Mathematics

Going back to the foundation work of von Neuman there is a question of whether there are smooth models of the models of classical ergodic theory. When both measure and map are required to be smooth there is only one known obstruction but essentially no general results. Within the class of zero entropy transformation we have a method called conjugation by approximation that can be used to realize many interesting properties. I will describe the method and some of the classical and modern consequences of this.

Many systems exhibit a digit bias. For example, the first digit base 10 of the Fibonacci numbers or of 2^n equals 1 about 30% of the time; the IRS uses this digit bias to detect fraudulent corporate tax returns. This phenomenon, known as Benford's Law, was first noticed by observing which pages of log tables were most worn from age- it's a good thing there were no calculators 100 years ago! We'll discuss the general theory and application, talk about some fun examples (ranging from the 3x + 1 problem to the Riemann zeta function), and discuss some current open problems suitable for undergraduate research projects.

For any $r\geq 2$, an $r$-uniform hypergraph $\mathcal{H}$, and integer $n$, the \emph{Tur\'{a}n number} for $\mathcal{H}$ is the maximum number of hyperedges in any $r$-uniform hypergraph on $n$ vertices containing no copy of $\mathcal{H}$. While the Tur\'{a}n numbers of graphs are well-understood and exact Tur\'{a}n numbers are known for some classes of graphs, few exact results are known for the cases $r \geq 3$. I will present a construction, using quadratic residues, for an infinite family of hypergraphs having no copy of the $4$-uniform hypergraph on $5$ vertices with $3$ hyperedges, with the maximum number of hyperedges subject to this condition. I will also describe a connection between this construction and a `switching' operation on tournaments, with applications to finding new bounds on Tur\'{a}n numbers for other small hypergraphs.

Which real numbers arise as the entropies of continuous, multimodal, postcritically finite self-maps of real intervals? This is the "one-dimensional" analogue of a more famous open question: which real numbers arise as the dilatations of pseudo-Anosov surface diffeomorphisms? In "Entropy in Dimension One," W. Thurston answers this one-dimensional version of the question. We'll discuss a small subset of the many beautiful ideas and questions in this paper.

The PIMS-UBC Math Job Forum is an annual Forum to help graduate students and postdoctoral fellows in Mathematics and related areas with their job searches. The session is divided in two parts: short presentations from our panel followed by a discussion.

Learn the secrets of writing an effective research statement, developing an outstanding CV, and giving a winning job talk. We will address questions like: Who do I ask for recommendation letters? What kind of jobs should I apply to? What can I do to maximize my chances of success?