Date

Speaker 
Title 

Jan 12

everyone

Open problem session

at noon in UHall W565 

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 19

Nathan Ng

A subconvexity bound for modular $\pmb{L}$functions

at noon in UHall W565 

Let $L(s)$ be the $L$function associated to a modular form of weight $k$ for the full modular group.
Using spectral theory, Anton Good proved that $L(k/2+it) \ll t^{1/3} (\log t)^{5/6}$. Matti Jutila
discovered a simpler proof which makes use of the Voronoi summation formula, exponential integrals, and Farey fractions. We shall present Jutila's argument.


Jan 26

Nathan Ng

A subconvexity bound for modular $\pmb{L}$functions, part 2

at noon in UHall W565 

This is a continuation of last week's talk. I will sketch Matti Jutila's proof
of a subconvex bound for a modular $L$function on the critical line.
The main ideas are an approximate functional equation, the use
of Farey fractions, Voronoi's summation formula, and exponential integral and sum estimates.


Feb 2

no seminar

(PIMS Distinguished Visitor will speak at noon)





Feb 9

Nathan Ng

Gaps between the zeros of the Riemann zeta function

at noon in UHall W565 

In this talk we will show how to exhibit large and small gaps
between the zeros of the Riemann zeta function, assuming
the Riemann hypothesis. This is based on a technique of
Montgomery and Odlyzko. The problem of finding small gaps
between the zeros leads to a very interesting optimization problem.


Feb 23

Gabriel Verret

Automorphism groups of vertex transitive graphs

at 9am in UHall D632 
(University of Western Australia) 
no abstract available


Mar 11

Peter J. Cho

Zeros of $\pmb{L}$functions

Wednesday at 9am in UHall D632 
(University of Buffalo) 
In 20th century, one of the most striking discoveries in number theory is Montgomery's paircorrelation. It says that paircorrelation of zeros of the Riemann zeta function is the same with that of eigenvalues of unitary matrices. In 1990's, Rudnick, Katz and Sarnak studied the zeros of $L$functions more systematically. Moreover, Katz and Sarnak proposed the $n$level density conjecture which claims that distributions of lowlying zeros of $L$functions in a family is predicted by one of compact matrix groups, which are $U(N)$, $SO(\text{even})$, $SO(\text{odd})$, $O(N)$, and $Sp(2N)$. At the end of the talk, I will state an $n$level density theorem for some families of Artin $L$functions and talk about counting number fields with local conditions. I will start with a friendly definition of $L$functions and give some examples. No background or knowledge for $L$functions are required for this talk.


Mar 13

Daniel Vallieres

Abelian Artin $\pmb{L}$functions at zero

Friday at 9am in UHall C620 
(Binghamton University) 
In the early 1970s, Harold Stark formulated a conjecture about the first nonvanishing Taylor coefficient at zero of Artin $L$functions. About 10 years later, he refined his conjecture for abelian $L$functions having order of vanishing one at zero, under certain hypotheses. In 1996, Karl Rubin extended this last refinement of Stark to the higher order of vanishing setting. In this expository talk for a general audience, we will give a survey of this area of research and present a more general conjecture, which we formulated in the past few years. At the end, we will present evidence for our conjecture and indicate one possible direction for further research.


Mar 23

Tristan Freiberg

Square totients

at noon in UHall W565

(University of Missouri) 
A wellknown conjecture asserts that there are infinitely
many primes $p$ for which $p  1$ is a perfect square.
We obtain upper and lower bounds of matching order on the
number of pairs of distinct primes $p,q \le x$ for which
$(p1)(q1)$ is a perfect square.
This is joint work with Carl Pomerance (Dartmouth College).


May 8

Ram Murty

Consecutive squarefull numbers

Friday at 10:30am in UHall C640

(Queen's University) 
A number $n$ is called squarefull if for every prime
$p$ dividing $n$, we have $p^2$ also dividing $n$.
Erdos conjectured that the number of pairs
of consecutive squarefull numbers $(n, n+1)$
with $n < N$ is at most $(log N)^A$ for some $A >0$.
This conjecture is still open. We will show that
the abc conjecture implies this number is at most
$N^e$ for any $e>0$. We will also discuss a related
conjecture of Ankeny, Artin and Chowla on fundamental
units of certain real quadratic fields and discuss
its connection with the Erdos conjecture.
This is joint work with Kevser Aktas.


June 1

Adam Felix

How close is the order of $\pmb{a}$ mod $\pmb{p}$ to $\pmb{p1}$?

at 11:00am in UHall C630

(KTH Royal Institute of Technology, Sweden) 
Let $a \in \mathbb{Z} \setminus \{0,\pm 1\}$, and let
$f_{a}(p)$ denote the order of $a$ modulo $p$, where $p \nmid a$ is
prime. There are many results that suggest $p1$ and $f_{a}(p)$ are
close. For example, Artin's conjecture and Hooley's subsequent proof
upon the Generalized Riemann Hypothesis. We will examine questions
related to the relationship between $p1$ and $f_{a}(p)$.
