Plenary talks

Quantum physics brings a new mathematical framework for physical theories and fundamentally changes what is possible and what is feasible when it comes to storing, processing and extracting information. For example, a quantum computer can solve certain mathematical problems with astronomically fewer computational steps than was previously thought possible, including the factorization of integers and finding discrete logarithms. This has profound implications for cybersecurity since these problems underpin the security of the public key cryptography used to protect the integrity and confidentiality of information in most of the digital systems we rely on today. The race to fix these fundamental building blocks includes the development of new mathematical algorithms for public-key cryptography as well as quantum cryptography, which leverages features of quantum information. The race to develop useful quantum computers includes finding quantum algorithms that solve useful problems, optimizing the compilation of algorithms to run on realistic quantum hardware, the design of practical quantum error correcting codes, and building robust quantum hardware. I will highlight some examples of the fundamental role mathematicians play across this spectrum of interesting and important activities.

Regularization is often used for variable selection in regression models when there are several potential predictors and interest lies in identifying drivers of the response. Here, we focus on modelling complex systems that can be represented by an undirected graph and discuss novel optimization methods that involve iteratively making local node-by-node inferences to perform variable selection.  For multivariate count data, where the underlying graph is a bipartite network, each node corresponds to one of two sets of actors and edges represent their interactions.  We can move through one set of actors, node-by-node, to infer which covariates are predictive of interactions.  For multivariate Gaussian data, where the structure among the predictors is known, we can move through the predictor graph, node-by-node, to exploit its structure and infer edges with the response.  However, when the regularizing penalty function induces sparsity among the regression coefficients we must resort to proximal distance or proximal gradient methods for optimization.  We will use modelling aquatic health through macroinvertebrate compositions and modelling the gene network associated with boar taint as two motivating problems.