
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES 
Number
Theory Seminar 20132014
Fields Institute,
Stewart Library,
Mondays
at 3:30 p.m.
Organizing
Committee:
Leo Goldmakher, JingJing Huang



Upcoming Seminars 

TBA

Past
Seminars 
April 7 
Damien Roy (University of Ottawa)
On Schmidt and Summerer parametric geometry of numbers
In a series of recent papers, W.M. Schmidt and L. Summerer
develop a remarkable theory of parametric geometry of numbers
which enables them to recover many results about simultaneous
rational approximation to families of Qlinearly independent
real numbers, or about the dual problem of forming small
linear integer combinations of such numbers. They recover
classical results of Khintchine and Jarnik as well as more
recent results by Bugeaud and Laurent. They also find many
new results of Diophantine approximation.
Their theory provides constraints on the behavior of the
successive minima of a natural family of one parameter convex
bodies attached to a given ntuple of real numbers, in terms
of this varying parameter. In this talk, we are interested
in the converse problem of constructing ntuples of numbers
for which the corresponding successive minima obey given
behavior. We will present the general theory of Schmidt
and Summerer, mention some applications, and report on recent
progress concerning the above problem.

Mar. 24 
Wentang Kuo (University of Waterloo)
On Erd\H{o}sPomerance conjecture for rank one Drinfeld
modules
(tentative abstract)
Let $\phi$ be a sgnnormalized rank one Drinfeld
$A$module defined over $\mathcal{O}$, the integral closure
of $A$ in the Hilbert class field of $A$. We prove an analogue
of a conjecture of Erd\H{o}s and Pomerance for $\varphi$.
Given any $0 \neq \alpha \in \mathcal{O}$ and an ideal $\frak{M}$
in $\mathcal{O}$, let $f_{\alpha}\left(\frak{M}\right) = \left\{f
\in A \mid \phi_{f}\left(\alpha\right) \equiv 0 \pmod{\frak{M}}
\right\}$ be the ideal in $A$. We denote by $\omega\big(f_\alpha\left(\frak{M}\right)\big)$
the number of distinct prime ideal divisors of $f_\alpha\left(\frak{M}\right)$.
If $q \neq 2$, we prove that there exists a normal distribution
for the quantity
$$
\frac{\omega\big(f_\alpha\left(\frak{M}\right)\big)\frac{1}{2}
\left(\log\deg\frak{M}\right)^2}{\frac{1}{\sqrt{3}}
\left(\log\deg\frak{M}\right)^{3/2}}.
$$
This is the jointed work with YenLiang Kuan and WeiChen
Yao

Mar. 17 
Kevin Hare (University of Waterloo)
Base $d$ expansions with digits $0$ to $q1$
Let $d$ and $q$ be positive integers, and consider representing
a positive integer $n$ with base $d$ and digits $0, 1, \cdots,
q1$. If $q < d$, then not all positive integers can
be represented. If $q = d$, every positive integer can be
represented in exactly one way. If $q > d$, then there
may be multiple ways of representing the integer $n$. Let
$f_{d,q}(n)$ be the number of representations of $n$ with
base $d$ and digits $0, 1, \cdots, q1$. For example, if
$d = 2$ and $q = 7$ we might represent 6 as $(110)_2 = 1
\cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0$ as well as $(102)_2
= 1 \cdot 2^2 + 0 \cdot 2^1 + 2 \cdot 2^0$. In fact, there
are six representations in this case $(110)_2, (102)_2,
(30)_2, (22)_2, (14)_2$ and $(6)_2$, hence $f_{2,7}(6) =
6$.
In this talk we will discuss the asymptotics of $f_{d,q}(n)$
as $n\to \infty$.
This depends in a rather strange way on the Generalized
ThueMorse sequence. While many results are computationally/experimentally
true, only partial results are known.

Mar. 10 
Julian Rosen (University of Waterloo)
Multiple zeta values and their truncations
The multiple zeta values are real numbers generalizing
the values of the Riemann zeta function at positive integers.
They are known to satisfy certain algebraic relations, but
there are many conjectured transcendence results that have
proven to be quite difficult. Truncations of the defining
series are called multiple harmonic sums. These rational
numbers have interesting arithmetic properties, and are
viewed as a finite analogue of the multiple zeta values.
We will discuss the parallels between the two theories,
as well as some recent results concerning multiple harmonic
sums.

Mar. 3 
YuRu Liu (University of Waterloo)
Equidistribution of polynomial sequences in function fields
We prove a function field analog of Weyl's classical theorem
on equidistribution of polynomial sequences. Our result
covers the case when the degree of the polynomial is greater
than or equal to the characteristic of the field, which
is a natural barrier when one tries to apply the Weyl differencing
process to function fields. We also discuss applications
to Sakozy's theorem in function fields. This is a joint
work with Thai Hoang Le.

