Speaker: Caner Kazanci
Title: Alternative mathematical representations for ecosystems
The highly connected structure and feedback cycles make ecosystems mathematically interesting to study. For example, it has been shown that the existence of some predators (e.g. American alligator) actually improve the population of their prey, which sounds counter intuitive at first. Ecosystems are represented as weighted digraphs, where the nodes and vertices represent species and feeding relations respectively.
We will investigate two alternative representations for ecosystem models: Pathways and Fluxes. Pathways are ordered lists of species, that represent all possible pathways that, say a carbon atom can take once it enters an ecosystem. Fluxes represent the smallest process within an ecosystem that can theoretically sustain itself, such as a food-chain or a nutrient cycle. Fluxes correspond to the cycle basis of the digraph (including the environment as a node).
In this research group, we will investigate two main problems:
(1) How to define the transfer function from one representation to another?
(2) How to derive insightful information about ecosystems using various representations?
Speaker: Joseph Fu
Title: Algebraic Integral Geometry
Integral geometry is pretty much synonymous with geometric probability, the study of how to calculate the expectations of certain measurements as applied to the intersections of objects in random relative positions. Recently we have started to decode the basic structure of this subject in the case when the objects lie in complexprojective spaces. It all boils down to understanding the structure of some specific commutative algebras. These algebras are a rich sourceof weird phenomena, involving unexpected formal power series, bizarre numerical/ combinatorial identities, and strange transforms, all of which encode the geometry in mysterious ways. So it should be fruitful to spend some time playing around with these algebras, so we can get a better grip on the phenomena we already know about, and with luck find new relations as well.