Schedule for: 21w2505 - Alberta Number Theory Days XIII

Note: the Lecture rooms are available after 16:00.

\begin{align*} &\text {Come and join, meet people, chat, talk math, play games etc.} \\ &\text{Icebreakers:} \\ &\text{1. What is your name and your preferred pronouns?} \\ &\text{2. Where did you get your last degree?} \\ &\text{3. Where are you studying/working now? with whom (if applicable)?} \\ &\text{4. What is your topic of interest (general theme)?} \\ &\text{5. Have you participated to ANTD before? If so when?} \\ &\text{6. Beside math, what else you are in interested in? } \\ &\text{7. Tell us about an unusual thing about you (skill/hobby) or what is the thing you are the most proud of?} \\ &\text{8. Tell us something interesting about a book you have read or a show/movie you have watched?} \\ &\text{9. Can you tell us something new you really want to do/try in the future? } \\ &\text{10. Show us an object/pet of importance to you (at reach in your room).} \\ &\text{11. Tell us the best place to eat your favorite food or the best place to hike?} \\ &\text{12. If applicable, tell us one positive thing that changed for you with Covid.} \end{align*}

A buffet breakfast is served daily between 7:00am and 9:00am in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Loop groups are certain infinite-dimensional versions of Lie groups. In this talk, we will try to motivate how the study of these groups and their representations can be useful for some concrete questions in (analytic, combinatorial) number theory. We will also turn things around and explain how some arithmetically motivated constructions on these groups can be used to shed light on questions in ‘algebraic’ representation theory.

Vertex Operator Algebras (VOAs) are the most recent attempt at mathematically rigorizing quantum field theory, however, these beasts are unruly and difficult to work with in practice. As a result, their category of representations, a distillation of a VOA’s core information, is often used for practical matters in its stead. Such categories have been proven to have remarkable structure and are modular tensor categories. This leads to the question asking if all such categories come from VOAs, this is conjectured to be so. Reconstructing a VOA from its category is currently a dauting, if not completely impractical, process. Multi-Vectored Jacobi Forms (MVJFs) are the tool I am currently developing to help expedite that process. This talk will be an introduction to these MVJFs, background, their definition, some examples, early results, and discussing potential future/ongoing work.

In a 2007 article, Takhtajan and Zograf proved a curvature formula for the determinant of an endomorphism bundle over a modular curve without elliptic points, relatively to a renormalization of the L2-metric using the first order derivative at 1 of the Selberg zeta function, which they called the Quillen metric. However, such a metric should be more than just smooth in family, as it should fit naturally in a Riemann-Roch type theorem, like the one proved by Deligne in 1987. It does not here in general. In this talk, we will see the main ideas required to get a Deligne-Riemann-Roch isometry for certain flat unitary vector bundles over modular curves. The resulting Quillen metric, as a renormalization of the L2-metric by an explicit factor involving the first non-zero derivative of the Selberg zeta function, will be compatible with the work of Takhtajan and Zograf.

In this presentation, I will talk about the representing integers by quadratic forms. The formulas for the number of representations will be obtained as a sum of an Eisenstein part and a cusp part. I will discuss two separate cases. The first one will be the study of the problem for binary quadratic forms of discriminant $-D<0$, where the number field $\mathbb{Q}(\sqrt{-D})$ has class number 3, and in the second one, I will discuss the problem of representing integers as a sum of $k$ triangular numbers, denoted by $\delta_{k}(n)$, for even values of $k$.

A buffet lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

Use one of the five Gather Town meeting rooms to continue a conversation (there are white boards there!), meet speakers and ask them questions you did not have the time to ask after their talk.

Let $k\geq 2$ be a square-free integer. In a joint work with Z. S. Aygin, we prove that the number of square-free integers $m\in [1,N]$ such that $(k,m)=1$ and $\mathbb{Q}(\sqrt[3]{k^2m})$ is monogenic is $\gg N^{1/3}$ and $\ll N/(\log N)^{1/3-\epsilon}$ for any $\epsilon>0$. Assuming ABC, the upper bound can be improved to $O(N^{(1/3)+\epsilon})$. Let $F$ be the finite field of order $q$ with $(q,3)=1$ and let $g(t)\in F[t]$ be non-constant square-free. We prove unconditionally the analogous result that the number of square-free $h(t)\in F[t]$ such that $\deg(h)\leq N$, $(g,h)=1$ and $F(t,\sqrt[3]{g^2h})$ is monogenic is $\gg q^{N/3}$ and $\ll N^2q^{N/3}$.

