June  7, 2023

Symposium Celebrating New Fellows of the Royal Society of Canada
November 14, 2005


Peter Abrams, University of Toronto
Problems related to the coexistence and diversification of species

A large fraction of current research in ecology seeks to explain geographic differences or temporal changes in the number of species of a given group of organisms that can be found together in a given place. One of the fundamental processes that affect species diversity is competition between species that depend on similar resources. In spite of many decades of work on the question of coexistence of competitors, we still have a relatively poor understanding of conditions that promote or inhibit coexistence. I will discuss some recent models on the role of variable environments in allowing coexistence or promoting evolution of new forms. Many of these focus on the ability of adaptive evolution to produces both generalist and specialist types that are able to coexist in environments with only two types of resource.

David M.R. Jackson, University of Waterloo
Maps in surfaces, and the moduli space of curves

This talk is about the use of algebraic combinatorics and the theory of maps in algebraic geometry and mathematical physics.

A map is a collection of points joined by lines that is embedded in a surface so that lines meet only at the points. The most familiar theorem about maps is perhaps the Four Colour Theorem, which states that the faces of a planar map can be coloured with at most four colours so that edges separate faces of different colours. However, maps appear in many other contexts in mathematics and mathematical physics. Of central interest in these contexts is a map generating series, which contains information about the number of maps with prescribed properties.

Part I gives an introduction to maps, illustrated with drawings of maps in the sphere, torus and Klein Bottle, a non-orientable surface. This culminates in an algebraic encoding of these combinatorial objects as transitive factorisations of permutations.

In Part II I shall show in general terms how a generating series (or partition function in the terms of mathematical physics) can be constructed algebraically for both orientable surfaces and all surfaces through representation theory. This has application to firming up 't Hooft's Conjecture about the origin of quark confinement from quantum chromo-dynamics, and also concerns the strong interaction between quarks and gluons. I shall also mention briefly a connexion of maps to the moduli spaces of real and complex algebraic curves.

Part III concerns the notion of a transitive factorization of a permutation into transpositions, the unifying theme of this talk. Such factorizations arise in determining, through algebraic combinatorics, the (top) intersection numbers in the moduli space of smooth curves, and the number of branched covers of the sphere. Localisation theory can be used to express the intersection numbers in combinatorial terms involving a restricted class of maps. This reveals the kind of Lagrangian structure that pervades branched covers of the sphere. I shall report recent progress towards a proof of Faber's Top Intersection Number Conjecture for all genera and at most three points.

E.A. Sudicky, University of Waterloo
On the Challenge of Simulating Integrated Surface-subsurface Flow and Contaminant Transport at Multiple Catchment Scales

Over the past several years, substantial hydrologic research has been directed towards understanding flow and contaminant transport processes occurring at the interface between surface water and groundwater. To date, however, very few numerical models have been formulated that couple these processes in a holistic, physically-based framework in three dimensions. In this lecture, we will examine these coupling strategies in the context of the HydroGeoSphere model, a surface-subsurface control-volume finite element model. HydroGeoSphere is a fully-integrated 3D model that can simulate water flow, heat flow and advective-dispersive solute transport on the 2D land surface and in the 3D subsurface under variably-saturated, heterogeneous geologic conditions. Full coupling of the surface and subsurface flow regimes is accomplished implicitly by simultaneously solving one system of non-linear discrete equations describing flow and transport in both flow regimes. The model capabilities and main features are demonstrated with several high-resolution 3D numerical simulations performed for catchments of various scales, ranging from the scale of an intensively-monitored rainfall-runoff tracer experiment (~ 2000 m2), to a regional-scale watershed of about 1000 km2, to the continental scale that comprises the entire Canadian land mass. The simulations highlight the difficulties and challenges for representing water flow and solute flux in complex natural systems, and stresses the advantage of using a process-based model such as HydroGeoSphere for prediction of current and future water management scenarios. Among the challenges associated with such simulations is the discrepancy in spatial and temporal resolutions needed for the surface and subsurface flow domains, the forms of the needed constitutive relations and effective parameter values at various scales, and dealing with data uncertainty. New algorithms such as sub-timing and sub-gridding, together with domain decomposition for parallel processing, are being implemented to alleviate some of the computational demands and are demonstrated here.



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