Date 
Speaker 
Title 

Jan 11 
everyone 
Open problem session 
at noon
in UHall C630


Please bring your
favourite (math) problems. Anyone with a problem to share will be
given about 5 minutes to present it. We will also choose most of
the speakers for the rest of the semester.


Jan 25 
Francesco
Pappalardi

On never primitive points on elliptic curves 
at noon
in UHall C630

(Universitŕ Roma Tre) 
The LangTrotter
Conjecture for primitive points predicts an expression for the
density of primes $p$ for which a fixed rational point (not
torsion) of a fixed elliptic curve defined on $\mathbb{Q}$ is a
generator of the curve reduced modulo $p$. After providing the
definition of such a density in terms of Galois representations
associated with torsion points of the curve, we will tell the
short story of the contributions to the conjecture and provide
examples of families of elliptic curves for which the conjecture
holds for trivial reasons. This is the notion of "never primitive
point." The case of elliptic curves in complex multiplication will
be discussed in greater detail. Part of the work is in
collaboration of N. Jones.


Jan 27 
Francesco
Pappalardi

The distribution of multiplicatively dependent vectors 
Wednesday
at 10am
in in UHall C630

(Universitŕ Roma Tre) 
Let $n$ be a positive
integer, $G$ be a group and let $\mathbf{\nu}=(\nu_1,\dots,\nu_n)$
be in $G^n.$ We say that $\mathbf{\nu}$ is a multiplicatively
dependent $n$tuple if there is a nonzero vector
$(k_1,\dots,k_n)$ in $\mathbb{Z}^n$ for which $\nu^{k_1}_1\cdots
\nu^{k_n}_n=1.$
Given a finite extension $K$ of $\mathbb Q$, we denote by
$M_{n,K}(H)$ the number of multiplicatively dependent $n$tuples
of algebraic integers of $K^*$ of naive height at most $H$ and
we denote by $M^*_{n,K}(H)$ the number of multiplicatively
dependent $n$tuples of algebraic numbers of $K^*$ of height at
most $H.$ In this seminar we discuss several estimates and
asymptotic formulas for $M_{n,K}(H)$ and for $M^*_{n,K}(H)$ as
$H\rightarrow\infty$.
For each $\nu$ in $(K^*)^n$ we define $m,$ the multiplicative
rank of $\nu,$ in the following way. If $\nu$ has a
coordinate which is a root of unity we put $m=1.$ Otherwise let
$m$ be the largest integer with $2\leq m\leq n+1$ for which
every set of $m1$ of the coordinates of $\nu$ is a
multiplicatively independent set. We also consider the sets
$M_{n,K,m}(H)$ and $M^*_{n,K,m}(H)$ defined as the number of
multiplicatively dependent $n$tuples of multiplicative rank $m$
whose coordinates are algebraic integers from $K^*,$
respectively algebraic numbers from $K^*,$ of naive height at
most $H$ and will consider similar questions for them.


Feb 1 
Micah
Milinovich

Fourier Analysis and the zeros of the Riemann zetafunction

at noon
in UHall C630

(University of Mississippi) 
I will show how the
classical BeurlingSelberg extremal problem in harmonic analysis
arises naturally when studying the vertical distribution of the
zeros of the Riemann zetafunction and other Lfunctions. Using
this relationship, along with techniques from Fourier analysis and
reproducing kernel Hilbert spaces, we can prove the sharpest known
bounds for the number of zeros in an interval on the critical line
and we can also study the pair correlation of zeros. Our results
on pair correlation extend earlier work of
P. X. Gallagher and give some evidence for the
wellknown conjecture of H. L. Montgomery. This talk is
based on a series of papers which are joint with E. Carneiro,
V. Chandee, and F. Littmann.


Feb 8 
Alexey Popov 
Operator Algebras with reduction properties 
at noon
in UHall C630


An algebra is a vector
space with a welldefined multiplication. An operator algebra is
an algebra of operators acting on a Hilbert space, typically
assumed closed in the norm topology. An easy example of an
operator algebra is the algebra $M_n(\mathbb{C})$ of all the
complex $n \times n$ matrices. In this colloquiumstyle talk, we
will discuss operator algebras $A$ with the following property:
every $A$invariant subspace is complemented by another
$A$invariant subspace. This property is called the Reduction
property and is a kind of semisimplicity. We will discuss the
connections of this property to some classical problems, such as
Kadison Similarity Problem and the structure of amenable operator
algebras.


Feb 22 
Nathan Ng 
Linear combinations of zeros of Lfunctions 
at noon
in UHall C630


The linear independence
conjecture asserts that the multiset of positive ordinates of the
zeros of automorphic Lfunctions is linearly independent over the
field of rational numbers. This deep conjecture implies that if
$1/2+i \gamma$ is a zero of the Riemann zeta function, then
$1/2+2i \gamma$ is a not a zero of the zeta function. I will show
that on the Riemann hypothesis this is true infinitely often. I
will also discuss variants of this phenomenon. This is joint work
with Greg Martin and Micah Milinovich.