February 10 
Yuanlin Li (Brock University)
On The Davenport Constant
Let G be a finite abelian group. The Davenport constant
D(G) of G is defined to be the smallest positive integer
d such that every sequence of d elements in G contains a
nonempty subsequence with the product of all its elements
equal to 1 the identity of G. The problem of finding D(G)
was proposed by H. Davenport in 1966, and it was pointed
out that D(G) is connected to the algebraic number theory
in the following way. Let K be an algebraic number field
and G be its class group. Then D(G) is the maximal number
of the prime ideals (counting multiplicity) that can occur
in the decomposition of an irreducibleinteger in K. In this
talk, we will review some known results regarding the Davenport
constant of abelian groups and discuss a few methods which
can be used to find the exact value of $D(G). Some recent
new results will also be presented.

February 3 
Stanley Xiao (University of Waterloo)
Powerfree values of polynomials
In this talk I will give an overview of the progress made
on the powerfree values of polynomial problem. In particular,
I intend to discuss the determinant method of HeathBrown
and Salberger, which so far is the most promising technique
on this problem.

January 27 
JingJing Huang (University of Toronto)
Rational points near manifold and Diophantine approximation
We will discuss the two topics mentioned in the title.

December 2
Monday 
Lluis Vena (University of Toronto)
The removal lemma for homomorphisms in abelian groups
The triangle removal lemma states that if a graph has a
subcubic number of triangles, then removing a subquadratic
number of edges suffices to make G free of triangles. One
of its most famous applications is a simple proof of Roth's
theorem, which asserts that any subset of the integers with
positive upper density contains a 3term arithmetic progression.
In 2005, Green showed an analogous result for linear equations
in finite abelian groups, the socalled removal lemma for
groups. In this talk, we will discuss a combinatorial proof
of Green's result, as well as a generalization to homomorphism
systems in finite abelian groups. In particular, our results
imply a multidimensional version of Szemeredi's theorem.

November 25 
Jonathan Bober (University of Bristol)
Conditionally bounding analytic ranks of elliptic curves
I'll describe how to use the explicit formula for the
Lfunction of an elliptic curve to compute upper bounds
for the analytic rank, assuming GRH. This method works particularly
well for elliptic curves of large rank and (relatively)
small conductor, and can be used to compute exact upper
bounds for the curves of largest known rank, assuming BSD
and GRH.

November 18 
Kevin McGown (Ursinus College)
Euclidean Number Fields and Ergodic Theory
When does a number field possess a Euclidean algorithm?
We will discuss how generalizations of this question lead
us to studying the SEuclidean minimum of an ideal class,
which is a real number attached to some arithmetic data.
Generalizing a result of Cerri, we show that this number
is rational under certain conditions. We also give some
corollaries and discuss the relationship with Lenstra's
notion of a normEuclidean ideal class and the conjecture
of Barnes and SwinnertonDyer on quadratic forms. The proof
involves using techniques of Berend from ergodic theory
and topological dynamics on the appropriate compact group.

November 11 
No seminar 
October 28 
Chantal David (Concordia University)
Onelevel density for zeroes in famlies of elliptic curves
Using the ratios conjectures as introduced by Conrey, Farmer
and
Zirnbauer, we obtain closed formulas for the onelevel density
for some families of Lfunctions attached to elliptic curves,
and we can then determine the underlying symmetry types
of the families. The onelevel density for some of those
families was studied in the past for test functions with
Fourier transforms of small support, but since the Fourier
transforms of the three orthogonal distributions (O, SO(even)
and SO(odd)) are undistinguishable for small support, it
was not possible to identify the distribution with those
techniques. This can be done with the ratios conjectures.
The results confirm the conjectures of Katz and Sarnak,
and shed more light on the phenomenon of "independent"
and "nonindependent" zeroes, and the repulsion
phenomenon. This is joint work with Duc Khiem Huynh and
James Parks. We also present some work in progress in collaboration
with Sandro Bettin where we obtain general formulas for
the onelevel density of oneparameter families of elliptic
curves in term of the rank over Q(t) and the average root
number.

October 14 
No seminar (Reading Week)

October 7
4:305:30
**please note time change for this week only

Alex Iosevich, University of Rochester
Group actions and Erdos type problems in vector spaces
over finite fields
We shall use group invariances to study the distribution
of simplexes in vector spaces over finite fields. It turns
out that the most convenient way to study repeated simplexes
is via appropriate norms of the natural "measure"
on the set $EgE$, where $E$ is a subset of the ${\Bbb F}_q^d$,
$d \ge 2$, and $g$ is an element of the orthogonal group
$O_d({\Bbb F}_q)$.

September 30 
No seminar (Fields Medal
Symposium)

September 23

Leo Goldmakher (University of Toronto)
On the least quadratic nonresidue
I will discuss the relationship between bounds on long
character sums and bounds on the least quadratic nonresidue.
In particular, I will show how small savings on one leads
to massive savings in the other. This is joint work with
Jonathan Bober.

September 16 
Giorgis Petridis,
University of Rochester
Higher sumsets with
different summands 


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