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)

Title: Some unprovable statements in number theory

Abstract: 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

systems.

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

Abstract: 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

Abstract: 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).

Time: 3:00-3:50 PM

Speaker: Brandon Fodden (PIMS/University of Lethbridge)

Title: Some unprovable statements in number theory

Abstract: 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

systems.

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

Abstract: 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

Abstract: 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).