Number Theory and Combinatorics Seminar

Fall 2007, Spring 2008, Fall 2008, Spring 2009, Fall 2009, Spring 2010, Fall 2010

Wednesday,
September 30, 2009

Room: TH204

Time: 12:00-12:50 PM

Speaker: Dragos Ghioca (University of Lethbridge)

Title: Arithmetic Dynamics

Abstract: Starting from a fundamental question regarding roots of unity we present a generalization of the classical Manin-Mumford conjecture in the context of algebraic dynamics.

Friday, October 16, 2009

Room: D630

Time: 12:00-12:50 PM

Speaker: Brandon Fodden (University of Lethbridge)

Title: An explicit inequality equivalence of the generalized Riemann hypothesis for a member of the Selberg class

Abstract: Given a member F of the Selberg class, we find a property P of the natural numbers such that the generalized Riemann hypothesis holds for F if and only if P holds for all natural numbers. P is given as an explicit inequality. If one can show that P is a decidable property, then the generalized Riemann hypothesis for F is equivalent to the unsolvability of a particular Diophantine equation. We discuss variants of P for which proving decidability is more practical. Finally, we apply this result to L-functions related to elliptic curves.

Wednesday, October 28, 2009

Room: A580

Time: 12:00-12:50 PM

Speaker: Amir Akbary (University of Lethbridge)

Title: Analytic Problems for Elliptic Curves (I); Titchmarsh Divisor Problem

Abstract: Titchmarsh Divisor Problem concerns the asymptotic behavior of the sum (on primes up to x) of the number of divisors of shifted primes. In 1930 Titchmarsh studied this problem and conjectured an asymptotic formula for such a sum. In 1961 Linnik proved that Titchmarsh's conjecture is true.

In this talk we will review the results given in the original paper of Titchmarsh. Our goal here is to describe an analogue of the Titchmarsh Divisor Problem in the context of elliptic curves.

Wednesday, November 4, 2009

Room: A580

Time: 12:00-12:50 PM

Speaker: Amir Akbary (University of Lethbridge)

Title: Analytic Problems for Elliptic Curves (II); An Elliptic Analogue of the Titchmarsh Divisor Problem

Abstract: We continue our discussion on the Titchmarsh Divisor Problem. Recall that this problem concerns the asymptotic behavior of the sum $$\sum_{{p\leq x}\atop{p~{\rm prime}}} \tau(p-1)$$ as $x\rightarrow \infty$, where $\tau(p-1)$ is the number of divisors of $p-1$.

Our goal is to describe an analogue of this problem in the context of elliptic curves.

Wednesday, November 18, 2009

Room: A580

Time: 12:00-12:50 PM

Speaker: Nathan Ng (University of Lethbridge)

Title: A Brief History of the Riemann Hypothesis

Abstract: In 1859, Riemann introduced the zeta function to the theory of prime numbers. Riemann proved some basic properties regarding the behaviour of this function: functional equation, approximate functional equation, number of zeros in a box, and explicit formula. Moreover, he introduced a profound conjecture, the Riemann hypothesis, which concerns the location of zeros of the zeta function. In this talk I will discuss Riemann's conjecture and it's great influence on analytic number theory.

Friday, November 27, 2009

Room: D630

Time: 12:00-12:50 PM

Speaker: Howard Cheng (University of Lethbridge)

Title: Time- and Space-efficient Computation of Hypergeometric Constants

Abstract: Hypergeometric series are infinite series of the form \[ \sum_{n=0}^\infty a(n) \prod_{i=0}^{n-1} \frac{p(i)}{q(i)} \]where $a$, $p$, and $q$ are polynomials with integer coefficients. Many elementary functions evaluated at rational points may be approximated to high precision (millions of digits) by using hypergeometric series with a technique commonly known as ``binary splitting.'' Furthermore, well-known constants such as $\pi$ and $\zeta(3)$ can also be evaluated in this manner.

Although the binary splitting method is relatively efficient in terms of time, it is not optimal in terms of the amount of space (memory) used. In this talk, we look at the development of an

algorithm that is space-efficient---it uses only $O(N)$ extra space where $N$ is the number of digits desired. Its time complexity is the same as binary splitting but is faster in practice. Elementary properties of integer factorization and prime numbers are used to obtain the resulting algorithm. To the best of my knowledge, this is also the fastest algorithm for this type of calculations.

Most of this talk should be accessible to undergraduate students.

Wednesday, December 2, 2009

Room: A580

Time: 12:00-12:50 PM

Speaker: Dave Morris (University of Lethbridge)

Title: Some arithmetic groups that cannot act on the line

Abstract: It is known that finite-index subgroups of the arithmetic group SL(3,Z) have no interesting actions on the real line. This naturally led to the conjecture that most other arithmetic groups (of higher real rank) also cannot act on the line (except by linear-fractional transformations). This problem remains open, but my joint work with Lucy Lifschitz (University of Oklahoma) and Vladimir Chernousov (University of Alberta) has verified the conjecture for many examples. The proofs are based on the fact, proved by D.Carter, G.Keller, and E.Paige, that if A is the ring of integers of an algebraic number field, and A has infinitely many units, then every element of SL(2,A) is a product of a bounded number of elementary matrices.

