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.

at noon
in UHall C630

This talk will be in two parts. It will begin with a short discussion of recent developments in the Sage Mathematics Software and some of the implications these developments may have for research in number theory world-wide. The second portion of this talk will focus on the structure of multiquadratic number fields, what we do and do not know about their integral bases, and how this knowledge may extend to understanding other rings of integers.

at noon
in UHall C630

(University of Lethbridge)

In combinatorial geometry (and engineering), it is important to know that certain scaffold-like geometric structures are rigid. (They will not collapse, and, in fact, have enough bracing that they cannot be deformed at all.) Replacing the geometric structure with an algebraic structure (namely, a group) leads to the following question: given a homomorphism that is defined on the elements of a subgroup, is it possible to extrapolate the homomorphism to the rest of the elements of the group? It is fairly obvious that every additive homomorphism from the group $\mathbb{Z}$ of integers to the real line $\mathbb{R}$ can be extended to a homomorphism that is defined on all of $\mathbb{R}$, and we will see some other examples.

at noon
in UHall C630

(University of Lethbridge)

It a classical open problem in Operator Theory whether every bounded linear operator T on a Hilbert space H has a non-trivial invariant subspace (that is, a subspace Y of H such that TY is contained in Y; nontrivial means not {0} and not H). This is called the Invariant Subspace Problem; it is almost 100 years old.

In this talk we will show that any bounded operator on an infinite-dimensional Hilbert space admits a rank one perturbation which has an invariant subspace of infinite dimension and co-dimension. Moreover, the norm of the perturbation can be chosen as small as needed.

This is a joint work with Adi Tcaciuc.

at noon
in UHall C630

(University of Lethbridge)

In this talk, we discuss some results on the non-vanishing in $p$-adic families of the central values of certain Rankin-Selberg $L$-functions (namely, anticyclotomic twists of $L$-functions) associated to automorphic forms on GL(2).

at noon
in UHall C630

(University of Lethbridge)

The Riemann Hypothesis predicts that all zeros of the Riemann zeta function are located on the line $\Re(s)=\tfrac{1}{2}$. Also, we have that the number of zeros with imaginary parts located between $T$ and $2T$ is approximately $(2\pi)^{-1}T\log T$. Therefore the average gap is about $\displaystyle \frac{2\pi}{\log T}$. It has been conjectured that there are gaps that are smaller than $\displaystyle \frac{2\pi c}{\log T}$, for every $c>0$. This has been proven for $c$ slightly larger than $\tfrac{1}{2}$.

Proving that $c$ can be taken less than $\tfrac{1}{2}$ seems to be a very hard problem, despite being far from the conjecture. In this talk we discuss the connection between Chowla's conjecture on the shifted convolution sums of the Liouville's function and the size of $c$.

at noon
in UHall C630

(University of Lethbridge)

Let $h$ be a natural number and $f$ an arithmetic function. The autocorrelation of $f$ is the sum $$C_{f}(x,h) = \sum_{n \le x} f(n) f(n+h). $$ Such sums play an important role in analytic number theory. For instance, consider the classical arithmetic functions $\Lambda(n)$ (the von Mangoldt function), $\lambda(n)$ (Liouville's function), and $\tau_k(n)$ (the $k$-th divisor function). The sums $C_{\Lambda}(x,h)$, $C_{\lambda}(x,h)$, and $C_{\tau_k}(x,h)$ are related to the Twin Prime Conjecture, Chowla's conjecture, and to the moments of the Riemann zeta function, respectively. In this talk I will present a heuristic probabilistic method for deriving a conjecture for $C_{f}(x,h)$ in the case $f$ is a multiplicative function.

at noon
in UHall C630

(University of Lethbridge)

For a prime $p$, let $n(p)$ be the number of solutions $(x, y)$ of $y^2= x^3+ax+b$ over the finite field $\mathbb{F}_p$ and let $a(p)=p-n(p)$. In 1976, Serge Lang and Hale Trotter formulated a conjecture regarding the distribution of primes $p$ for which $a(p)=A$ for a fixed integer $A$. In this talk we give an exposition of this conjecture as it is given in the introduction of a paper of Katz (N. Katz, Lang-Trotter Revisited, Bulletin of the AMS, Vol. 46, No. 3, July 2009, pp. 413–457).

at noon
in UHall C630

For a field $F$ and a quadratic form $Q$ defined on an $n$-dimensional vector space $V$ over $F$, let $G_Q$, called the quadratic graph associated to $Q$, be the graph with the vertex set $V$ where vertices $v$, $w$ in $V$ form an edge if and only if $Q(v - w) = 1$. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger-Nelson problem. In the present talk, we will prove that for a local field $F$ of characteristic zero, the Borel chromatic number of $G_Q$ is infinite if and only if $Q$ represents zero non-trivially over $F$. The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, combined with an analysis of certain oscillatory integrals over local fields. As an application, we will also answer a variant of question 525 proposed in the 22nd British Combinatorics Conference 2009.

at noon
in UHall C630

(University of Lethbridge)

The prime number theorem in arithmetic progressions establishes that, for $a$ and $q$ fixed coprime integers, then $\psi(x;q,a)$ is asymptotic to $\displaystyle\frac{x}{\phi(q)}$ when $x$ is large. We discuss new explicit bounds for the error term $\displaystyle\left|\frac{\psi(x;q,a)-\frac{x}{\phi(q)}}{\frac{x}{\phi(q)}}\right|$, which provide an extension and improvement over the previous work of Ramaré and Rumely. Such results depend on the zeros of the Dirichlet $L$-functions: a numerical verification of the Generalized Riemann Hypothesis up to a given height and explicit zero-free regions. We use the latest results of respectively Platt and Kadiri. In addition our method makes use of smooth weights. This is joint work with Allysa Lumley.

at noon
in UHall C630

A Cayley graph Cay($G;S$) on a group $G$ with connection set $S=S^{-1}$ is the graph whose vertices are the elements of $G$, with $g\sim h$ if and only if $g^{-1}h \in S$. If we assign a colour $c(s)$ to each $s \in S$ so that $c(s)=c(s^{-1})$ and $c(s) \neq c(s’)$ when $s’ \neq s, s^{-1}$, this is a natural (but not proper) edge-colouring of the Cayley graph.

The most natural automorphisms of any Cayley graph are those that come directly from the group structure: left-multiplication by any element of $G$; and group automorphisms of $G$ that fix $S$ setwise. It is easy to see that these graph automorphisms either preserve or permute the colours in the natural edge-colouring defined above. Conversely, we can ask: if a graph automorphism preserves or permutes the colours in this natural edge-colouring, need it come from the group structure in one of these two ways?

I will show that in general, the answer to this question is no. I will explore the answer to this question for a variety of families of groups and of Cayley graphs on these groups. I will touch on work by other authors that explores similar questions coming from closely-related colourings.

This is based on joint work with Ademir Hujdurović, Klavdija Kutnar, and Dave Witte Morris.