Date 
Speaker 
Title 

Jan 15 
everyone 
Open problem session 
at noon
in B543


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 29 
Nathan Ng 
Mean values of long Dirichlet polynomials 
at noon
in B543

(University of Lethbridge)

A Dirichlet polynomial is a function of the form $A(t)=\sum_{n \le N} a_n n^{it}$
where $a_n$ is a complex sequence, $N \in \mathbb{N}$, and $t \in \mathbb{R}$.
For $T \ge 1$, the mean values
$$\int_{0}^{T} A(t)^2 \, dt$$
play an important role in the theory of Lfunctions. I will discuss work of
Goldston and Gonek on how to evaluate
these integrals in the case that $T < N < T^2$. This will depend on the correlation sums
\[
\sum_{n \le x} a_n a_{n+h} \text{ for } h \in \mathbb{N}.
\]
If time permits, I will discuss a conjecture
of Conrey and Keating in the case that $a_n$ corresponds to a generalized divisor
function and $N > T$.


Feb 12 
Ha Tran 
Reduced Ideals from the Reduction Algorithm 
at noon
in B543

(University of Calgary)

Reduced ideals of a number field $F$ have inverses of small norms and they form
a finite and regularly distributed set in the infrastructure of $F$. Therefore,
they can be used to compute the regulator and the class number of a number
field [5, 3, 2, 1, 4]. One usually applies the reduction algorithm (see
Algorithm 10.3 in [4]) to find them. Ideals obtained from this algorithm are
called 1reduced. There exist reduced ideals that are not 1reduced. We will
show that these ideals have inverses of larger norms among reduced ones.
Especially, we represent a sufficient and necessary condition for reduced
ideals of real quadratic fields to be obtained from the reduction algorithm.
References
 Johannes Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In Séminaire de Théorie des Nombres, Paris 19881989, volume 91 of Progr. Math., pages 2741. Birkhäuser Boston, Boston, MA, 1990.
 Johannes Buchmann and H. C. Williams. On the infrastructure of the principal ideal class of an algebraic number field of unit rank one. Math. Comp., 50(182):569579, 1988.
 H. W. Lenstra, Jr. On the calculation of regulators and class numbers of quadratic fields. In Number theory days, 1980 (Exeter, 1980), volume 56 of London Math. Soc. Lecture Note Ser., pages 123150. Cambridge Univ. Press, Cambridge, 1982.
 René Schoof. Computing Arakelov class groups. In Algorithmic number theory: lattices, number fields, curves and cryptography, volume 44 of Math. Sci. Res. Inst. Publ., pages 447495. Cambridge Univ. Press, Cambridge, 2008.
 Daniel Shanks. The infrastructure of a real quadratic field and its applications. In Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972), pages 217224. Univ. Colorado, Boulder, Colo., 1972.


Feb 26 
Andrew Fiori 
A Geometric Description of Arthur Packets 
at noon
in B543

(University of Lethbridge)

In this talk I will discuss joint work with Clifton Cunningham, Ahamed
Moussaui, James Mracek and Bin Xu.
I will begin by giving a brief overview of the (conjectural) Langlands
Correspondence, focusing in particular on Vogan's geometric reformulation of
the local Langlands Correspondence. We will then discuss some geometric
objects that arise as part of several conjectures which give geometric
interpretations to Arthur packets and their associated stable distributions
under the LLC. More specifically we shall discuss equivariant perverse
sheaves and their vanishing cycles.


Mar 5 
Amir Akbary 
On the size of the gcd of $\boldsymbol{a^n1}$ and $\boldsymbol{b^n1}$ 
at noon
in B543

(University of Lethbridge)

We review some results, from the last twenty years, on the problem of bounding $${\rm gcd}(a^n1, b^n1),$$
as $n$ varies. Here either $a$ and $b$ are integers or $a$ and $b$ are polynomials with coefficients in certain fields.
In spite of elementary nature of the problem, the results are depended on tools from Diophantine approximation and Diophantine geometry.


Mar 12 
Steve Wilson 
The BGCG Construction 
at noon
in B543

(Northern Arizona University)

Well, it's not really a construction yet — it's more like a template for constructions. It's a way to take many copies of one tetravalent graph B, the 'base graph', and identify each edgemidpoint with one other according to another graph C, the 'connection graph' to produce a bipartite tetravalent graph. If the identifying is done with caution, wisdom and, um, insouciance, the resulting graph will have lots of symmetry.
The cunning of the identifications is related to edgecolorings of the base graph which are themselves nicely symmetric, and we will give several examples where the symmetry can actually be achieved.


