Department of Mathematics and Computer Science Number Theory and Combinatorics Seminar Spring/Summer 2020 Until further notice, all talks this summer will be given online For more information, contact Nathan Ng or Dave Morris . Talks are usually at noon on Monday. All times are Mountain Time.
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 Date Speaker Title Jan 13 everyone Open problem session at noon in B543 (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. Jan 20 Joy Morris Regular Representations of Groups at noon in C620 (University of Lethbridge) A natural way to understand groups visually is by examining objects on which the group has a natural permutation action. In fact, this is often the way we first show groups to undergraduate students: introducing the cyclic and dihedral groups as the groups of symmetries of polygons, logos, or designs. For example, the dihedral group $D_8$ of order $8$ is the group of symmetries of a square. However, there are some challenges with this particular example of visualisation, as many people struggle to understand how reflections and rotations interact as symmetries of a square. Every group $G$ admits a natural permutation action on the set of elements of $G$ (in fact, two): acting by right- (or left-) multiplication. (The action by right-multiplication is given by $\{\tau_g: g \in G\}$, where $\tau_g(h)=hg$ for every $h \in G$.) This action is called the right- (or left-) regular representation of $G$. It is straightforward to observe that this action is regular (that is, for any two elements of the underlying set, there is precisely one group element that maps one to the other). If it is possible to find an object that can be labelled with the elements of $G$ in such a way that the symmetries of the object are precisely the right-regular representation of $G$, then we call this object a regular representation of $G$. A Cayley (di)graph Cay$(G,S)$ on the group $G$ (with connection set $S \subset G$) is defined to have the set $G$ as its vertices, with an arc from $g$ to $sg$ for every $s \in S$. It is straightforward to see that the right-regular representation of $G$ is a subset of the automorphism group of this (di)graph. However, it is often not at all obvious whether or not Cay$(G,S)$ admits additional automorphisms. For example, Cay$(\mathbb Z_4, \{1,3\})$ is a square, and therefore has $D_8$ rather than $\mathbb Z_4$ as its full automorphism group, so is not a regular representation of $\mathbb Z_4$. Nonetheless, since a regular representation that is a (di)graph must always be a Cayley (di)graph, we study these to determine when regular representations of groups are possible. I will present results about which groups admit graphs, digraphs, and oriented graphs as regular representations, and how common it is for an arbitrary Cayley digraph to be a regular representation. Jan 27 Habiba Kadiri Explicit results about primes in Chebotarev's density theorem at noon in W561 (University of Lethbridge) Let $L/K$ be a Galois extension of number fields with Galois group $G$, and let $C \subset G$ be a conjugacy class. Attached to each unramified prime ideal $\mathfrak{p}$ in $\mathcal{O}_K$ is the Artin symbol $\sigma_{\mathfrak{p}}$, a conjugacy class in $G$. In 1922 Chebotarev established what is referred to his density theorem (CDT). It asserts that the number $\pi_C(x)$of such primes with $\sigma_{\mathfrak{p}}=C$ and norm $N_{\mathfrak{p}} \le x$ is asymptotically $\frac{|C|}{|G|}Li(x)$ as $x\to \infty$, where $Li(x)$ is the usual logarithmic integral. As such, CDT is a generalisation of both the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. In light of Linnik's result on the least prime in an arithmetic progression, one may ask for a bound for the least prime ideal whose Artin symbol equals $C$. In 1977 Lagarias and Odlyzko proved explicit versions of CDT and in 1979 Lagarias, Montgomery and Odlyzko gave bounds for the least prime ideal in the CDT. Since 2012 several unconditional explicit results of these theorems have appeared with contributions by Zaman, Zaman and Thorner, Ahn and Kwon, and Winckler. I will present several recent results we have proven with Das, Ng, and Wong. Feb 3 Selcuk Aygin On the eta quotients whose derivatives are also eta quotients at noon in C620 (University of Lethbridge) 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). Recently, 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 introduce a beautiful interplay between eta quotients, their derivatives and Eisenstein series. Then we share our work in progress (joint with A. Akbary) in proving the completeness of our list beyond squarefree levels. Feb 10 Amir Akbary Reciprocity Laws at noon in W561 (University of Lethbridge) The Artin reciprocity law provides a solution to Hilbert's ninth problem (9. Proof of the Most General Law of Reciprocity in any Number Field). In this talk we provide an exposition of this theorem with emphasis on its relation with the classical law of quadratic reciprocity and describe its motivating role in the far reaching Langlands Reciprocity Conjecture. Feb 24 Peng-Jie Wong Cyclicity of CM Elliptic Curves Modulo $p$ at noon in W561 (University of Lethbridge) Let $E$ be a CM elliptic curve defined over $\Bbb{Q}$. In light of the Lang-Trotter conjecture, there is a question asking for an asymptotic formula for the number of primes $p\le x$ for which the reduction modulo $p$ of $E$ is cyclic. This has been studied by Akbary, Cojocaru, Gupta, M.R. Murty, V.K. Murty, and