at noon
in E575

Please bring your favourite (math) problems. Anyone with a problem to share will be given about 5 minutes to present it. We will also choose most of the speakers for the rest of the semester.

at noon
in E575

(University of Lethbridge)

A natural number \(s\) is said to be a square modulo a composite number \(q\) if it is a square modulo each of the prime numbers dividing \(q\). Let \(p\) be a prime number, then \[\textbf{Prob}(s \text{ is a square mod }p)=\displaystyle{\frac{p+1}{2p} \approx \frac{1}{2}}.\] Roughly speaking, the probability of a number to be a square modulo \(q\) is \(\displaystyle{ \frac{1}{2^{\omega(q)}}}\), where \(\omega(q)\) is the number of prime divisors of \(q\).

Fix \(h\) and let \(\mathcal{ X}: \{1, 2, \cdots , q\} \rightarrow \mathbb{N}\) be a random variable, given by \[\mathcal{ X}(i)= \#\{s \in [i, i+h] : s \text{ is a square modulo } q\}.\] For the mean, we have \({\rm \bf E}(\mathcal{ X})\approx h/ 2^{\omega(q)}\), and, in this talk, we show the following bound for the variance: \[{\rm \bf Var}(\mathcal{ X}) \leq {\rm \bf E}(\mathcal{ X}) \approx \frac{h}{2^{\omega(q)}} . \]

at noon
in B660

(University of Lethbridge)

In April 2013, Yitang Zhang announced one of the great theorems in the history of number theory. He showed there exists an absolute constant C such that infinitely many consecutive primes differ by C. This theorem goes a long way towards proving the twin prime conjecture. In this talk I will give an overview of Zhang's theorem and some of the main ideas in the proof.

at noon
in B660

(University of Lethbridge)

In this talk, I will focus on the Goldston-Pintz-Yildirim (GPY) method for detecting small gaps between primes. In particular, I will discuss the choice of weight function in their optimization argument and the role of primes in arithmetic progressions. Finally, we will consider the Motohashi-Pintz/Zhang variant of the GPY argument which yields bounded gaps between primes.

at noon
in B660

(University of Lethbridge)

In 1948, Morgan Ward introduced the concept of an Elliptic Divisibility Sequence (EDS) as an integer sequence \((W_{n})\) which satisfies the recurrence relation $$W_{m+n}W_{m-n}W_{1}^{2} = W_{m+1}W_{m-1}W_{n}^{2} - W_{n+1}W_{n-1}W_{m}^{2},$$ and satisfies the additional property that \(W_{m}|W_{n}\) whenever \(m|n\). Of particular interest to Ward, were what he called symmetries of an EDS. Ward showed that if \((W_{n})\) is an EDS with \(W_{r} = 0\), then we have $$W_{r+i} = ab^{i}W_{i},$$ for some \(a\) and \(b\). In her Ph.D. thesis in 2008, Kate Stange generalized the concept of an EDS to an \(n\)-dimensional array called an Elliptic Net.

We will discuss the connections between EDS's, Elliptic Nets, and elliptic curves, and give a generalization of Ward's symmetry theorem for elliptic nets.

at noon
in B660

(University of Lethbridge)

We will discuss a paper of Chen and Kuan, in which they study the distribution of torsion points modulo primes over several different commutative algebraic groups. They demonstrate that the average is related to some generalized divisor function for these groups.

at noon
in B660

(University of Lethbridge)

Coxeter groups arise in a wide variety of areas, so every mathematician should know some basic facts about them, including their connection to "Dynkin diagrams." Proofs about these "groups generated by reflections" mainly use group theory, geometry, and combinatorics.

at noon
in B660

(University of Lethbridge)

Let \(S\) be a finite collection of prime numbers. We say a number is an \(S\)-unit if it is a product of powers of primes in \(S\). For instance \(-3/8\) is an example of a \(\{2,3\}\)-unit. Many interesting Diophantine equations are reduced to solving equations of the form \[ x+y=1 \] with \(x\) and \(y\) both being an \(S\)-unit. Using linear forms of logarithms, we can show that there only finitely many solutions to these \(S\)-unit equations. In this talk, I will explain an algorithm (due primarily to Smart and Wildanger) on how we can actually enumerate all these solutions.

at noon
in B660

Let \(f\) be an endomorphism of \(N\)-dimensional projective space. In complex dynamics, it has been known for a century (at least when \(N = 1\)) that the orbits of the critical points determines much of the dynamics of \(f\). Morphisms for which all of these critical orbits are finite (so-called PCF maps) turn out to be an important class to understand. Thurston proved, when \(N = 1\), that there are no algebraic families of PCF maps, except for a small number of easy-to-understand examples. I will discuss some recent research into the arithmetic properties of these maps, as well as a partial extension of Thurston's result to arbitrary dimension.

at noon
in B660

(University of Lethbridge)

Let \(E\) be an elliptic curve over \(\mathbb{Q}\). In this talk we consider several open conjectures about the distribution of local invariants associated with the reductions of \(E\) modulo \(p\) as \(p\) varies over the primes. In order to gain evidence for the conjectures, we consider them on average over a family of elliptic curves.

at noon
in B660

(University of Lethbridge)

In this talk we present some new Chebyshev bounds for the function \(\psi(x)\). In 1962, Rosser and Schoenfeld provided a method to estimate the error term in the approximation \(|\psi(x)-x|\). Since then, progress on the numerical verification of the Riemann Hypothesis and widening the zero-free region have allowed to improve numerically these bounds. In this talk we present a new method by introducing a smooth weight and by using the first explicit zero density estimate for the Riemann zeta function. We also present new results for primes in short intervals, based on this zero density estimate.

at noon
in B660

(University of Lethbridge)

A set of unit vectors \(V \subset \mathbb{C}^n\) is called biangular if for any \(u,v \in V, u \neq v\), $$|\langle u,v\rangle| \in \left\{0,\alpha\right\}$$ for some \(0 < \alpha < 1\). There are well-known upper bounds on the size of these sets of vectors. We will discuss these upper bounds, and the implications when they are met, including the generation of combinatorial objects such as strongly regular graphs and association schemes.

at noon
in B660

It is well-known that the continued fraction expansion of a quadratic irrational is horizontally symmetric about its centre. However, an additional vertical symmetry is exhibited by the continued fraction expansions arising from a certain one-parameter family of positive integers known as Schinzel sleepers. This talk provides a method for generating any Schinzel sleeper and investigates their period lengths as well as both their horizontal and vertical symmetries.

This is joint work with Kell Cheng (Hongkong Institute of Education) as well as Richard Guy and Hugh Williams (University of Calgary). The talk is geared toward an audience with a background corresponding to no more than a first number theory course.