Ecologists have been debating the best way to measure diversity for more than 50 years. The concept of diversity is relevant not only in ecology, but also in other fields such as genetics and economics, as well as being closely related to entropy. The question of how best to quantify diversity has surprising mathematical depth. I will argue that the best approach is axiomatic: to enable us to reason logically about diversity, the measures we use must satisfy certain mathematical conditions, and those conditions dramatically limit the choice of measures. This point will be illustrated with a theorem: using a simple model of ecosystems, the only diversity measures that behave logically are the Hill numbers, which are very closely related to the Rényi entropies of information theory.

In living tissues where cells migrate, spatial distribution of mechanical properties, especially matrix stiffness, are generally heterogenous with cell-scales ranging from 10 to 1000 μm. Since the cell migration in our body plays critical role in morphogenesis, wound healing, and cancer metastasis, it is essential to understand the migratory dynamics on the matrix with cell-scale stiffness heterogeneity. However, while cellular responses to homogeneous matrix have been extensively explored, studies of the cell motility with stiffness heterogeneity have been limited to the directional movement (durotaxis) driven by a simple stiffness gradient. Thus, we need to elucidate how cell migration is determined through the interaction among cell-scale stiffness heterogeneity and cellular responses such as dynamics of the cell-matrix adhesion site, the intracellular prestress, and cell shape.

In this talk, we introduce our experiments on cell motility, shaping, adhesion, and traction forces at long time scales using microelastically patterned hydrogels that enable us to systematically control the cell-scale heterogeneity of the matrix-stiffness. Using microelastically patterned hydrogels, we showed that the cell exhibited a general mode of durotaxis depending on the shape and size of the stiff domains, which was coincided with the extraordinarily large fluctuation of the traction force. We proposed a cell migration model based on equations of a deformable self-propelled particle adopting an amoeboid swimmer-like velocity-shape relationship. By considering the cellular response to stiffness gradients, the model can reproduce general durotaxis driven by cell-scale heterogeneity of the matrix-stiffness.

Traditional measures of the diversity of an ecological community depend only on how abundant the species are, not the similarities or differences between them. To better reflect biological reality, species similarity should be incorporated. Mathematically, this corresponds to moving from probability distributions on sets to probability distributions on metric spaces. I will explain how to do this and how it can change ecological judgements. Finally, I will describe a surprising theorem on maximum diversity (joint with Meckes and Roff), which reveals close connections between maximum diversity and invariants of geometric measure.

The Fermi Bubbles are giant, gamma-ray emitting lobes emanating from the nucleus of the Milky Way discovered in ~1-100 GeV data collected by the Fermi Gamma-Ray Space Telescope. Previous work has revealed substructure within the Fermi Bubbles that has been interpreted as a signature of collimated outflows from the Galaxy's super-massive black hole. In this talk, I will show that much of the gamma-ray emission associated to the brightest region of substructure -- the so-called cocoon -- is likely due to the Sagittarius dwarf spheroidal (Sgr dSph) galaxy. This large Milky Way satellite is viewed through the Fermi Bubbles from the position of the Solar System. As a tidally and ram-pressure stripped remnant, the Sgr dSph has no on-going star formation, but I will demonstrate that the dwarf's millisecond pulsar (MSP) population can plausibly supply the observed gamma-ray signal. This finding plausibly suggests that MSPs produce significant gamma-ray emission amongst old stellar populations, potentially confounding indirect dark matter searches in regions such as the Galactic Centre, the Andromeda galaxy, and other massive Milky Way dwarf spheroidals.

In this lecture, basic concepts in quantum measurement theory are introduced, including measurement operators and POVM's. The related topics are also picked up.

Lecture 1: Nov. 10, 10:30 - 12:00
Lecture 2: Nov. 10, 13:30 - 15:00
Lecture 3: Nov. 10, 15:30 - 17:00
Lecture 4: Nov. 11, 10:30 - 12:00
Lecture 5: Nov. 11, 13:30 - 15:00
Lecture 6: Nov. 12, 10:30 - 12:00