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.


June 28 
Tim Trudgian 
Grosswald's
conjecture on primitive roots 
at noon
in C630

(Australian National University)

Very little is known about
the distribution of primitive roots of a prime $p$. Grosswald
conjectured that the least primitive root of a prime p is less
than $\sqrt{p}  2$ for all $p> 409$. While this is certainly
true for all $p$ sufficiently large, Grosswald's conjecture in
still open. I shall outline some recent work which resolves the
conjecture completely under the Generalised Riemann Hypothesis and
which almost resolves the conjecture unconditionally.


June 28 
Vijay Patankar 
Pairs of elliptic curves and their Frobenius fields 
at 2pm
in C630

(Jawaharlal Nehru University)

Given an elliptic curve $E$ over a number field $K$. The Frobenius field attached to $E$ at a prime $p$ is the splitting field of the characteristic polynomial of the Frobenius endomorphism acting on the $\ell$adic Tate module of $E$ ($\ell$ a prime different from $p$) over the rationals. Thus, the splitting field is either of degree $1$ or degree $2$ over the rationals.
Let $E_1$ and $E_2$ be elliptic curves defined over a number field $K$, with at least one of them without complex multiplication. We prove that the set of places $v$ of $K$ of good reduction such that the corresponding Frobenius fields are equal has positive upper density if and only if $E_1$ and $E_2$ are isogenous over some extension of $K$.
For an elliptic curve $E$ defined over a number field $K$, we show that the set of finite places of $K$ such that the Frobenius field at $v$ equals a fixed imaginary quadratic field $F$ has positive upper density if and only if $E$ has complex multiplication by $F$.
Time permits we will provide a sketch of a result about two dimensional $\ell$adic Galois representations that we will need using an algebraic density theorem due to Rajan.
