Department of Mathematics and Computer Science Number Theory and Combinatorics Seminar Fall 2017 Talks are at noon on Monday in C630 of University Hall For more information, or to receive an email announcement of each week's seminar, contact Nathan Ng < ng AT cs DOT uleth DOT ca > or Dave Morris .
 The next talk: Oct 2 at noon in C630 Andrew Fiori The average number of quadratic Frobenius pseudoprimes Primality testing has a number of important applications. In particular in cryptographic applications the complexity of existing deterministic algorithms causes increasing latency as the size of numbers we must test grow and the number of tests we must run before finding a prime grows aswell. These observations lead one to consider potentially non-deterministic algorithms which are faster, and consequently leads one to consider the false positives these algorithms yield, which we call pseudoprimes. In this talk I will discuss my recent work with Andrew Shallue where we study Quadratic Frobenius Pseudoprimes. I shall describe our results on an asymptotic lower bounds on the number of false positives. These results represent a generalization of those Erdos-Pomerance concerning similar problems for (Fermat) pseudoprimes.
 Talks in the series this semester: (Click on any title for more info, including the abstract. Then click on it again to hide the info.)

 Date Speaker Title Sep 11 everyone Open problem session at noon in C630 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. Sep 18 Peng-Jie Wong Nearly supersolvable groups and Artin's conjecture at noon in C630 (University of Lethbridge) Let $K/k$ be a Galois extension of number fields with Galois group $G$, and let $\rho$ be a non-trivial irreducible representation of $G$ of dimension $n$. The Artin holomorphy conjecture asserts that the Artin $L$-function attached to $\rho$ extends to an entire function. It is well-known that when $n=1$, this conjecture follows from Artin reciprocity. Also, by the works of Langlands and many others, we know that this conjecture is valid for $n=2$ under certain conditions. However, in general, the Artin holomorphy conjecture is wildly open. In this talk, we will discuss how elementary group theory plays a role in studying the Artin holomorphy conjecture and introduce the notion of "nearly supersolvable groups". If time allows, we will explain how such groups lead to a proof of the Artin holomorphy conjecture for Galois extensions of degree less than 60. Sep 25 Muhammad Khan The contact graphs of totally separable packings at noon in C630 (University of Lethbridge) Contact graphs have emerged as an important tool in the study of translative packings of convex bodies and have found numerous applications in materials science. The contact number of a packing of translates of a convex body is the number of edges in the contact graph of the packing, while the Hadwiger number of a convex body is the maximum vertex degree over all such contact graphs. In this talk, we investigate the Hadwiger and contact numbers of totally separable packings of convex bodies, known as the separable Hadwiger number and the separable contact number, respectively. We show that the separable Hadwiger number of any smooth strictly convex body in dimensions $d = 2, 3, 4$ is $2d$ and the maximum separable contact number of any packing of $n$ translates of a smooth strictly convex domain is $\lfloor 2n - 2\sqrt{n} \rfloor$. Our proofs employ a characterization of total separability in terms of hemispherical caps on the boundary of a smooth convex body, Auerbach bases of finite dimensional real normed spaces, angle measures in real normed planes, minimal perimeter polyominoes and an approximation of smooth $o$-symmetric strictly convex domains by, what we call, Auerbach domains. This is joint work with K. Bezdek (Calgary) and M. Oliwa (Calgary). Oct 2 Andrew Fiori The average number of quadratic Frobenius pseudoprimes at noon in C630 (University of Lethbridge) Primality testing has a number of important applications. In particular in cryptographic applications the complexity of existing deterministic algorithms causes increasing latency as the size of numbers we must test grow and the number of tests we must run before finding a prime grows aswell. These observations lead one to consider potentially non-deterministic algorithms which are faster, and consequently leads one to consider the false positives these algorithms yield, which we call pseudoprimes. In this talk I will discuss my recent work with Andrew Shallue where we study Quadratic Frobenius Pseudoprimes. I shall describe our results on an asymptotic lower bounds on the number of false positives. These results represent a generalization of those Erdos-Pomerance concerning similar problems for (Fermat) pseudoprimes. Oct 16 Lee Troupe Title TBA at noon in C630 (University of British Columbia) Abstract TBA Oct 23 Sam Broadbent, Kirsten Wilk, and Habiba Kadiri Title TBA at noon in C630 (University of Lethbridge) Abstract TBA Oct 30 Akshaa Vatwani Title TBA at noon in C630 (University of Waterloo) Abstract TBA Nov 6 Forrest Francis Title TBA at noon in C630 (University of Lethbridge) Abstract TBA Nov 20 Kirsty Chalker Title TBA at noon in C630 (University of Lethbridge) Abstract TBA Nov 27 Sara Sasani Title TBA at noon in C630 (University of Lethbridge) Abstract TBA Dec 4 Joy Morris Title TBA at noon in C630 (University of Lethbridge) Abstract TBA
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