Scientific

Solaleh Bolvardizadeh (University of Lethbridge, Canada)

The quality of the triplet $(a,b,c)$, where $\gcd(a,b,c) = 1$, satisfying $a + b = c$ is defined as
$$
q(a,b,c) = \frac{\max\{\log |a|, \log |b|, \log |c|\}}{\log \mathrm{rad}(|abc|)},
$$
where $\mathrm{rad}(|abc|)$ is the product of distinct prime factors of $|abc|$. We call such a triplet an $ABC$-solution. The $ABC$-conjecture states that given $\epsilon > 0$ the number of the $ABC$-solutions $(a,b,c)$ with $q(a,b,c) \geq 1 + \epsilon$ is finite.

In the first part of this talk, under the $ABC$-conjecture, we explore the quality of certain families of the $ABC$-solutions formed by terms in Lucas and associated Lucas sequences. We also introduce, unconditionally, a new family of $ABC$-solutions that has quality $> 1$.

In the remaining of the talk, we prove a conjecture of Erd\"os on the solutions of the Brocard-Ramanujan equation
$$
n! + 1 = m^2
$$
by assuming an explicit version of the $ABC$-conjecture proposed by Baker.

Julie Desjardins (University of Toronto, Canada)

The blow up of the anticanonical base point on X, a del Pezzo surface of degree 1, gives rise to a rational elliptic surface E with only irreducible fibers. The sections of minimal height of E are in correspondence with the 240 exceptional curves on X.

A natural question arises when studying the configuration of those curves: If a point of X is contained in “many” exceptional curves, is it torsion on its fiber on E?

In 2005, Kuwata proved for del Pezzo surfaces of degree 2 (where there is 56 exceptional curves) that if “many” equals 4 or more, then yes. In a joint paper with Rosa Winter, we prove that for del Pezzo surfaces of degree 1, if “many” equals 9 or more, then yes. Moreover, we find counterexamples where a torsion point lies at the intersection of 7 exceptional curves.

Elchin Hasanalizade (University of Lethbridge, Canada)

The Fibonacci sequence \(F(n) : (n\geq 0) is the binary recurrence sequence defined by

$$
F(0) = F(1) = 1 \qquad \mbox{and} \\
F(n+2) = F(n+1) + F(n) \qquad \forall n \geq 0.
$$

There is a broad literature on the Diophantine equations involving the Fibonacci numbers. In this talk, we will study the Diophantine inequality

$$
\left\lvert F(n) + F(m) − 2a\right\rvert < 2a/2
$$

in positive integers n,m and a with $n \geq m$. The main tools used are lower bounds for linear forms in logarithms due to Matveev and Dujella-Petho version of the Baker-Davenport reduction method in Diophantine approximation.

Modern machine learning has had remarkable success in all kinds of AI applications, and is also poised to change fundamentally the way we do research in traditional areas of science and engineering. In this talk, I will give an overview of some of the recent progress made in this exciting new direction and the theoretical and practical issues that I consider most important.

We compute extreme values of the Riemann Zeta function at the critical points of the zeta function in the critical strip. i.e. the points where $\zeta'(s) = 0$ and $\mathfrak{R}s< 1.$. We show that the values taken by the zeta function at these points are very similar to the extreme values taken without any restrictions. We will show geometric significance of such points.

We also compute extreme values of Dirichlet L-functions at the critical points of the zeta function, to the right of $\mathfrak{R}s=1$. It shows statistical independence of L-functions and zet function in a certain way as these values are very similar to the values taken by L-functions without any restriction.