Quantum Hall Effects, Berry's Phase and Superconductivity
DateTuesday, April 2, 2019 - 12:30pm
AbstractThis talk will provide some physical and mathematical motivations underlying the idea of topological phases of matter. Time permitting we will also introduce the Chern-Simons-Ginzburg-Landau mean field model for Quantum Hall effects which motivates what will be discussed in this week's Mathematical Physics seminar.
Discrete time Darwinian dynamics and the evolution of semelparous (annual) versus iteroparous (perennial) life history strategies
DateTuesday, April 9, 2019 - 12:30pm
AbstractOne of life history theory's oldest problems (pondered by the likes of Aristotle and Linneaus) is identifying the reasons why organisms are either semelparous or iteroparous. Semelparity is the life history strategy defined by a single bout of reproduction and is contrasted with iteroparity, defined by repeated bouts of reproduction throughout life. These reproductive strategies are exemplified by annuals and perennials in the plant world, but both parities are found throughout virtually all taxa. Seminal theoretical studies last century suggested semelparity should be favored by evolution, despite the ubiquity of iteroparous species throughout the world. However, subsequent contemporary studies have shown there is no simple answer to this question and that many factors can be in play, including density dependence, variable environmental conditions, and many others. Recent studies have further proposed that reproductive parity should not be binary, but instead should be a continuous variable. In that it involves continuous phenotypic traits subject to Darwinian evolution, the methodology of evolutionary game theoretic modeling is suitable for this approach. In this talk I will discuss Darwinian dynamic versions of some standard discrete time population models and their implications with regard to the evolution of semelparous and iteroparous life history strategies. A particular focus is on the role of density dependent reproduction and survival. The mathematical analysis revolves around equilibrium bifurcations and stability and involves multiple attractor scenarios.
Stochastic persistence and extinction
DateTuesday, April 16, 2019 - 12:30pm
AbstractA key question in population biology is understanding the conditions under which the species of an ecosystem persist or go extinct. Theoretical and empirical studies have shown that persistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by n dimensional piecewise deterministic Markov processes. These are processes (X(t), r(t)) where the vector X denotes the density of the n species and r(t) is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we show how the random switching can `rescue' species from extinction. The talk is based on joint work with Dang H. Nguyen (University of Alabama).
Discrete geometry and mechanics of leaves, flowers, and sea slugs
DateTuesday, April 23, 2019 - 12:30pm
AbstractThe edges of growing leaves, blooming flowers, torn plastic sheets, and frilly sea slugs all exhibit intricate wrinkled patterns. Why is this so? We argue that the mechanics of these so-called non-Euclidean elastic sheets are influenced by non-trivial geometric considerations (i.e., non-smooth defects) which may be explored by new methods using discrete differential geometry (DDG). Wrinkled morphologies appear as an optimization of many topological/geometric degrees of freedom underlying its microstructure. I will motivate the need for DDG-inspired methods to study the mechanics of hyperbolic sheets, i.e., soft/thin objects with negative Gauss curvature. And, I will share results obtained from them, including energetic impacts from non-smooth defects, the role of weak external forces, and associated scaling laws. Ultimately, these modeling techniques have the potential to explain rippled shapes in leaves, flowers, etc. and to enable the control/design of slender elastic materials, e.g., for soft robotics. This is joint work with Shankar Venkataramani.