Let $N(T)$ denote the number of non-trivial zeros $\rho$, with $0<\Im(\rho)\le T$, of the Riemann zeta function $\zeta(s)$. We give an explicit estimate for $N(T)$. Namely, for $T\geq e$, we show that $$\Big| N (T) - \frac{T}{2\pi} \log \Big( \frac{T}{2\pi e}\Big)\Big| \le 0.1038\log T +0.2573\log\log T +9.3675.$$ This improves the previous result of Trudgian for sufficiently large $T$. The improvement comes from the use of various improved subconvexity bounds and ideas from the work of Bennett et al. on counting zeros of Dirichlet $L$-functions. This talk is based on the joint work with Quanli Shen and Peng-Jie Wong.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In 1976, Victor Guillemin published a paper discussing geometric scattering theory, in which he related the Lax-Phillips Scattering matrices (associated to a noncompact hyperbolic surface with cusps) and the sojourn times associated to a set of geodesics which run to infinity in either direction. Later, the work of Guillemin was generalized to locally symmetric spaces by Lizhen Ji and Maciej Zworski. In the case of a $\mathbb{Q}$-rank one locally symmetric space $\Gamma \backslash X$, they constructed a class of scattering geodesics which move to infinity in both directions and are distance minimizing near both infinities. An associated sojourn time was defined for such a scattering geodesic, which is the time it spends in a fixed compact region. One of their main results was that the frequencies of oscillation coming from the singularities of the Fourier transforms of scattering matrices on $\Gamma \backslash X$ occur at sojourn times of scattering geodesics on the locally symmetric space. In this talk I will review the work of Guillemin, Ji and Zworski as well as discuss the work from my doctoral dissertation on analogous results for higher rank locally symmetric spaces. In particular, I will describe higher dimensional analogues of scattering geodesics called \textbf{Scattering Flat} and study these flats in the case of the locally symmetric space given by the quotient $SL(3,\mathbb{Z}) \backslash SL(3,\mathbb{R}) / SO(3)$. A parametrization space is discussed for such scattering flats as well as an associated vector valued parameter (bearing similarities to sojourn times) called \textbf{sojourn vector} and these are related to the frequency of oscillations of the associated scattering matrices coming from the minimal parabolic subgroups of $\text{SL}(3,\mathbb{R})$. The key technique is the factorization of higher rank scattering matrices.

In 1927, Emil Artin proposed a conjecture for the density of primes $p$ for which a given integer $a$ is a primitive root modulo $p$, i.e., $\langle a \bmod p\rangle=\mathbb{F}_p^{\times}$. Let $a$ be a non-zero integer that is not $\pm1$. Then, the integer $a$ is a primitive root modulo prime $p$ for $p\nmid2a$ if and only if $p$ does not split completely in $K_q=\mathbb{Q}(\zeta_q,\sqrt[q]{a})$ for all primes $q \mid p-1$. Let $e$ be the largest integer for which $a$ is a perfect $e$-th power. The initial version of Artin's conjecture states that for a given integer $a$ ($\neq0,\;\pm1$), the density of primes $p$ such that $a$ is a primitive root modulo $p$ is \[ A_a=\displaystyle\prod_{\substack{q \: \text{prime} \\ q\mid e}}\left(1-\frac{1}{q-1}\right)\displaystyle\prod_{\substack{q \: \text{prime} \\ q\nmid e}}\left(1-\frac{1}{q(q-1))}\right).\] In 1957, computer calculations of the density for various values of $a$, by D. H. Lehmer and E. Lehmer revealed some discrepancies from the conjectured value $A$. The reason for these inconsistencies is the dependency between the splitting conditions in Kummer fields $K_q$'s. To deal with these dependencies, Artin introduced a correction factor. An entanglement correction factor appears when $a_{sf}\equiv1\bmod{4}$, where $a_{sf}$ is the square-free part of $a$. Hence, for the corrected conjectured density $\delta_a$, we have \[\delta_a= \begin{cases} A_a &\quad\quad\text{if }a_{sf}\not\equiv 1\bmod{4},\\ A_a\cdot E_a&\quad\quad\text{if }a_{sf}\equiv1\bmod{4} \end{cases} \] where \[ E_a=1-\mu(|a_{sf}|)\displaystyle\prod_{\substack{q \mid e\\ q\mid a_{sf}} }\frac{1}{q-2}\displaystyle\prod_{\substack{q \nmid e\\ q\mid a_{sf}} }\frac{1}{q^2-q-1}. \] Here $\mu(.)$ is the M\"obius function. The modified conjecture was proved by Hooley in 1967 under the assumption of the Generalized Riemann Hypothesis. In 2014, Lenstra, Moree, and Stevenhagen introduced a method involving character sums to deduce the formula for the product in the density for Artin's conjecture. The method applies in similar problems such as the density of primes of cyclic reduction for Serre curves. In this talk, we introduce a generalization of this method which yields product expressions for a large family of Artin's type problems in which densities can be stated by summations involving the orders of certain finite groups. As a consequence, the product expressions of some Artin's type problems, such as the Titchmarsh Divisor Problem in Kummer families, are computed.

