
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 
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.

Past Seminars 
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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