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.