What’s Happening in the Mathematical Sciences

Dana Mackenzie

What’s Happening in the Mathematical Sciences looks at some highlights of the most recent developments in mathematics. These include the mathematics behind stories that made headlines and fascinating stories that never made it into the newspapers.

In 2009, a flu pandemic, the world’s first in more than 40 years, tested a new generation of mathematical models that take some of the guesswork out of public health decisions. As health officials rushed to quell the outbreak of H1N1 flu, mathematicians were working just as hurriedly to answer questions like these: Was the epidemic severe enough to justify school closings or quarantines? Who should be vaccinated first, the elderly or the young? Their findings substantially affected the response of local governments, national governments, and the World Health Organization.

Mathematics can also help society prepare for other kinds of natural and human-made disasters. A major tsunami in 2011 in Japan, like the one seven years earlier in the Indian Ocean, highlighted flaws in our understanding of these catastrophic events and inadequacies in our early warning systems. Geoscientists are working together with mathematicians to improve our short-term forecasting ability and quantify tsunamis’ long-term risks. Meanwhile, in California, another group of mathematicians succeeded in adapting earthquake prediction algorithms to forecast criminal activity. Their “predictive policing” software was tested in Los Angeles and adopted by other cities across the United States.

Fortunately, not all mathematics has to do with emergencies. Pure mathematicians have been busy cleaning out their closets of long-standing open problems. In 2012, two conjectures about different kinds of minimizing surfaces were solved: the Willmore Conjecture (minimizing energy) and the Lawson Conjecture (minimizing area). In 2012, following up on the extraordinary proofs of the Poincare Conjecture and Thurston’s Geometrization Conjecture, topologists proved a collection of conjectures that ensure that three-dimensional spaces can all be constructed uniformly. Meanwhile, for the last ten years, a new understanding of algebraic curves and surfaces has developed, leading to a subject now known as tropical geometry. With the new ideas, specific challenging problems in algebraic geometry suddenly become manageable, and certain “mathematical mysteries” of string theory begin to make sense.

In physics, the nine-billion-dollar search for the elusive Higgs boson finally begged its quarry in 2012. This discovery, one of the most widely publicized science stories of the year, provides experimental evidence for the “Higgs mechanism,” a nearly 50-year-old mathematical argument that explains how certain subatomic particles acquire mass.

Rounding out this volume are chapters on a new statistical technique called topic modeling, which breaks down the academic barriers between math and the humanities, and discoveries about mathematicians’ (and a lot of other people’s) favorite toy: the Rubik’s Cube.