During World War II Hedy Lamarr, a striking Hollywood actress, together with George Antheil, a radical composer, invented and patented a secret signaling system for the remote control of torpedoes. The ideas in this patent have since developed into one of the ingredients in modern digital wireless communications. The unlikely biography of these two characters, along with some of the more modern developments in wireless communications will be described.
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).
This will be a general talk about the role of dilation theory in studying operators on Hilbert space, illustrated in part by some recent work of mine with Raphaël Clouâtre on multivariable operator theory.
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.
In this seminar we will discuss a new model for strategic investment model for a merchant energy storage facility. The facility's actions impact market-clearing outcomes, and thus it is a price-maker facility. We consider the uncertainties associated with other generation units offering strategies and future load levels in the proposed model. Thestrategic investment decisions include the sizes of charging device,discharging device, and energy reservoir. The proposed model is astochastic bi-level optimization problem where planning and operation decisions of the energy storage facility are made in the upper level, and market clearing is modeled in the lower level under different operating conditions. To make the proposed model computationally tractable, an iterative solution technique based on Benders¹ decomposition is implemented. This provides a master problem and a set of subproblems for each scenario. Each subproblem is recast as a Mathematical Programs with Equilibrium Constraints (MPEC). Numerical results based on real-lifemarket data from Alberta's electricity market will be provided.
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.
PIMS CRG in Explicit Methods for Abelian Varieties
Before Mazur proved his theorem on the possible torsion groups on elliptic curves over Q, Manin had shown in 1969 that for a fixed number field K and prime p, the p-primary torsion of elliptic curves over K was uniformly bounded. We will explain a result in this direction for more general abelian varieties, subject to Lang's conjecture on rational points on varieties of general type. This is joint work with Dan Abramovich.
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.
Beginning with Aristotle, almost every major philosopher and mathematician before the nineteenth century rejected the notion of the actual infinite. They all argued that the only sensible notion is that of potential infinity. The list includes some of the greatest mathematical minds ever. Due to Georg Cantor’s influence, the situation is almost the opposite nowadays (with some intuitionists as notable exceptions). The received view is that the notion of a merely potential infinity is dubious: it can only be understood if there is an actual infinity that underlies it.
After a sketch of some of the history, Prof. Shapiro will analyze the notion of potential infinity, in modal terms, and assess its scientific merits. This leads to a number of more specific questions. Perhaps the most pressing of these is whether the conception of potential infinity can be explicated in a way that is both interesting and substantially different from the now-dominant conception of actual infinity. One might suspect that, when metaphors and loose talk give way to precise definitions, the apparent differences will evaporate.
A number of differences still remain. Some of the most interesting and surprising differences concern consequences that one’s conception of infinity has for higher-order logic. Another important question concerns the relation between potential infinity and mathematical intuitionism. In fact, as will be shown, potential infinity is not inextricably tied to intuitionistic logic. There are interesting explications of potential infinity that underwrite classical logic, while still differing in important ways from actual infinity. However, on some more stringent explications, potential infinity does indeed lead to intuitionistic logic.
(The lecture is based on joint work with Øystein Linnebo.)
Aside from games of chance and a handful of textbook topics (e.g. opinion polls) there is little overlap between the content of an introductory course in mathematical probability and our everyday perception of chance. In this mostly non-mathematical talk I will give some illustrations of the broader scope of probability.
Why do your friends have more friends than you do, on average? How can we judge someone’s ability to assess probabilities of future geopolitical events, where the true probabilities are unknown? Were there unusually many candidates for the 2012 and 2016 Republican Presidential Nominations whose fortunes rose and fell? Why, in a long line at airport security, do you move forward a few paces and then wait half a minute before moving forward again? In what everyday contexts do ordinary people perceive uncertainty/unpredictability in terms of chance?