at 12:15pm
in MH1060

(University of Lethbridge)

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 12:15pm
in MH1060

Let $d>k$ be positive integers. Motivated by an earlier result of Bugeaud and Nguyen, we let $E_{k,d}$ be the set of $(c_1,\ldots,c_k)\in\mathbb{R}_{\geq 0}^k$ such that $\vert\alpha_0\vert\vert\alpha_1\vert^{c_1}\cdots\vert\alpha_k\vert^{c_k}\geq 1$ for any algebraic integer $\alpha$ of degree $d$, where we label its Galois conjugates as $\alpha_0,\ldots,\alpha_{d-1}$ with $\vert\alpha_0\vert\geq \vert\alpha_1\vert\geq\cdots \geq \vert\alpha_{d-1}\vert$. First, we give an explicit description of $E_{k,d}$ as a polytope with $2^k$ vertices. Then we prove that for $d>3k$, for every $(c_1,\ldots,c_k)\in E_{k,d}$ and for every $\alpha$ that is not a root of unity, the strict inequality $\vert\alpha_0\vert\vert\alpha_1\vert^{c_1}\cdots\vert\alpha_k\vert^{c_k}>1$ holds. We also provide a quantitative version of this inequality in terms of $d$ and the height of the minimal polynomial of $\alpha$.

at 12:15pm
in MH1060

(University of Lethbridge)

The Polya group $Po(K)$ of a Galois number field $K$ coincides with the subgroup of the ideal class group $Cl(K)$ of $K$ consisting of all strongly ambiguous ideal classes. We prove that there are only finitely many imaginary abelian number fields $K$ whose `Polya index' $\left[Cl(K):Po(K)\right]$ is a fixed integer. Accordingly, under GRH, we completely classify all imaginary quadratic fields with the Polya indices 1 and 2. Also, we unconditionally classify all imaginary biquadratic and imaginary tri-quadratic fields with the Polya index 1. In another direction, we classify all real quadratic fields $K$ of extended R-D type (with possibly only one more field $K$) for which $Po(K)=Cl(K)$. Our result generalizes Kazuhiro's classification of all real quadratic fields of narrow R-D type whose narrow genus numbers are equal to their narrow class numbers. This is a joint work with Amir Akbary (University of Lethbridge).

at 12:15pm
in MH1060

(University of Lethbridge)

We will explore how a Fourier optimization framework may be used to study two classical problems in number theory involving Dirichlet characters: The problem of estimating the least character non-residue; and the problem of estimating the least prime in an arithmetic progression. In particular, we show how this Fourier framework leads to subtle, but conceptually interesting, improvements on the best current asymptotic bounds under the Generalized Riemann Hypothesis, given by Lamzouri, Li, and Soundararajan. Based on joint work with Emanuel Carneiro, Micah Milinovich, and Antonio Ramos.

at 12:15pm
in MH1060

In 1770 Euler observed that $3^3 + 4^3 + 5^3 = 6^3$ and asked if there was another perfect power that equals the sum of consecutive cubes. This captivated the attention of many important mathematicians, such as Cunningham, Catalan, Genocchi and Lucas. In the last decade, the more general equation $x^k + (x+1)^k + \cdots + (x+d)^k = y^n$ began to be studied. In this talk we will focus on this equation. We will see some known results and one of the most used tools to attack this kind of problems. At the end we will show some new results that appear in arXiv:2404.03457.

at 12:15pm
in MH1060

Let $\rm{ord}_p(a)$ be the order of $a$ in $( \mathbb{Z} / p \mathbb{Z} )^*$. In 1927, Artin conjectured that the set of primes $p$ for which an integer $a\neq -1,\square$ is a primitive root (i.e. $\rm{ord}_p(a)=p-1$) has a positive asymptotic density among all primes. In 1967 Hooley proved this conjecture assuming the Generalized Riemann Hypothesis (GRH). In this talk we will study the behaviour of $\rm{ord}_p(a)$ as $p$ varies over primes, in particular we will show, under GRH, that the set of primes $p$ for which $\rm{ord}_p(a)$ is “$k$ prime factors away” from $p-1$ has a positive asymptotic density among all primes except for particular values of $a$ and $k$. We will interpret being “$k$ prime factors away” in three different ways, namely $k=\omega(\frac{p-1}{\rm{ord}_p(a)})$, $k=\Omega(\frac{p-1} {\rm{ord}_p(a)})$ and $k=\omega(p-1)-\omega(\rm{ord}_p(a))$, and present conditional results analogous to Hooley's in all three cases and for all integer $k$. From this, we will derive conditionally the expectation for these quantities. Furthermore we will provide partial unconditional answers to some of these questions. This is joint work with Leo Goldmakher and Greg Martin.

at 12:15pm
in MH1060

It has been known since the 80s, thanks to Conrey and Ghosh, that the average of the square of the Riemann zeta function, summed over the extreme points of zeta up to a height $T$, is $\frac{1}{2} (e^2-5) \log T$ as $T \rightarrow \infty$. This problem and its generalisations are closely linked to evaluating asymptotics of joint moments of the zeta function and its derivatives, and for a time was one of the few cases in which Number Theory could do what Random Matrix Theory could not. RMT then managed to retake the lead in calculating these sorts of problems, but we may now tell the story of how Number Theory is fighting back, and in doing so, describe how to find a full asymptotic expansion for this problem, the first of its kind for any nontrivial joint moment of the Riemann zeta function. This is joint work with Chris Hughes and Solomon Lugmayer.

at 12:15pm
in MH1060

(University of Lethbridge)

A bijection $f$ of a metric space is "distance-permuting" if the distance from $f(x)$ to $f(y)$ depends only on the distance from $x$ to $y$.

For example, it it is known that every distance-permuting bijection of the real line is the composition of an isometry and a dilation ($x \mapsto kx$). So they are affine maps.

We study the analogue in which $G$ is any (finite or infinite) group, and the "distance" from $x$ to $y$ is the "absolute value" of the unique element $s$ of $G$, such that $xs = y$. We determine precisely which groups have the property that every distance-preserving bijection is an affine map. The smallest exception is the quaternion group of order 8, and all other exceptions are constructed from this one.

It is natural to state the problem in the language of graph-theory: construct a graph by joining each pair of points $(x,y)$ with an edge, and label (or "colour") this edge with its length. Then we are interested in bijections that permute the colours of the edges: i.e., the colour of the edge from $f(x)$ to $f(y)$ depends only on the colour of the edge from $x$ to $y$.

This is joint work with Shirin Alimirzaei.

at 12:15pm
in MH1060

(University of Lethbridge)

Abstract TBA

at 12:15pm
in MH1060

(University of Lethbridge)

Abstract TBA