Use one of the five Gather Town meeting rooms to continue a conversation (there are white boards there!), meet speakers and ask them questions you did not have the time to ask after their talk.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

Come and join, meet people, chat, talk math, play games etc.

For over 100 years, $I_k(T)$, the $2k$-th moments of the Riemann zeta function on the critical line, have been extensively studied. In 1918 Hardy-Littlewood established an asymptotic formula for the second moment ($k=1$) and in 1926 Ingham established an asymptotic formula for the fourth moment ($k=2$). Since then, no other moments have been asymptotically evaluated. In the late 1990's Keating and Snaith gave a conjecture for the size of $I_k(T)$ based on a random matrix model. In this talk I will give a historical overview of the advances on $I_k(T)$ and the techniques used to study them since the beginning of the twentieth century.

I will discuss recent work completed by JC Saunders, Dang Khoa Nguyen and myself where we resolve the algebraicity problem for the Artin-Mazur zeta function of all surjective endomorphisms of d-dimensional positive characteristic tori.

2-day workshop participants are welcome to use BIRS facilities (TCPL) until 15:00 on Sunday, although participants are still required to checkout of the guest rooms by 11 M. There is no coffee break service on Sunday afternoon, but self-serve coffee and tea are always available in the 2nd floor lounge, Corbett Hall.

In 2009, Florian Luca and Florin Nicolae proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are $1,2$, and $3$. In the first half of the talk we give a survey of such results on Diophantine equations involving the Euler totient function and binary recurrence sequences. For instance, in 2015, Bernadette Faye and Florian Luca proved that the only Pell numbers whose Euler totient function is another Pell number are $1$ and $2$. In the second half of the talk, we provide an outline of a proof for a result that generalises the results of Luca, Nicolae, and Faye. More specifically, for any fixed natural number $P\geq 3$, if we define the sequence $\left(u_n\right)_n$ 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 $\varphi\left(u_n\right)=u_m$ is $\varphi\left(u_1\right)=\varphi(1)=1=u_1$.

In classical q-series studies there are examples of eta quotients whose derivatives are also eta quotients. The most famous examples can be found in works of S. Ramanujan and N. Fine. In 2019, in a joint work with P. C. Toh, we have given 203 pairs of such eta quotients, which we believe to be the complete list (see "When is the derivative of an eta quotient another eta quotient?", J. Math. Anal. Appl. 480 (2019) 123366). Around the same time, D. Choi, B. Kim and S. Lim have given a complete list of such eta quotients with squarefree levels (see "Pairs of eta-quotients with dual weights and their applications", Adv. Math. 355 (2019) 106779). Their findings support the idea that our list is complete. In this talk we present a beautiful interplay between eta quotients, their derivatives and Eisenstein series. Then we use this interplay to extend the results of latter paper to prime power levels. This is a joint work with A. Akbary.

Use one of the five Gather Town meeting rooms to continue a conversation (there are white boards there!), meet speakers and ask them questions you did not have the time to ask after their talk.