## Symposium Celebrating New Fellows of the Royal Society of Canada

November 14, 2005

**Abstracts**

**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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