I will then discuss extensions and refinements of these results, obtained in recent joint work with B.
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In this talk I will explain how they allow to express the characteristic classes of a foliation solely in terms of the transverse holonomy data, without resorting to the ambient manifold. Hopf algebras and their Hopf cyclic cohomology emerged as a geometric tool in the work of A. Connes and myself computing the local index theorem for transversely (hypo)elliptic operators on foliations. Hopf algebras and transverse characteristic classes HENRI MOSCOVICI, The Ohio State University, Columbus, OH 43210 In this talk, I will highlight mathematical approaches that deal with multiple scales, and provide methods to understand the emergence of properties of aggregated systems from the properties of its constituents.Įxamples of application are taken from my research, ranging from the investigation of oscillations that play important roles in physiological phenomena such as breathing and the control of blood sugar to the investigation of spatial animal group patterns that result from different communication mechanisms. A significant challenge in modelling biological systems is to bridge the wide range of spatial and temporal scales on which biological processes operate. The role of mathematics in biology is to make sense of the complexity of our living world, to provide insight into mechanisms that may underlie observed phenomena. Mathematics for multiple-scale modelling and emergent behaviour: case studies from biology There are certainly implications here for the education of our students, but I will leave these for the many critical discussions that will continue to take place in our community. I will argue that outreach of various kinds is essential for our future success, and that we have to find ways and means to have some parts of our discipline aimed at solving today's problems in multi-disciplinary teams. On the math side, I will argue that we need to see ourselves more clearly as part of the quantitative sciences (and engineering) and include CS, Stats, OR, etc., under this umbrella. I will talk about the university and the situation in which we find ourselves this year, with some mention of research funding, particularly MITACS and NSERC.
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H. E. A. EDDY CAMPBELL, President, University of New Brunswick-Fredericton