Understanding mechanisms of coexistence is a central topic in ecology. Mathematical analysis of models of competition between two identical species moving at different rates of symmetric diffusion in heterogeneous environments show that the slower mover excludes the faster one. The models have not been tested empirically and lack inclusions of a component of directed movement toward favorable areas. To address these gaps, we extended previous theory by explicitly including exploitable resource dynamics and directed movement. We tested the mathematical results experimentally using laboratory populations of the nematode worm, Caenorhabditis elegans. Our results not only support the previous theory that the species diffusing at a slower rate prevails in heterogeneous environments but also reveal that moderate levels of a directed movement component on top of the diffusive movement allow species to coexist. Additionally, we have expanded our work to test the outcomes of different movement strategies in a various of fragmented and toxincant environments. For instance, we combine mechanistic mathematical modeling and laboratory experiments to disentangle the impacts of habitat fragmentation and locomotion. Our theoretical and empirical results found that species with a relatively low motility rate maintained a moderate growth rate and high population abundance in fragmentation. Alternatively, fragmentation harmed fast-moving populations through a decrease in the populations’ growth rate by creating mismatch between the population distribution and the resource distribution. Our study will advance our knowledge of understanding habitat fragmentation's impacts and potential mitigations, which is a pressing concern in biodiversity conservation.

A subset Z of R^n is said to be u-convex if for any two points z1, z2 ∈ Z, the arc of the circle through u, z1 and z2, lying between z1, z2 and not containing u, is contained in Z. We call u a pole of Z and Z a polar convex set if this happens. The notion of polar convexity was developed for the complex plane to study the geometry of univariate complex polynomials. This talk discusses some motivations for the theory and presents the extension of polar convexity to higher dimension Euclidean spaces. The introduction of a pole creates a richer theory full of properties that classical convexity has no analogues for. We study the geometrical properties of polar convex sets and develop analogues of theorems from classical convexity. Finally, we define polar derivatives for multivariate polynomials and demonstrate the use of polar convex sets in studying the roots of multivariate polynomials with several analogues of the Gauss-Lucas theorem.

In this talk I will introduce the notion of polynomial and rational convexity and present other relevant material in order to formulate a result of Cieliebak and Eliashberg (Invent. Math. 2015) concerning the topology of smoothly bounded domains with polynomially, resp. rationally, convex closure in complex Euclidean spaces.

Fix $g \in \mathbb{N}$ such that $g$ is not a perfect square. Artin's primitive root conjecture (1927) states that there exist infinitely many primes $p \in \mathbb{N}$ such that $g$ generates the finite cyclic group $(\mathbb{Z}/p\mathbb{Z})^{\times}$. Nearly a century later, Artin's conjecture remains wide-open: in fact there is no known specified $g$ for which the conjecture has been resolved.

In this talk, we survey the interesting history of Artin's conjecture and introduce several new variants to the problem. Specifically, we discuss an "Artin Twin Primes Conjecture"; and prove an appropriate analogue of Artin's conjecture for algebraic function fields.