Department of Mathematics and Computer Science Number Theory and Combinatorics Seminar Spring 2016 Talks are at noon on Monday in room 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 .
 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 Jan 11 everyone Open problem session at noon in UHall 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. Jan 25 Francesco Pappalardi On never primitive points on elliptic curves at noon in UHall C630 (Universit Roma Tre) The Lang-Trotter Conjecture for primitive points predicts an expression for the density of primes $p$ for which a fixed rational point (not torsion) of a fixed elliptic curve defined on $\mathbb{Q}$ is a generator of the curve reduced modulo $p$. After providing the definition of such a density in terms of Galois representations associated with torsion points of the curve, we will tell the short story of the contributions to the conjecture and provide examples of families of elliptic curves for which the conjecture holds for trivial reasons. This is the notion of "never primitive point." The case of elliptic curves in complex multiplication will be discussed in greater detail. Part of the work is in collaboration of N. Jones. Jan 27 Francesco Pappalardi The distribution of multiplicatively dependent vectors Wednesday at 10am in in UHall C630 (Universit Roma Tre) Let $n$ be a positive integer, $G$ be a group and let $\mathbf{\nu}=(\nu_1,\dots,\nu_n)$ be in $G^n.$ We say that $\mathbf{\nu}$ is a multiplicatively dependent $n$-tuple if there is a non-zero vector $(k_1,\dots,k_n)$ in $\mathbb{Z}^n$ for which $\nu^{k_1}_1\cdots \nu^{k_n}_n=1.$ Given a finite extension $K$ of $\mathbb Q$, we denote by $M_{n,K}(H)$ the number of multiplicatively dependent $n$-tuples of algebraic integers of $K^*$ of naive height at most $H$ and we denote by $M^*_{n,K}(H)$ the number of multiplicatively dependent $n$-tuples of algebraic numbers of $K^*$ of height at most $H.$ In this seminar we discuss several estimates and asymptotic formulas for $M_{n,K}(H)$ and for $M^*_{n,K}(H)$ as $H\rightarrow\infty$. For each $\nu$ in $(K^*)^n$ we define $m,$ the multiplicative rank of $\nu,$ in the following way. If $\nu$ has a coordinate which is a root of unity we put $m=1.$ Otherwise let $m$ be the largest integer with $2\leq m\leq n+1$ for which every set of $m-1$ of the coordinates of $\nu$ is a multiplicatively independent set. We also consider the sets $M_{n,K,m}(H)$ and $M^*_{n,K,m}(H)$ defined as the number of multiplicatively dependent $n$-tuples of multiplicative rank $m$ whose coordinates are algebraic integers from $K^*,$ respectively algebraic numbers from $K^*,$ of naive height at most $H$ and will consider similar questions for them. Feb 1 Micah Milinovich Fourier Analysis and the zeros of the Riemann zeta-function at noon in UHall C630 (University of Mississippi) I will show how the classical Beurling-Selberg extremal problem in harmonic analysis arises naturally when studying the vertical distribution of the zeros of the Riemann zeta-function and other L-functions. Using this relationship, along with techniques from Fourier analysis and reproducing kernel Hilbert spaces, we can prove the sharpest known bounds for the number of zeros in an interval on the critical line and we can also study the pair correlation of zeros. Our results on pair correlation extend earlier work of P. X. Gallagher and give some evidence for the well-known conjecture of H. L. Montgomery. This talk is based on a series of papers which are joint with E. Carneiro, V. Chandee, and F. Littmann. Feb 8 Alexey Popov Operator Algebras with reduction properties at noon in UHall C630 An algebra is a vector space with a well-defined multiplication. An operator algebra is an algebra of operators acting on a Hilbert space, typically assumed closed in the norm topology. An easy example of an operator algebra is the algebra $M_n(\mathbb{C})$ of all the complex $n \times n$ matrices. In this colloquium-style talk, we will discuss operator algebras $A$ with the following property: every $A$-invariant subspace is complemented by another $A$-invariant subspace. This property is called the Reduction property and is a kind of semisimplicity. We will discuss the connections of this property to some classical problems, such as Kadison Similarity Problem and the structure of amenable operator algebras. Feb 22 Nathan Ng Linear combinations of zeros of L-functions at noon in UHall C630 The linear independence conjecture asserts that the multiset of positive ordinates of the zeros of automorphic L-functions is linearly independent over the field of rational numbers. This deep conjecture implies that if $1/2+i \gamma$ is a zero of the Riemann zeta function, then $1/2+2i \gamma$ is a not a zero of the zeta function. I will show that on the Riemann hypothesis this is true infinitely often. I will also discuss variants of this phenomenon. This is joint work with Greg Martin and Micah Milinovich. Feb 29 Rob Craigen Survey of Negacyclic Weighing Matrices at noon in UHall C630 (University of Manitoba) A square or rectangular matrix is circulant if every row after the first is a right circular shift of its predecessor. Negacyclic matrices are defined the same way except that the first entry of each row is negated after circulating the preceding row. A partial Hadamard matrix is a rectangular $k \times n$ $(1,-1)$-matrix M satisfying $MM^T = nI$. In the summer of 2013 I hired four sharp undergraduate students to tackle a problem about circulant partial Hadamard matrices. The question of existence of certain negacyclic weighing matrices kept coming up, so we devoted some energy to exploring this largely uncultivated territory. In the end we produced, apparently for the first time, a fairly comprehensive survey of these objects, their structure, why certain classes exist and others cannot. The flavour of the existence questions for this class