Events

Amir Akbary, University of Lethbridge

The Dedekind eta function is defined by the infinite product
\[
\eta(z) = e^{\pi i z/12}\prod_{n=1}^\infty (1 - e^{2 \pi i z}) = q^{1/24}\prod_{n=1}^\infty (1 - q^n).
\]
and
\[
f(z) = \prod_{t\mid N} \eta^{r_t}(tz),
\]
where the exponent r_t are integers. Let k be an even positive integer, p be a prime, and m be a nonnegative integer. We find an upper bound for orders of zeros (at cusps) of a linear combination of classical Eisenstein series of weight k and level p^m. As an immediate consequence, we find the set of all eta quotients that are linear combinations of these Eisenstein series and, hence, the set of all eta quotients of level p^m whose derivatives are also eta quotients.

This is joint work with Zafer Selcuk Aygin (Northwestern Polytechnic).

MC 5417

Akash Sengupta, Department of Pure Mathematics, University of Waterloo

Given a real number x, how well can we approximate it using rational numbers? This question has been classically studied by Dirichlet, Liouville, Roth et al, and the approximation exponent of a real number x measures how well we can approximate x. Similarly, given an algebraic variety X over a number field k and a point x in X, we can ask how well can we approximate x using k-rational points? McKinnon and Roth generalized the approximation exponent to this setting and showed that several classical results also generalize to rational points algebraic varieties.

In this talk, we will define a new variant of the approximation constant which also captures the geometric properties of the variety X. We will see that this geometric approximation constant is closely related to the behavior of rational curves on X. In particular, I’ll talk about a result showing that if the approximation constant is larger than the dimension of X, then X must be isomorphic to projective space. This talk is based on joint work with David McKinnon.

MC 5417

Micah Milinovich, University of Mississippi

There are many situations where one imposes certain conditions on a function and its Fourier transform and then wants to optimize a certain quantity. I will describe two such Fourier optimization frameworks that can be used to study classical problems in number theory: bounding the maximum gap between consecutive primes assuming the Riemann hypothesis and bounding for the size of the least quadratic non-residue modulo a prime assuming the generalized Riemann hypothesis (GRH) for Dirichlet L-functions. The resulting extremal problems can be stated in accessible terms, but finding the exact answer appears to be rather subtle. Instead, we experimentally find upper and lower bounds for our desired quantity that are numerically close. If time allows, I will discuss how a similar Fourier optimization framework can be used to bound the size of the least prime in an arithmetic progression on GRH. This is based upon joint works with E. Carneiro (ICTP), E. Quesada-Herrera (TU Graz), A. Ramos (SISSA), and K. Soundararajan (Stanford). 

MC 5417

Jakub Krásenský, Czech Technical University in Prague

By the celebrated 15 theorem of Conway and Schneeberger, a classical positive definite quadratic form over Z is universal if it represents each element of {1,2,3,5,6,7,10,14,15}. Moreover, this is the minimal set with this property. In 2005, B.M. Kim, M.-H. Kim and B.-K. Oh showed that such a finite criterion set exists in a much general setting, but the uniqueness of the criterion set is lost. Since then, the question of uniqueness for particular situations has been studied by several authors.

We will discuss the analogous questions for totally positive definite quadratic forms over totally real number fields. Here again, the existence of criterion sets for universality is known, and Lee determined the set for Q(sqrt5). We will show the uniqueness and a strong connection with indecomposable integers. A part of our uniqueness result is (to our best knowledge) new even over Z. This is joint work with G. Romeo and V. Kala.

Zoom link: https://uwaterloo.zoom.us/j/98937322498?pwd=a3RpZUhxTkd6LzFXTmcwdTBCMWs0QT09