Number Theory and Combinatorics Seminar
                              Fall 2007, Spring 2008, Fall 2008, Spring 2009, Fall 2009, Spring 2010, Fall 2010

Wednesday, September 12, 2007
Room: L1114
Time: 3:00-3:50 PM

Speaker: Brandon Fodden (PIMS/University of Lethbridge)
  Some unprovable statements in number theory

:  We will discuss some number theoretic statements which are
unprovable with respect to either Peano arithmetic or the ZFC axioms of
set theory. We will cover Goodstein's theorem as well as the connection
that Diophantine equations have with the consistency of formalized

Wednesday, September 26, 2007
Room: L1112
Time: 3:00-3:50 PM

Speaker: Brandon Fodden (PIMS/University of Lethbridge)
Title:   Diophantine equations and the generalized Riemann hypothesis

:  We show that the statement "for all number fields K, the generalized
Riemann hypothesis for K holds" is equivalent to a statementof the form "for all
natural numbers n, property P holds" where P is a decidable property of the natural
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.

Wednesday, October 10, 2007
Room: L1114
Time: 3:00-3:50 PM

Speaker: Dragos Ghioca (University of Lethbridge)
Title:  Polynomial dynamics in number theory

  We 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.

Friday, October 26, 2007
Room: C610
Time: 12:00-12:50 PM

Speaker: Nathan Ng (University of Ottawa)
Title:  Chebyshev's Bias, Galois groups, and L-functions

Abstract:  Chebyshev 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 there are
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, splits
into a linear factor and a quadratic factor, or splits into three linear factors.
I will explain which of these cases occurs the most frequently and I will
 explain why the "bias" arises.

Wednesday, November 7, 2007
Room: L1114
Time: 3:00-3:50 PM

Speaker:  Habiba Kadiri (University of Lethbridge)
Title:  About Vinogradov's bound for the three primes Goldbach's conjecture

Abstract:  Can any odd number greater than 5 be written as a sum of
3 primes ? In 1922, Hardy and Littlewood were the first to
give a substantial answer to this question: using their Circle
Method, they proved that, under the condition of the Generalized Riemann
Hypothesis, it was true for sufficiently large integers.
Fifteen years later, Vinogradov removed completely this hypothesis.
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.

Wednesday, November 21, 2007
Room: L1114
Time: 3:00-3:50 PM

Speaker: Kerri Webb (University of Lethbridge)
Title:  Lattice Path Bijections

Abstract:  Two opponents play 2n  head to head games, and each player
wins a total of  n  games. In how many ways can this be done, if the
second player never has more wins than her  opponent?

This puzzle can be solved with lattice paths: paths in the plane from the
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 are
also discussed.

Wednesday, November 28, 2007
Room: TH 343
Time: 3:00-3:50 PM

Speaker:  Radan Kucera (Masaryk University)
Title:  On circular units and the class group of an abelian field

Abstract:  The aim of this talk is to show that circular units can be used to study the
class group of an abelian field. At the beginning we recall the definition
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 quadratic
fields unramified at 2, we derive a lower bound for the divisibility of the
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 abelian
field K and a prime p, two Galois modules appear here naturally, namely the
p-th part of the group of all units modulo the subgroup of circular units
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.

Wednesday, December 5, 2007
Room: L1114
Time: 3:00-3:50 PM

Speaker: Amir Akbary (University of Lethbridge)
Title:  Sharp Upper Bounds for Divisor Sums

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 Alberta) and
Robert Juricevic (University of Waterloo).