of weighing matrices is decidedly different from that of group-developed form, even though much of the theory is the same. We discuss some situations in which negacyclic weighing matrices naturally appear, and conclude with some tantalizing new open questions arising from the work. Mar 7 Alia Hamieh Determining Hilbert modular forms by the central values of Rankin-Selberg convolutions at noon in UHall C630 In this talk, we give a brief overview of adelic Hilbert modular forms. Then, we show that the central values of the Rankin-Selberg convolutions, $L(g\otimes f, s)$, uniquely determine an adelic Hilbert modular form $g$, where $f$ varies in a carefully chosen infinite family of adelic Hilbert modular forms. We prove our results in both the level and weight aspects. This is a joint work with Naomi Tanabe. Mar 14 Joy Morris Automorphisms of circulant graphs at noon in UHall C630 Determining the full automorphism group of a graph is a hard problem with a long history. I will discuss some of the major results that involve finding graphs with a given automorphism group. I will then focus on circulant graphs, and describe some structural results and algorithms that help us determine the full automorphism group of the graph. I will also give some asymptotic results about how many circulant graphs fall into different categories. Mar 21 Arnab Bose Investigations on some Exponential Congruences at noon in UHall C630 Around 1981, Selfridge asked for what positive integers $a$ and $b$ does $2^a -2^b$ divide $n^a - n^b$ for all $n \in \mathbb{N}$. The problem was independently solved by various people in different contexts. In this talk, we study their ideas and prove a generalization of the problem, in the elementary number theoretic sense and also in algebraic number fields. Further, we develop ideas to give a conditional resolution and generalizations to another problem by H.Ruderman which is closely related to Selfridge’s problem. Apr 4 Brandon Fuller CCA groups and graphs at noon in UHall C630 An automorphism of a Cayley graph that preserves its natural edge-colouring is called colour-preserving. We study groups $G$ with the property that every automorphism on every connected Cayley graph on $G$ is the composition of a left-translation and a group automorphism. We call this class of groups CCA groups and we look at classifying which groups are not CCA. More precisely, we look at abelian groups, groups of odd order and direct or semidirect products of groups. Apr 11 Asif Zaman The least prime ideal in the Chebotarev Density Theorem at noon in UHall C630 (University of Toronto) In 1944, Linnik famously showed unconditionally that the least prime in an arithmetic progression $a \pmod{q}$ with $(a,q) = 1$ is bounded by $q^L$ for some absolute effective constant $L > 0$, known as “Linnik’s constant”. Many authors have computed explicit admissible values of $L$ with the current world record at $L = 5$ by Xylouris (2011), refining techniques of Heath-Brown (1992). We consider a broad generalization of this problem in the Chebotarev Density Theorem (CDT), which is concerned with the splitting behaviour of prime ideals in number fields. Namely, what is the least norm of a prime ideal occurring in CDT? Papers of Lagarias-Montgomery-Odlyzko (1979) and Weiss (1983) give different unconditional field-uniform bounds but without any explicit exponents analogous to the subsequent work on Linnik’s constant. I will report on our recent work establishing such explicit estimates along with some applications related to primes represented by binary integral quadratic forms and congruences for Fourier coefficients of cuspidal Hecke eigenforms. This is joint work with Jesse Thorner. June 17 Ram Murty Twin Primes at noon in C630 (Queen's University) We will discuss recent progress towards the twin prime conjecture as well as highlight some recent joint work with Akshaa Vatwani the connects the parity problem with the twin prime conjecture. The talk will be accessible to a wide audience. June 28 Tim Trudgian Grosswald's conjecture on primitive roots at noon in C630 (Australian National University) Very little is known about the distribution of primitive roots of a prime $p$. Grosswald conjectured that the least primitive root of a prime p is less than $\sqrt{p} - 2$ for all $p> 409$. While this is certainly true for all $p$ sufficiently large, Grosswald's conjecture in still open. I shall outline some recent work which resolves the conjecture completely under the Generalised Riemann Hypothesis and which almost resolves the conjecture unconditionally. June 28 Vijay Patankar Pairs of elliptic curves and their Frobenius fields at 2pm in C630 (Jawaharlal Nehru University) Given an elliptic curve $E$ over a number field $K$. The Frobenius field attached to $E$ at a prime $p$ is the splitting field of the characteristic polynomial of the Frobenius endomorphism acting on the $\ell$-adic Tate module of $E$ ($\ell$ a prime different from $p$) over the rationals. Thus, the splitting field is either of degree $1$ or degree $2$ over the rationals. Let $E_1$ and $E_2$ be elliptic curves defined over a number field $K$, with at least one of them without complex multiplication. We prove that the set of places $v$ of $K$ of good reduction such that the corresponding Frobenius fields are equal has positive upper density if and only if $E_1$ and $E_2$ are isogenous over some extension of $K$. For an elliptic curve $E$ defined over a number field $K$, we show that the set of finite places of $K$ such that the Frobenius field at $v$ equals a fixed imaginary quadratic field $F$ has positive upper density if and only if $E$ has complex multiplication by $F$. Time permits we will provide a sketch of a result about two dimensional $\ell$-adic Galois representations that we will need using an algebraic density theorem due to Rajan.
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