well-established
ones
in
free
probability.

Zoom Meeting: https://us02web.zoom.us/j/87274747278?pwd=RG1Bak5lbk1GaHdIL0dtSzlBbjdiUT09