Mathematics

Furstenberg's conjecture on the dimension of the intersection of x2,x3-invariant Cantor sets can be restated as a bound on the dimension of linear slices of the product of x2,x3-Cantor sets, which is a self-affine set in the plane. I will discuss some older and newer variants of this, where the self-affine set is replaced by a self-similar set such as the Sierpinski triangle, Sierpinski carpet or (support of) a complex Bernoulli convolution. Among other things, I will show that the intersection of the Sierpinski carpet with circles has small dimension, but on the other hand the Sierpinski carpet can be covered very efficiently by linear tubes (neighborhoods of lines). The latter fact is a recent result joint with A. Pyörälä, V. Suomala and M. Wu.

Multiscale multicellular models combine representations of subcellular biological networks, cell behaviors, tissue level effects and whole body effects to describe tissue outcomes during development, homeostasis and disease. I will briefly introduce these simulation methodologies, the CompuCell3D simulation environment and their applications, then focus on a multiscale simulation of an acute primary infection of an epithelial tissue infected by a virus like SARS-CoV-2, a simplified cellular immune response and viral and immune-induced tissue damage. The model exhibits four basic parameter regimes: where the immune response fails to contain or significantly slow the spread of viral infection, where it significantly slows but fails to stop the spread of infection, where it eliminates all infected epithelial cells, but reinfection occurs from residual extracellular virus and where it clears the both infected cells and extracellular virus, leaving a substantial fraction of epithelial cells uninfected. Even this simplified model can illustrate the effects of a number of drug therapy concepts. Inhibition of viral internalization and faster immune-cell recruitment promote containment of infection. Fast viral internalization and slower immune response lead to uncontrolled spread of infection. Existing antivirals, despite blocking viral replication, show reduced clinical benefit when given later during the course of infection. Simulation of a drug which reduces the replication rate of viral RNA, shows that a low dosage that provides only a relatively limited reduction of viral RNA replication greatly decreases the total tissue damage and extracellular virus when given near the beginning of infection. However, even a high dosage that greatly reduces the rate of RNA replication rapidly loses efficacy when given later after infection. Many combinations of dosage and treatment time lead to distinct stochastic outcomes, with some replicas showing clearance or control of the virus (treatment success), while others show rapid infection of all epithelial cells (treatment failure). This switch between a regime of frequent treatment success and frequent failure occurs is quite abrupt as the time of treatment increases. The model is open-source and modular, allowing rapid development and extension of its components by groups working in parallel.

Binocular rivalry explores the question of how the brain copes with contradictory information. A subject is shown two different pictures – one to each eye. What images does the subject perceive? Results from rivalry experiments usually lead to alternation of percepts and are often surprising. Hugh Wilson proposed modeling rivalry in the brain by using structured networks of differential equations. We use Wilson networks as modeling devices and equivariant Hopf bifurcation as a tool to both post-dict and predict experimentally observed percepts. This work is joint with Casey Diekman, Zhong-Lin Lu, Tyler McMillen, Ian Stewart, Yunjiao Wang, and Yukai Zhao.

This talk will present Veech's criterion for an ergodic probability measure preserving system to be prime. It will define factors of measure preserving systems, prime and self-joinings and provide examples. It uses disintegration of measures, the ergodic decomposition and Haar's Theorem. It will state these results and have examples of disintegration of measures and the ergodic decomposition, but wont discuss their proofs.

The COVID-19 global pandemic has led to unprecedented public interest in mathematical modelling as a tool to understand the dynamics of disease spread and predict the impact of public health interventions. In this pair of talks, we will describe how mathematical models are being used, with particular reference to the British Columbia epidemic.

In the first talk, Prof. Caroline Colijn (Dept. of Mathematics, Simon Fraser University) will outline the key features of the British Columbia data and focus on how modelling has allowed us to estimate the effectiveness of the provincial response. In the second talk, Prof. Daniel Coombs (Dept. of Mathematics and Inst. of Applied Mathematics, University of British Columbia) will describe forward-looking modelling approaches that can provide some guidance as the province moves towards partial de-escalation of measures. Each talk will be 30 mins in length and followed by a question and discussion period.

For more details on the group's work and to contact the team, please visit https://bccovid-19group.ca/