Feb 29 
Rob
Craigen 
Survey of Negacyclic Weighing Matrices 
at noon
in UHall C630

(University of Manitoba) 
A square or rectangular
matrix is circulant if every row after the first is a right
circular shift of its predecessor. Negacyclic matrices are defined
the same way except that the first entry of each row is negated
after circulating the preceding row. A partial Hadamard matrix is
a rectangular $k \times n$ $(1,1)$matrix M satisfying $MM^T =
nI$.
In the summer of 2013 I hired four sharp undergraduate students to
tackle a problem about circulant partial Hadamard matrices. The
question of existence of certain negacyclic weighing matrices kept
coming up, so we devoted some energy to exploring this largely
uncultivated territory. In the end we produced, apparently for the
first time, a fairly comprehensive survey of these objects, their
structure, why certain classes exist and others cannot. The
flavour of the existence questions for this class of weighing
matrices is decidedly different from that of groupdeveloped form,
even though much of the theory is the same.
We discuss some situations in which negacyclic weighing matrices
naturally appear, and conclude with some tantalizing new open
questions arising from the work.


Mar 7 
Alia Hamieh 
Determining Hilbert modular forms by the central values of
RankinSelberg convolutions 
at noon
in UHall C630


In this talk, we give a
brief overview of adelic Hilbert modular forms. Then, we show that
the central values of the RankinSelberg convolutions, $L(g\otimes
f, s)$, uniquely determine an adelic Hilbert modular form $g$,
where $f$ varies in a carefully chosen infinite family of adelic
Hilbert modular forms. We prove our results in both the level and
weight aspects. This is a joint work with Naomi Tanabe.


Mar 14 
Joy Morris 
Automorphisms of circulant graphs 
at noon
in UHall C630


Determining the full
automorphism group of a graph is a hard problem with a long
history. I will discuss some of the major results that involve
finding graphs with a given automorphism group. I will then focus
on circulant graphs, and describe some structural results and
algorithms that help us determine the full automorphism group of
the graph. I will also give some asymptotic results about how many
circulant graphs fall into different categories.


Mar 21 
Arnab Bose 
Investigations on some Exponential Congruences 
at noon
in UHall C630


Around 1981, Selfridge
asked for what positive integers $a$ and $b$ does $2^a 2^b$
divide $n^a  n^b$ for all $n \in \mathbb{N}$. The problem was
independently solved by various people in different contexts. In
this talk, we study their ideas and prove a generalization of the
problem, in the elementary number theoretic sense and also in
algebraic number fields. Further, we develop ideas to give a
conditional resolution and generalizations to another problem by
H.Ruderman which is closely related to Selfridgeâ€™s problem.


Apr 4 
Brandon Fuller 
CCA groups and graphs 
at noon
in UHall C630


An automorphism of a Cayley
graph that preserves its natural edgecolouring is called
colourpreserving. We study groups $G$ with the property that
every automorphism on every connected Cayley graph on $G$ is the
composition of a lefttranslation and a group automorphism. We
call this class of groups CCA groups and we look at classifying
which groups are not CCA. More precisely, we look at abelian
groups, groups of odd order and direct or semidirect products of
groups.


Apr 11 
Asif Zaman 
The least prime ideal in the Chebotarev Density Theorem 
at noon
in UHall C630

(University of Toronto)

In 1944, Linnik famously
showed unconditionally that the least prime in an arithmetic
progression $a \pmod{q}$ with $(a,q) = 1$ is bounded by $q^L$ for
some absolute effective constant $L > 0$, known as
â€śLinnikâ€™s constantâ€ť. Many authors have computed explicit
admissible values of $L$ with the current world record at $L = 5$
by Xylouris (2011), refining techniques of HeathBrown (1992).
We consider a broad generalization of this problem in the
Chebotarev Density Theorem (CDT), which is concerned with the
splitting behaviour of prime ideals in number fields. Namely, what
is the least norm of a prime ideal occurring in CDT? Papers of
LagariasMontgomeryOdlyzko (1979) and Weiss (1983) give different
unconditional fielduniform bounds but without any explicit
exponents analogous to the subsequent work on Linnikâ€™s constant.
I will report on our recent work establishing such explicit
estimates along with some applications related to primes
represented by binary integral quadratic forms and congruences for
Fourier coefficients of cuspidal Hecke eigenforms.
This is joint work with Jesse Thorner.


June 17 
Ram Murty 
Twin Primes 
at noon
in C630

(Queen's University)

We will discuss recent
progress towards the twin prime conjecture as well as highlight
some recent joint work with Akshaa Vatwani the connects the parity
problem with the twin prime conjecture. The talk will be
accessible to a wide audience.