Mar 19 
PengJie Wong 
On Generalisations of the Titchmarsh divisor problem 
at noon
in B543

(University of Lethbridge)

The study of the asymptotic behaviour of the summatory function of the number
of divisors of shifted primes was initiated by Titchmarsh, who showed that
under the generalised Riemann hypothesis, one has
\[
\sum_{p \le x} \tau(pa) = x \prod_{p\nmid a} \left( 1+ \frac{1}{p(p1)}\right)\prod_{p a} \left( 1 \frac{1}{p}\right) + O\left(\frac{x \log \log x}{\log x}\right),
\]
where $\tau$ denotes the divisor function.
The above formula was first proved unconditionally by Linnik via the dispersion method.
Moreover, applying the celebrated BombieriVinogradov theorem, Halberstam and Rodriguez
independently gave another proof.
In this talk, we shall study the Titchmarsh divisor problem in arithmetic progressions
by considering the sum
\[
\sum_{\substack{ p \le x \\ p\equiv b \ (\mathrm{mod}\, r)}} \tau(pa).
\]
Also, we will try to explain how to obtain an asymptotic formula for the same, uniform in
a certain range of the modulus $r$. If time allows, we will discuss a number field analogue
of this problem by considering the above sum over primes satisfying Chebotarev conditions.
(This is joint work with Akshaa Vatwani.)


Mar 26 
Joy Morris 
Cayley index and Most Rigid Representations (MRRs) 
at noon
in B543

(University of Lethbridge)

For any finite group $G$, a natural question to ask is the order of the
smallest possible automorphism group for a Cayley graph on $G$. A particular
Cayley graph whose automorphism group has this order is referred to as an MRR
(Most Rigid Representation), and its Cayley index is the index of the
regular representation of $G$ in its automorphism group. Study of GRRs
(Graphical Regular Representations, where the full automorphism group is the
regular representation of $G$) showed that with the exception of two infinite
families and ten individual groups, every group admits a Cayley graph whose
MRRs are GRRs, so that the Cayley index is 1. I will present results that
complete the determination of the Cayley index for those groups whose Cayley
index is greater than 1. This is based on joint work with Josh Tymburski, who
was an undergraduate student here at the time.


Apr 9 
JeanMarc Deshouillers 
Values of arithmetic functions at consecutive arguments 
at noon
in B543

(University of Bordeaux)

We shall place in a general context the following result recently (*) obtained
jointly with Yuri Bilu (Bordeaux), Sanoli Gun (Chennai) and Florian Luca
(Johannesburg).
Theorem. Let $\tau(\cdot)$ be the classical Ramanujan $\tau$function and let $k$ be a
positive integer such that $\tau(n) \neq 0$ for $1 \le n \le k/2$. (This is known to be
true for $k < 10^{23}$, and, conjecturally, for all $k$.) Further, let $\sigma$ be a
permutation of the set $\{1, \ldots, k\}$. We show that there exist infinitely many
positive integers $m$ such that
$$\bigl\tau \bigl( m + \sigma(1) \bigr)\bigr < \bigl\tau \bigl(m + \sigma(2)
\bigr)\bigr < \cdots < \bigl \tau \bigl( m + \sigma(k) \bigr)\bigr .$$
The proof uses sieve method, SatoTate conjecture, recurrence relations for the values
of $\tau$ at prime power values.
(*) Hopefully to appear in 2018.


May 7 
Alia Hamieh 
Nonvanishing of $L$functions of Hilbert Modular Forms inside the Critical Strip 
at noon
in C630

(University of Northern British Columbia)

In this talk, I will discuss recent joint work with Wissam Raji. We show that,
on average, the $L$functions of cuspidal Hilbert modular forms with
sufficiently large weight $k$ do not vanish on the line segments $\Im(s)=t_{0}$,
$\Re(s)\in(\frac{k1}{2},\frac{k}{2}\epsilon)\cup(\frac{k}{2}+\epsilon,\frac{k+1}{2})$.
The proof follows from computing the Fourier expansion of a certain kernel
function associated with Hilbert modular forms and estimating its first Fourier
coefficient. This result is analogous to the case of classical modular forms
which was proved by W. Kohnen in 1997.

