From Notices of the AMS

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Contact Geometry and the Mapping Class Group

by Joan E. Licata | MR Author Profile

This article introduces the mapping class group of a surface; describes techniques for building three-manifolds from mapping classes; and explores a relationship between algebraic structures in the mapping class group and some geometric properties of the resulting manifolds.

Three-manifolds are tantalizing objects. With physical senses adapted to a three-dimensional physical world, we might expect to have geometric intuition, yet few of us can easily visualize how to rotate the Poincaré homology sphere or navigate $\mathbb{R}P^3$. One trick for making three-manifolds more accessible is cutting them into pieces: a lens space may be hard to imagine, but anyone who's enjoyed a donut or two can picture a pair of solid tori. Decomposing large complicated objects into smaller, simpler pieces is ubiquitous in topology, but ultimately, the infamous Pottery Barn aphorism kicks in: if you break it, you fix it. Once we cut, we need to be able to glue these pieces back together.

From the Feature Column

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The Origins of Ordinary Least Squares Assumptions

Some Are More Breakable Than Others

by Sara Stoudt - Bucknell University

Fitting a line to a set of points . . . how hard can it be? When those points represent the temperature outside and a town's ice cream consumption, I'm really invested in that line helping me to understand the relationship between those two quantities. (What if my favorite flavor runs out?!) I might even want to predict new values of ice cream consumption based on new temperature values. A line can give us a way to do that too. But when we start to think more about it, more questions arise. What makes a line "good"? How do we tell if a line is the "best"?