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 .
 The next talk: Feb 22 at noon in UHall C630 Nathan Ng Title TBA Abstract TBA

 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 (Università 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 (Università 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 Title TBA at noon in UHall C630 Abstract TBA Feb 29 Rob Craigen Title TBA at noon in UHall C630 (University of Manitoba) Abstract TBA Mar 7 Alia Hamieh Title TBA at noon in UHall C630 Abstract TBA Mar 14 Joy Morris Title TBA at noon in UHall C630 Abstract TBA Mar 21 Arnab Bose Title TBA at noon in UHall C630 Abstract TBA Apr 4 Brandon Fuller Title TBA at noon in UHall C630 Abstract TBA Apr 11 Asif Zaman Title TBA at noon in UHall C630 (University of Toronto) Abstract TBA Apr 18 Speaker TBA Title TBA at noon in UHall C630 Abstract TBA
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