Talk #1 Valentio Iverson: Powerfree Sieve on Prime Inputs
Abstract: In recent years, there have been several results on the density of integer tuples (a_1,...,a_n) such that f(a_1,...,a_n), for some fixed integer polynomial f, is squarefree. In this talk, we restrict to the case of prime inputs and prove similar results.
Talk #2 Gian Sanjaya: Density of Restricted Polynomials with Squarefree Discriminant
Abstract:
In
2007,
Ash,
Brakenhoff,
and
Zarrabi
conjectured
the
density
of
general
monic
integer
polynomials
of
fixed
degree
with
squarefree
discriminant.
They
also
conjectured
the
density
of
monic
irreducible
integer
polynomials
f
of
fixed
degree
such
that
Z[X]/(f(X))
is
integrally
closed.
These
conjectures
have
been
proven
in
2022
by
Bhargava,
Shankar,
and
Wang.
In
this
talk,
we
explore
these
problems
for
two
classes
of
polynomials:
1.
monic
integer
polynomials
of
fixed
degree
with
fixed
first
sub-coefficient,
and
2.
monic
integer
polynomials
of
fixed
degree
with
first
and
second
sub-coefficient
equal
to
zero.
We
give
a
conjecture
on
the
two
densities
and
describe
how
to
compute
these
densities
via
counting
over
function
fields.
It
is
to
be
noted
that
the
degree
4
case
for
the
second
class
has
been
solved
in
a
previous
joint
work
with
Xiaoheng
Wang.
Talk #3 Steven Piatkowski: Representation Theory and Brauer's Theorem
Abstract: We will provide an introduction to representation theory of finite groups with emphasis on character theory of representations. Our connection to number theory will be Brauer’s theorem. Every character of a group G is a linear combination with integer coefficients of characters induced from characters of elementary subgroups. Which we re-phrase to be each character is a linear combination with integers coefficients of monomial characters.
MC 5417