at noon
in UHall D632

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.

at noon
in UHall B650

By combining Green and Tao’s transference principle with a density increment argument, we show if \(A\) is a set of prime numbers satisfying $$ \sum_{a \in A} \frac{1}{a} = \infty ,$$ then \(A\) must contain a 3 term arithmetic progression. The previous methods of Helfgott and De Roton, and the speaker, used the transference principle to move from a set of primes to a dense subset of integers by considering the \(L^2\) and \(L^p\)-norms of a smoothed version of the indicator function of \(A\). Instead, we work directly with the \(L^\infty\)-norm, and exploit the structure of \(A\) to obtain increased density on a large subprogression when \(A\) contains no arithmetic progressions. By iterating we show that for any constant \(B > 0\), if \(A\) is a subset of primes contained in \(\{ 1,\ldots,N\}\) with size at least $$ |A| \gg_B \frac{N}{ (\log N) (\log \log N)^B} ,$$ then \(A\) contains a three term arithmetic progression.

at noon
in UHall B650

In this talk we introduce recursive algorithms for generation of three families of plane graphs. These algorithms start with small graphs and iteratively convert them to larger graphs. The families are k-angulations (plane graphs with whose faces are of size k), plane graphs with a given face size sequence and planar hypohamiltonian graphs. Hypohamiltonian graphs have the property that removing each vertex from the graph, there is a Hamiltonian cycle through all remaining vertices while the original graph does not have any such cycle. One of the problems in the literature since 1976 is to find the smallest planar hypohamiltonian graphs. The previous record by Weiner and Araya was a planar graph with 42 vertices. We improve this record by finding 25 planar hypohamiltonian graphs on 40 vertices while discovering many larger ones on 42 and 43 vertices.

We also introduce a very fast method for canonical embedding and isomorphism rejection of plane graphs. Most graph generators like plantri generate graphs isomorph-free up to isomorphism of the embedding, however our method does the isomorphism checking up to the underlining graph while taking advantage of the planarity and embeddings to speed up the computation.

at noon
in UHall B650

Standard numerical methods focus on algorithms for matrices over the real and complex numbers: finding singular value decompositions, eigenvalues and eigenvectors, QR and LU decompositions. Many of the same questions make sense for non-archimedian local fields, and some methods carry over. Moreover, the ultrametric properties of p-adic arithmetic allow for much better tracking of precision than in the real and complex cases. This is a report on early stages of joint work with Xavier Caruso, Tristan Vaccon, Jen Balakrishnan and Kiran Kedlaya.

at noon
in UHall B650

(University of Lethbridge)

A Cayley graph \(\Gamma={\rm Cay}(G;S)\) on a group \(G\) with connection set \(S\), is a graph whose vertices are labelled with the elements of \(G\), with vertices \(g_1\) and \(g_2\) adjacent if \(g_1^{-1}g_2 \in S\). We say that an automorphism \(\alpha\) of \(\Gamma\) respects the partition \(\mathcal C\) of the edge set of \(\Gamma\) if for every \(C \in \mathcal C\), we have \(\alpha(C) \in \mathcal C\). I will discuss some obvious partitions of the edge set of a Cayley graph \(\Gamma\), and find conditions under which a graph automorphism of \(\Gamma\) that respects these partitions and fixes a vertex, must be an automorphism of the group \(G\).

at noon
in UHall B650

This talk will retrace the main steps of the modern theory of prime numbers and in particular how the combinatorial sieve combined with the Dirichlet series theory to give birth to the modern representation of the primes via a linear combination of terms, some of which being "linear", while the other ones are "bilinear". This will lead us to the recent developments of Green & Tao, Mauduit & Rivat, Tao, Helfgott, and Bourgain, Sarnak & Ziegler.

at noon
in UHall B650

While the two fields named in the title seem unrelated, there is a strong link between them. Indeed, random matrix theory makes predictions in both fields, and some of these predictions can be verified rigorously on the number theory side. This amazing connection came to life during a meeting between Freeman Dyson and Hugh Montgomery at the Institute for Advanced Study. Random matrices are now known to predict many statistics about zeta functions, such as moments, low-lying zeros and correlations between zeros. The goal of this talk is to discuss this connection, focusing on number theory. We will cover both basic facts about the zeta functions and recent developments in this active area of research.

at noon
in UHall B650

(University of Lethbridge)

We will describe the celebrated ABC conjecture, due to Oesterle and Masser, formulated in 1985. We will explain one of the motivations behind this conjecture and discuss a refinement of this conjecture put forward by Baker in 1996.

at noon
in UHall B650

(University of Lethbridge)

Let \((G,\odot)\) be a group of order \(g\). A \((g,k;\lambda)\)- difference matrix is a \(k \times g\lambda\) matrix \(D = (d_{ij})\) with entries from \(G\), so that for each \(1 \leq i < j \leq k\), the multiset $$ \{ d_{i\ell} \odot d_{j\ell}^{-1}: 1 \leq \ell \leq g\lambda \} $$ (the difference list) contains every element of \(G\) \(\lambda\) times.

A generalized Hadamard matrix \(\mathrm{GH}(g,\lambda)\) is a \((g,g\lambda;\lambda)\)-difference matrix.

Some interesting examples of difference matrices and generalized Hadamard matrices with emphasis on their applications will be discussed.

at noon
in UHall B650

In 1983, Marušič posed the problem of determining which positive integers \(n\) have the property that every vertex-transitive graph of order \(n\) is isomorphic to a Cayley graph of some group. Such an integer \(n\) is called a Cayley number. Much work on this problem has been done, and, for example, it is known exactly which integers divisible by a square are Cayley numbers. These are \(p^2\), \(p^3\), and \(12\). Additionally, a fair amount is known via constructions about which square-free integers are not Cayley numbers. Much less is known about which square-free integers are Cayley numbers, and it is not even known if there is a Cayley number that is a product of five distinct primes. We answer a question posed by C. Praeger who asked if there was a Cayley number of order \(n\) where \(n\) has \(k\) distinct prime factors for every positive integer \(k\). We construct an infinite set of distinct prime numbers \(S\) with the property that the product of any \(k\) elements of \(S\) is always a Cayley number. This is joint work with Pablo Spiga.

at noon
in UHall B650

(University of Lethbridge)

Let \(a,q\) be relatively prime integers. Then consider $$ \psi (x; q, a) = \sum_{ \begin{matrix} n \le x \\ {n \,\equiv\, a} \ (\mathrm{mod}\, {q}) \end{matrix} } \Lambda(n) .$$ We discuss new explicit bounds for \(\psi(x; q, a)\), which provide an extension and improvement over the bounds given in the previous work of Ramaré and Rumely. This article introduces two new ideas. We smooth the prime counting function and use the partial verification of GRH by Platt along with an explicit zero-free region given by Kadiri. This is joint work with Habiba Kadiri.

at noon
in UHall B650

(University of Lethbridge)

Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\). A pair \((p,q)\) of distinct prime numbers is called an amicable pair of \(E\) if \(E\) has good reduction at \(p\) and \(q\) and \(\#{E}_{p}(\mathbb{F}_{p})=q\) and \(\#{E}_{q}(\mathbb{F}_q)=p\). In this talk we show a non-trivial upper bound for the number of amicable pairs on average over a family of elliptic curves.

at noon
in UHall B650

(University of Lethbridge)

‘Modular curves are the parameter space of elliptic curves with certain torsion structure.’ In this talk, I will explain what that sentence means, and how it is related to the usual definition of modular curves in terms of quotients of the complex upper half plane with certain matrix groups.