Mathematics

The Criminal Justice System is responsible for upholding public safety through the enforcement of laws, the apprehension, prosecution, and judging of suspects, and the administration of community and custodial sentences. It is highly complex, with interactions between police, prosecutors, judges, the court, and correctional services. Effective and efficient administration of justice is important for maintaining public safety.

We present an overview of operational research modelling applied to the Criminal Justice System. Two case studies are considered: a systems dynamics model of the impact of the 2010 impaired driving legislation in British Columbia and a queue network model of the impact of the Truth in Sentencing Act of Canada.

Experimental design is a branch of statistics focused upon designing experimental studies in a way that maximizes the amount of salient information produced by the experiment. It is a topic which has been well studied in the context of linear systems. However, many physical, biological, economic, financial and engineering systems of interest are inherently non-linear in nature. Experimental design for non-linear models is complicated by the fact that the optimal design depends upon the parameters that we are using the experiment to estimate. A Bayesian, often simulation-based, framework is a natural setting for such design problems. We will illustrate the use of such a framework by considering the design of an animal disease transmission experiment where the underlying goal is to identify some characteristics of the disease dynamics (e.g. a vaccine effect, or the infectious period).

Matrices contain combinatorial information. They may provide alternative representations of combinatorial ideas. Examples include permutation matrices as representations of permutations of a finite set, and adjacency matrices as representations of a finite graph. The linear algebraic properties of these matrices may provide useful combinatorial information, and combinatorial information about a matrix may impact its linear algebraic properties. At the same time, some combinatorial constructs are defined by matrices. A notable example is the alternating sign matrices which arise in a number of ways including from the partial order on permutations called the Bruhat order. In this talk we will explore various connections between combinatorics and matrices, combinatorial matrices.

Description:

The “language" of mathematics has been developed since the dawn of humanity to describe and comprehend the surrounding world and its phenomena. Mathematics as a science and a school subject is widely identified with the mechanical rules of algebra or calculus, and with the symbolic writings native to this discipline. In this talk I will attempt to bring a perspective of mathematics as a natural, everyday endeavour, which each one of us lives and performs every day, most of the time without even realizing that we do so. I will present several examples in which the pervasiveness of mathematics in our everyday life will be illustrated, and I will also show some interesting mathematical applications through modelling. This particular feature, that the universe can be modelled "by the use of a minimum of primary concepts and relations” (A. Einstein) is one of the most puzzling properties of our universe. Finally, I will discuss some of these historic reflections on the nature of mathematics and its connection to the “real” world.

Thin sheets exhibit a daunting array of patterns. A key difficulty in their analysis is that while we have many examples, we have no classification of the possible "patterns." I have explored an alternative viewpoint in a series of recent projects with Peter Bella, Hoai-Minh Nguyen, and others. Our goal is to identify the *scaling law* of the minimum elastic energy (with respect to the sheet thickness, and the other parameters of the problem). Success requires proving upper bounds and lower bounds that scale the same way. The upper bounds are usually easier, since nature gives us a hint. The lower bounds are more subtle, since they must be ansatz-independent. In many cases, the arguments used to prove the lower bounds help explain "why" we see particular patterns. My talk will give an overview of this activity, and details of some examples.