Time: 12:00-12:50 PM

Speaker: Dragos Ghioca (University of Lethbridge)

Title: Arithmetic Dynamics

Abstract: Starting from a fundamental question regarding roots of unity we present a generalization of the classical Manin-Mumford conjecture in the context of algebraic dynamics.

Friday, October 16, 2009

Room: D630

Time: 12:00-12:50 PM

Speaker: Brandon Fodden (University of Lethbridge)

Title: An explicit inequality equivalence of the generalized Riemann hypothesis for a member of the Selberg class

Abstract: Given a member F of the Selberg class, we find a property P of the natural numbers such that the generalized Riemann hypothesis holds for F if and only if P holds for all natural numbers. P is given as an explicit inequality. If one can show that P is a decidable property, then the generalized Riemann hypothesis for F is equivalent to the unsolvability of a particular Diophantine equation. We discuss variants of P for which proving decidability is more practical. Finally, we apply this result to L-functions related to elliptic curves.

Wednesday, October 28, 2009

Room: A580

Time: 12:00-12:50 PM

Speaker: Amir Akbary (University of Lethbridge)

Title: Analytic Problems for Elliptic Curves (I); Titchmarsh Divisor Problem

Abstract: Titchmarsh Divisor Problem concerns the asymptotic behavior of the sum (on primes up to x) of the number of divisors of shifted primes. In 1930 Titchmarsh studied this problem and conjectured an asymptotic formula for such a sum. In 1961 Linnik proved that Titchmarsh's conjecture is true.

In this talk we will review the results given in the original paper of Titchmarsh. Our goal here is to describe an analogue of the Titchmarsh Divisor Problem in the context of elliptic curves.

Wednesday, November 4, 2009

Room: A580

Time: 12:00-12:50 PM

Speaker: Amir Akbary (University of Lethbridge)

Title: Analytic Problems for Elliptic Curves (II); An Elliptic Analogue of the Titchmarsh Divisor Problem

Abstract: We continue our discussion on the Titchmarsh Divisor Problem. Recall that this problem concerns the asymptotic behavior of the sum $$\sum_{{p\leq x}\atop{p~{\rm prime}}} \tau(p-1)$$ as $x\rightarrow \infty$, where $\tau(p-1)$ is the number of divisors of $p-1$.

Our goal is to describe an analogue of this problem in the context of elliptic curves.

Wednesday, November 18, 2009

Room: A580

Time: 12:00-12:50 PM

Speaker: Nathan Ng (University of Lethbridge)

Title: A Brief History of the Riemann Hypothesis

Abstract: In 1859, Riemann introduced the zeta function to the theory of prime numbers. Riemann proved some basic properties regarding the behaviour of this function: functional equation, approximate functional equation, number of zeros in a box, and explicit formula. Moreover, he introduced a profound conjecture, the Riemann hypothesis, which concerns the location of zeros of the zeta function. In this talk I will discuss Riemann's conjecture and it's great influence on analytic number theory.

Friday, November 27, 2009

Room: D630

Time: 12:00-12:50 PM

Speaker: Howard Cheng (University of Lethbridge)

Title: Time- and Space-efficient Computation of Hypergeometric Constants

Abstract: Hypergeometric series are infinite series of the form \[ \sum_{n=0}^\infty a(n) \prod_{i=0}^{n-1} \frac{p(i)}{q(i)} \]where $a$, $p$, and $q$ are polynomials with integer coefficients. Many elementary functions evaluated at rational points may be approximated to high precision (millions of digits) by using hypergeometric series with a technique commonly known as ``binary splitting.'' Furthermore, well-known constants such as $\pi$ and $\zeta(3)$ can also be evaluated in this manner.

Although the binary splitting method is relatively efficient in terms of time, it is not optimal in terms of the amount of space (memory) used. In this talk, we look at the development of an

algorithm that is space-efficient---it uses only $O(N)$ extra space where $N$ is the number of digits desired. Its time complexity is the same as binary splitting but is faster in practice. Elementary properties of integer factorization and prime numbers are used to obtain the resulting algorithm. To the best of my knowledge, this is also the fastest algorithm for this type of calculations.

Most of this talk should be accessible to undergraduate students.

Wednesday, December 2, 2009

Room: A580

Time: 12:00-12:50 PM

Speaker: Dave Morris (University of Lethbridge)

Title: Some arithmetic groups that cannot act on the line

Abstract: It is known that finite-index subgroups of the arithmetic group SL(3,Z) have no interesting actions on the real line. This naturally led to the conjecture that most other arithmetic groups (of higher real rank) also cannot act on the line (except by linear-fractional transformations). This problem remains open, but my joint work with Lucy Lifschitz (University of Oklahoma) and Vladimir Chernousov (University of Alberta) has verified the conjecture for many examples. The proofs are based on the fact, proved by D.Carter, G.Keller, and E.Paige, that if A is the ring of integers of an algebraic number field, and A has infinitely many units, then every element of SL(2,A) is a product of a bounded number of elementary matrices.