Abstact: Suppose that infinitely many red and blue points occur at random locations in Euclidean space, and consider translation-invariant schemes for perfectly matching red points to blue points. (Translation- invariance can be interpreted as meaning that the matching is constructed without favouring one spatial location over another.) What is the best possible cost of such a matching, measured in terms of the edge lengths? What happens if we insist that the matching is constructed without extra randomness, or if we forbid edge crossings, or if the points act as selfish agents? This last restriction can be formalized via the concept of stable marriage, the topic of the 2012 Nobel Prize in Economics.

I will review recent progress and open problems on these questions, as well as variants including fair allocation, multi-colour matching and multi-edge matching.

Alexander Holroyd received his Ph.D. from the University of Cambridge in 2000. After post-doctoral positions at UCLA and Berkeley, he became a faculty member at UBC. Holroyd won the Rollo Davidson prize (awarded to an outstanding young probabilist) and the Andre-Aisenstadt Prize (for an outstanding young mathematician in Canada). In 2010, he moved to the Theory Group at Microsoft Research in Redmond, Washington.

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MARK LEWIS is a Professor in the Department of Mathematical and Statistical Sciences at the University of Alberta. He holds a Canada Research Chair in Mathematical Biology and a Killam Research Fellowship. His research is mathematical biology, with a focus in spatial ecology and his mathematical models include nonlinear partial differential equations, integrodifference equations and related stochastic spatial processes. Biological problems include modeling the process of territorial pattern formation in wolves, predicting population spread in biological invasions, calculating optimal strategies for biocontrol and assessing the effect of habitat fragmentation on species survival. A significant part of his research involves the formulation and verification of quantitative models, in collaboration with biologists. His mathematical approaches include analytical methods for dynamical systems, perturbation theory and computational methods.

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PIMS DISTINGUISHED SPEAKER: MARK LEWIS

Mathematics behind stream population dynamics

Human activities change the natural flow regimes in streams and rivers and this impacts ecosystems. In this talk I will mathematically investigate the impact of changes in water flow on biological populations.  The approach I will take is to develop process-oriented advection-diffusion-reaction equations that couple hydraulic flow to population growth, and then to analyze the equations so as to assess the effect of impacts of water flow on population dynamics. The mathematical framework is based on new theory for the net reproductive rate R_o as applied to advection-diffusion-reaction equations. I will then connect the theory to populations in rivers under various flow regimes. This work lays the groundwork for connecting R_o to more complex models of spatially structured and interacting populations, as well as more detailed habitat and hydrological data. This is achieved through explicit numerical simulation of two dimensional depth-averaged models for river population dynamics.

MARK LEWIS is a Professor in the Department of Mathematical and Statistical Sciences at the University of Alberta. He holds a Canada Research Chair in Mathematical Biology and a Killam Research Fellowship. His research is mathematical biology, with a focus in spatial ecology and his mathematical models include nonlinear partial differential equations, integrodifference equations and related stochastic spatial processes. Biological problems include modeling the process of territorial pattern formation in wolves, predicting population spread in biological invasions, calculating optimal strategies for biocontrol and assessing the effect of habitat fragmentation on species survival. A significant part of his research involves the formulation and verification of quantitative models, in collaboration with biologists. His mathematical approaches include analytical methods for dynamical systems, perturbation theory and computational methods.

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