This month's topics:

Why math majors change major

A story by Doug Lederman posted on Inside Higher Ed (December 18, 2017) examines a recent report from the U. S. Department of Education entitled "Beginning College Students Who Change Their Majors Within 3 Years of Enrollment," with data from the 2012/14 Beginning Postsecondary Students Longitudinal Study (BPS:12/14). He focuses on the conclusion "students who started out studying math were likeliest of all: 52 percent of those who initially declared as math majors ended up majoring in something else, followed by 40 percent of those in the natural sciences, 37 percent in education, 36 percent in humanities disciplines and 32 percent in engineering and general studies ..." and on the causes and consequences of this phenomenon.

 

Covering a sphere with zones

The press release from the Moscow Institute of Physics and Technology was posted on Eurekalert!. Their Alexandr Polyanskii, working with Zilin Jiang (Technion), has proved a 44-year-old conjecture due to László Fejes Tóth about covering the sphere with zones. These are areas, like the Equatorial Zone in geography, that extend an equal distance on both sides of a great circle. The Equatorial Zone has been defined as extending 12o on either side of the equator; in this context that zone would have width 24o or $2\pi/15$. Fejes Tóth's conjecture is that if a collection of zones, no matter how they are oriented, covers the sphere their widths must add up to at least $\pi$.

5-zones-cover-sphere
The five zones illustrated here cover the sphere; the sum $\omega_1 + \omega_2 + \omega_3 + \omega_4 + \omega_5$ of their widths must therefore be at least $\pi$. Image credit MIPT Press Office. [Actually, in this picture, the five zones do not quite cover the sphere. -TP]

So far "sphere" has meant the 2-dimensional $S^2$, but in their article Jiang and Polyanskii prove Fejes Tóth's conjecture for a sphere $S^n$ of any dimension, where zone has the same definition, except centered on a great $S^{n-1}$. They also give a necessary and sufficient condition for a covering set of zones to have total width exactly $\pi$. They have to be arranged like the ones in the illustration, straddling a collection of great circles sharing a common diameter.

Topological waves in the ocean

According to a report posted on physicsworld.com (Philip Ball, October 10, 2017), the same topological mechanism that produces topological insulators and related phenomena in condensed matter physics also accounts for some exceptional waves in the Earth's oceans, for example the Kelvin waves. A simple example: north of the equator, the Coriolis force bends northward motion to the east. This is what makes hurricanes --Northern Hemisphere-- rotate counter-clockwise; good explanation here. The same happens to southward motion in the Southern Hemisphere. So an eastward-traveling disturbance along the equator is trapped: if it strays up or down the Coriolis force nudges it back; this is an equatorial Kelvin wave. (It travels eastward until it reaches a coast, where it splits and travels away from the equator along the coastline). In the new point of view, this Kelvin wave is an edge phenomenon: its domain is a region (for instance, the North Pacific Ocean) together with its boundary (the equator and the shores of the continents).

Another edge phenomenon occurs in topological insulators where, Ball explains, "the topology of the electron bands causes the electrons of opposite spin to move in opposite directions -- resulting in circular motions. The effect is closely related to the so-called quantum Hall effect, which occurs in 2D conducting materials such as thin films in the presence of a magnetic field. The resulting circular motion permits no net flow of current through the material -- except at the edges, where the circular orbits are truncated so that electrons can move around the surface in a series of arcs."

Ball is reporting on work recently published (Science, October 5, 2017) by Pierre Delplace and Antoine Venaille (ENS-Lyon) and J. B. Marston (Brown). The two phenomena are equivalent, with the Coriolis force playing the role of the magnetic field; and "this equivalence shows up in the mathematics of the problem." Ball quotes Marston. "It's obvious once you're attuned to the concepts of topological insulators. All the recent candidates for a condensed-matter physics position at Brown immediately saw the link when shown the equations for Kelvin waves in an old geophysics textbook."

Tony Phillips
Stony Brook University
tony at math.sunysb.edu