Serre. In this talk, we will discuss their work and some variants of the question. Mar 2 Nathan Ng Moments of the Riemann zeta function and mean values of long Dirichlet polynomials at noon in W561 (University of Lethbridge) The $2k$-th moments $I_k(T)$ of the Riemann zeta function have been studied extensively. In the late 90's, Keating-Snaith gave a conjecture for the size of $I_k(T)$. At the same time Conrey-Gonek connected $I_k(T)$ to mean values of long Dirichlet polynomials with divisor coefficients. Recently this has been further developed by Conrey-Keating in a series of 5 articles. I will discuss my work relating $I_3(T)$ to smooth shifted ternary additive divisor sums and also recent joint work with Alia Hamieh on mean values of long Dirichlet polynomials with higher divisor coefficients. Mar 9 Nathan Ng The size of prime number error terms at noon in W561 (University of Lethbridge) Montgomery made a conjecture for the size of the error term in the prime number theorem. Similarly, Gonek made a conjecture for the size of the error term for the summatory function of the Mobius function. I will discuss these conjectures and the connection to large deviations of infinite sums of identically distributed random variables. This is joint work with Amir Akbary and Majid Shahabi. May 20 Daniel Fiorilli On the distribution of the error term in Chebotarev's density theorem and applications WED at noon (8pm in France) online (University of Ottawa and University of Paris - Saclay) We will discuss both extreme and generic values of the error term in Chebotarev's density theorem. This will allow us to deduce applications on a conjecture of K. Murty on the least unramified prime ideal in a given Frobenius set as well as on asymptotic properties of Chebyshev's bias. This is joint work with Florent Jouve, and builds on ideas of Rubinstein-Sarnak, Ng, K. Murty, R. Murty, and others. May 25 Olivier Ramaré Explicit Average Orders: News & Problems at noon online (Institut de Mathématiques de Marseille) We mix some of the novelties that have occurred recently in the field of explicit multiplicative number theory, together with some questions that have not been answered yet and with several new results. June 8 Forrest Francis Explicit Improvements to the Burgess Bound at 4pm online (UNSW Canberra, Australia) The standard estimate for the size of a (Dirichlet) character sum is the well-known Pólya–Vinogradov inequality. However, this inequality only uses the modulus of the character and does not consider the length of the sum. For a bound that does consider the length of the sum, we have a family of estimates known as the Burgess bound. Owing to the size of their effective ranges, the Burgess bound is sometimes considered a "better" estimate. Despite this, in an explicit setting, some improvements to the Pólya–Vinogradov inequality end up yielding improvements to the Burgess bound. In this talk, we will look at two such results. One involves the leading constant for the Burgess bound for characters of prime modulus, while the other makes improvements to the effective range of the Burgess bound for odd quadratic characters. The second topic is joint work with Matteo Bordignon. June 15 Ethan Lee Goldbach's Conjecture and Square-free Numbers at 4pm online (UNSW Canberra, Australia) We introduce the Goldbach conjecture and some heuristic evidence to support it. Goldbach's conjecture is hard, so we consider some relaxations of Goldbach's conjecture, which has lead to theorems by Helfgott and Dudek. Then, I give an account of the results and the main method used in joint work with F. Francis (submitted), which builds upon Dudek's result. Finally, we observe some conjectures, which would build upon the aforementioned results from myself and F. Francis. June 22 Asif Zaman An effective Chebotarev density theorem for fibres at noon online (University of Toronto) Let $K/k$ be a Galois extension of number fields. The Chebotarev density theorem asserts that, as $x$ tends to infinity, the primes of $k$ with norm at most $x$ equidistribute according to their splitting behaviour in $K$. How large must $x$ be to actually observe this equidistribution? Pierce, Turnage-Butterbaugh, and Wood have made remarkable progress on this central question for certain families of number fields. Their results rely on two key ingredients: (1) the Artin conjecture is known for every field in the family and (2) there are few common intermediate field extensions within the family. Dependence on (1) was removed in recent joint work with Thorner. I will discuss forthcoming work with Lemke Oliver and Thorner where we prove an unconditional result that allows both (1) and (2) to be bypassed in many cases of interest. For example, we prove that almost all degree $n$ $S_n$-extensions have GRH-quality bounds on the $\ell$-torsion subgroups of their class groups. June 29 Shabnam Akhtari How to count (cubic and quartic) binary forms at noon online (University of Oregon) In order to study interesting arithmetic, algebraic, analytic and geometric properties of binary forms, we often need a way to order them. I will discuss some natural ways to order binary forms F(x , y) with integer coefficients, especially those of degree 3 and 4. I will show some of my recent works as examples of the importance of understanding the invariant theory of integral binary forms in order to count these important arithmetic objects.
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