Events

Jerry Wang, Department of Pure Mathematics, University of Waterloo

Counting number fields of fixed degree and bounded discriminant is a classical question in arithmetic statistics. The problem is known in degrees at most 5. In this talk, we will talk about an improvement to Schmidt's upper bound for general degree n, which beats existing improvements for n up to 94. This is joint work with Manjul Bhargava and Arul Shankar.

This seminar will be held jointly online and in person:

• Room: MC 5479
• Zoom information: https://uwaterloo.zoom.us/j/92724311110?pwd=bWV6cjVqQ1Jra1pPMXZhbjhpaEFLQT09

Gian Cordana Sanjaya, Department of Pure Mathematics, University of Waterloo

A classical question in analytic number theory is to determine the density of integers $a_1, \ldots, a_n$ such that $P(a_1, \ldots, a_n)$ is squarefree, where $P$ is a fixed integer polynomial. In this talk, we consider the case $P(a, b) = a^4 + b^3$. When the pairs $(a, b)$ are ordered by $\max\{|a|^{1/3}, |b|^{1/4}\}$, we prove that this density equals the conjectured product of local densities. More generally, we prove the same result for $P(a, b) = \beta a^4 + \alpha b^3$, where $\alpha$ and $\beta$ are fixed nonzero integers such that $\gcd(\alpha, \beta)$ is squarefree. This is joint work with Xiaoheng Wang.

Zoom link: https://uwaterloo.zoom.us/j/99239659097?pwd=ZjNVRjQ1MWpuQmhTb01ZS0RGNDJjQT09

Julie Desjardins, University of Toronto Mississauga

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, it is 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. With Rosa Winter, we prove that for del Pezzo surfaces of degree 1, if « many » equals 9 or more, then yes, but we find counterexamples where a torsion point lies at the intersection lies at the intersection of 7 exceptional curves.

This seminar will be held jointly online and in person:

• Room: MC 5479
• Zoom information: https://uwaterloo.zoom.us/j/99239659097?pwd=ZjNVRjQ1MWpuQmhTb01ZS0RGNDJjQT09

Yu-Ru Liu, Department of Pure Mathematics, University of Waterloo

Waring's problem is about representations of integers as sums of fixed powers, and Hilbert-Kamke's problem is about a system of Diophantine equations of Waring's type. Motivated by the asymptotic estimates for multidimensional Waring's problem, we consider multidimensional analogues of Hilbert-Kamke's problem. We proved that the corresponding singular series is bounded below by an absolute positive constant without any nonsingular local solubility assumption. The number of variables we need is near-optimal. This is joint work with Wentang Kuo and Xiaomei Zhao.

This seminar will be held online only: https://uwaterloo.zoom.us/j/99239659097?pwd=ZjNVRjQ1MWpuQmhTb01ZS0RGNDJjQT09

Debanjana Kundu, University of Toronto

For an imaginary quadratic field, there are two natural $\mathbb{Z}_p$-extensions, the cyclotomic and the anticyclotomic. We'll start with a brief description of Iwasawa theory for the cyclotomic extensions, and then describe some computations for anticyclotomic  $\mathbb{Z}_p$ extensions, especially the fields and their class numbers. This is joint work with LC Washington.

This seminar will be held both online and in person

• Room: MC 5417
• Zoom link: https://uwaterloo.zoom.us/j/99239659097?pwd=ZjNVRjQ1MWpuQmhTb01ZS0RGNDJjQT09 (Please contact Kevin Hare - kghare@uwaterloo.ca - to confirm that this talk will be also available through Zoom.)

David McKinnon, Department of Pure Mathematics, University of Waterloo

Say you’re having a cup of tea with your favourite Gaussian integer.  The two of you get to discussing real and imaginary parts, you know, as you do.  The subject of the gcd of those two quantities arises, and you start wondering how big that gcd can be.  After you both realise how silly a question that is, you then figure out that there are nevertheless some interesting patterns in the answer that turn out to generalize to arbitrary number fields.

MC 5417

Michael Rubinstein, Department of Pure Mathematics, University of Waterloo

Keating, Rodgers, Roditty-Gershon, and Rudnick have given a conjecture for the asymptotic behaviour of the mean square of sums of the $k$-th divisor numbers over short intervals, and have proven formulas for the analogous problem over $\mathbb{F}_q[t]$. I will discuss their work and describe determinantal and differential equations related to their formulas.

This seminar will be held both online and in person

• Room: MC 5479
• Zoom link: https://uwaterloo.zoom.us/j/99239659097?pwd=ZjNVRjQ1MWpuQmhTb01ZS0RGNDJjQT09

J.C. Saunders, Middle Tennessee State University

In 2009, Luca and Nicolae proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are 1, 2, and 3. In 2015, Faye and Luca proved that the only Pell numbers whose Euler totient function is another Pell number are 1 and 2. Here we add to these two results and prove that for any fixed natural number $P\geq 3$, if we define the sequence $(u_n)_n$ as $u_0=0$, $u_1=1$, and $u_n=Pu_{n-1}+u_{n-2}$ for all $n\geq 2$, then the only solution to the Diophantine equation $\varphi(u_n)=u_m$ is $\varphi(u_1)=\varphi(1)=1=u_1$.

Zoom link: https://uwaterloo.zoom.us/j/99239659097?pwd=ZjNVRjQ1MWpuQmhTb01ZS0RGNDJjQT09

Liam Orovec, Department of Pure Mathematics, University of Waterloo

When looking at the representation of numbers in non-integer bases, $\beta$-expansions, we often find an infinite number of expansions for any given real number under any given base. We look at finding, given a fixed positive real number $x$, the smallest base $q_s(x)$ for which $x$ has a unique $q_s(x)$-expansion. Beginning with $x=1$ we find the ever present Thue-Morse sequence will be helpful throughout the talk. Having found our constant $q_{KL}=q_s(1)$, the Komornik-Loreti constant, we will explore when $q_s(x)<q_{KL}$. The majority of this talk will follow the Results due to Derong Kong which covers the case where are expansions have only digits 0 and 1, in what time that remains we will look at generalizing these results for larger alphabets.

MC 5479

Matthew Satriano, Department of Pure Mathematics, University of Waterloo

Bhargava and the speaker introduced a functorial Galois closure operation for finite-rank ring extensions, generalizing constructions of Grothendieck and Katz-Mazur. In this talk, we use Galois closures to construct new components of Hilbert schemes of points, which are fundamental objects in algebraic geometry whose component structure is largely mysterious. We answer a 35 year old open problem posed by Iarrobino by constructing an infinite family of low dimensional components. This talk is based on joint work with Andrew Staal.

This seminar will be held both online and in person:

• Room: MC 5479
• Zoom link: https://uwaterloo.zoom.us/j/99239659097?pwd=ZjNVRjQ1MWpuQmhTb01ZS0RGNDJjQT09