Brandon Fodden (PIMS/University of Lethbridge)
unprovable statements in number theory
will discuss some number theoretic statements which
unprovable with respect to either Peano arithmetic or the ZFC
set theory. We will cover Goodstein's theorem as well as the connection
that Diophantine equations have with the consistency of formalized
Brandon Fodden (PIMS/University of Lethbridge)
Title: Diophantine equations and the
generalized Riemann hypothesis
show that the statement "for all number fields K, the generalized
Riemann hypothesis for K holds" is equivalent to a statementof the form
natural numbers n, property P holds" where P is a decidable property of
numbers (that is, there is an algorithm which will tell if P holds for
an arbitrary natural
number in finitely many
steps). This in turn implies that the original statement is
equivalent to the unsolvability of a particular Diophantine
equation in the integers.
Dragos Ghioca (University of Lethbridge)
Title: Polynomial dynamics in
study orbits of complex
polynomials from the point of
view of arithmetic geometry. In particular we show that if two nonlinear
complex polynomials have orbits with infinite intersection, then the
polynomials have a common iterate.
Nathan Ng (University of Ottawa)
Title: Chebyshev's Bias, Galois
groups, and L-functions
observed the strange phenomenon that there
appear to be more primes congruent to three modulo four than to one
modulo four. This is counterintuitive since one expects that
an equal number of primes congruent to three modulo four than to
one modulo four. In this talk, I will explain this phenomenon
known as "Chebyshev's bias" and I will discuss generalizations.
For example, consider the polynomial x^3-x-1. If we consider this
polynomial modulo p a prime, this polynomial is either irreducible,
into a linear factor and a quadratic factor, or splits into three
I will explain which of these cases occurs the most frequently and I
explain why the "bias" arises.
Habiba Kadiri (University of Lethbridge)
Vinogradov's bound for the three primes Goldbach's conjecture
any odd number greater than 5 be written as a sum of
3 primes ? In 1922, Hardy and Littlewood were the
give a substantial answer to this question: using
Method, they proved that, under the condition of
the Generalized Riemann
Hypothesis, it was true for sufficiently large
Fifteen years later, Vinogradov removed completely
His theorem remains one of the strongest results in
the direction of
Goldbach's conjecture. In this lecture, we will go
through Vinogradov's proof,
and understand how the distribution of the zeros of
the Dirichlet L functions
enter into play.
Kerri Webb (University of Lethbridge)
opponents play 2n head to head games, and each player
wins a total of n games. In how many ways can this be done,
second player never has more wins than her opponent?
This puzzle can be solved with lattice paths: paths in the plane from
point (0,0) to (n,n), where each step in the path is in the direction
(1,0) or (0,1). We survey enumeration results for various modifications
of lattice paths. Classical bijections and a search for a new bijection
Room: TH 343
Radan Kucera (Masaryk University)
Title: On circular units and the
class group of an abelian field
The aim of this talk is to show that circular units can be used to
class group of an abelian field. At the beginning we recall the
and basic properties of the group of circular units of an abelian number
field (explicit generators, finite index in the full group of units,
Sinnott's formula containing the class number).
Then we show a concrete application: having a compositum K of real
fields unramified at 2, we derive a lower bound for the divisibility of
class number of K by a power of 2.
We also explain that circular units can be used to obtain an information
concerning the Galois module structure of the class group: For an
field K and a prime p, two Galois modules appear here naturally, namely
p-th part of the group of all units modulo the subgroup of circular
and the p-th part of the class group. Thaine's theorem states that any
annihilator of the former module is an annihilator of the latter one,
provided p is odd and relatively prime to the degree of the field.
Finally a joint result with C. Greither concerning the p-th part of the
class group of cyclic fields of p-power degree will be explained.
Amir Akbary (University of Lethbridge)
Title: Sharp Upper Bounds for
Abstract: We describe an algorithm
that provides explicit upper bounds
for a certain arithmetic function (which will be defined in the talk).
We use this algorithm to establish some explicit upper bounds for the
sum of divisors function.
This is a joint work with Zachary Friggstad (University of
Robert Juricevic (University of Waterloo).