# Mathematics

## Accessibility for partially hyperbolic systems

Accessibility is a fundamental tool when working with partially hyperbolic systems. For instance, in the 1970s it was used as a tool to show certain systems were transitive, and in the 1990s it was used to establish stable ergodicity. We will review the general notion and how it applies in these settings. We will also review the result from 2003 by Dolgopyat and Wilkinson on the C^1 density of stably transitive systems.

## Smooth Realization and Conjugation By Approximation

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.

## The Infinite HaPPY Code

I will construct an infinite-dimensional analog of the HaPPY code as a growing series of stabilizer codes defined respective to their Hilbert spaces. These Hilbert spaces are related by isometries that will be defined during this talk. I will analyze its system in various aspects and discuss its implications in AdS/CFT. Our result hints that the relevance of quantum error correction in quantum gravity may not be limited to the CFT context.

For other events in this series see the quanTA events website.

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## PIMS-UNBC Distinguished Colloquium: Benford's Law: Why the IRS might care about the 3x + 1 problem and zeta (s)

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.

## Shape Recognition of Convex Bodies

A broad class of convex geometry problems deals with questions on retrieval of information about (convex) sets from data about different types of their projections, sections, or both. Examples of such assumptions are volume estimates, rigidity of structure, symmetry conditions etc.

In this talk, we will discuss known results and recent developments regarding the dual notions of point-projections and non-central sections of convex bodies. In particular, we provide a partial affirmative answer to the question on a shape recognition posed by A. Kurusa, and discuss a generalization of V. Klee's theorem for polyhedra.

## Turán numbers for a 4-uniform hypergraph

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.

## What is a Foster-Lyapunov-Margulis Function?

We'll show how a simple idea from probability theory on the recurrence of random walks can be used in many important dynamical and geometric situations, building on work of Eskin-Margulis and others. No prior knowledge of probability theory, random walks, or geometry is required. If time permits, as an unrelated "dessert" of sorts, we'll give a brief proof of the Hopf ratio ergodic theorem using the Birkhoff ergodic theorem for flows.

## An invitation to "Entropy in Dimension One"

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 Cost of 2-Distinguishing Hypercubes

The distinguishing number of a graph is the smallest number of colors necessary to color the vertices so that no nontrivial automorphism preserves the color classes. If a graph can be distinguished with two colors, the distinguishing cost is the smallest possible size of a color class over all 2-distinguishing colorings. In this talk I will present the long-sought-after (at least by me, :-) ) cost of 2-distinguishing hypercubes. We will begin the talk with definitions and intuitive examples of distinguishing and of cost, cover a bit of history, and work our way to a new technique using binary matrices. Then will we be able to state and understand the new results on hypercubes.

## 2020 PIMS-UBC Math Job Forum for Postdoctoral Fellows and Graduate Students

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?