The chiral soliton lattice is an array of topological solitons realized as ground states of QCD at finite density under strong magnetic fields or rapid rotation, and chiral magnets with an easy-plane anisotropy. In such cases, topological solitons have negative energy due to topological terms originating from the chiral magnetic or vortical effect and the Dzyaloshinskii-Moriya interaction, respectively. We study quantum nucleation of topological solitons in the vacuum through quantum tunneling in 2+1 and 3+1 dimensions, by using a complex ϕ4 (or the axion) model with a topological term proportional to an external field, which is a simplification of low-energy theories of the above systems. In 2+1 dimensions, a pair of a vortex and an anti-vortex is connected by a linear soliton, while in 3+1 dimensions, a vortex is string-like, a soliton is wall-like, and a disk of a soliton wall is bounded by a string loop. Since the tension of solitons can be effectively negative due to the topological term, such a composite configuration of a finite size is created by quantum tunneling and subsequently grows rapidly. We estimate the nucleation probability analytically in the thin-defect approximation and fully calculate it numerically using the relaxation (gradient flow) method. The nucleation probability is maximized when the direction of the soliton is perpendicular to the external field.

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.

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