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Advancing Research. Creating Connections.
The American Mathematical Society is dedicated to advancing research and connecting the diverse global mathematical community through publications, meetings and conferences, MathSciNet, professional services, advocacy, and awareness programs.
We will discuss all possible closures of a Euclidean line in various geometric spaces. Imagine the Euclidean traveller, who travels only along a Euclidean line. She will be traveling to many different geometric worlds, and our question will be what places does she get to see in each world?
Here is the itinerary of our Euclidean traveller:
In 1884, she travels to the torus of dimension $n\ge 2$, guided by Kronecker.
In 1936, she travels to the world, called a closed hyperbolic surface, guided by Hedlund [Hed36].
In 1991, she then travels to a closed hyperbolic manifold of higher dimension $n\ge 3$ guided by Ratner [Rat91].
Finally, she adventures into hyperbolic manifolds of infinite volume guided by Dal'bo [Dal00] in dimension two in 2000, by McMullen-Mohammadi-O. [MMO16] in dimension three in 2016 and by Lee-O. [LO19] in all higher dimensions in 2019.
Rotations of the Circle
As a warm up, she will first do her exercise of jumping on the circle $\mathbb S^1$,
which may be considered as the one-dimensional torus $\mathbb T^1$.
With fifteen partner organizations offering paper sessions, panels, workshops, student activities, and invited addresses, each day of JMM 2023 will provide an array of offerings — truly something for everyone.
Stay up-to-date with mathematics in current events and get ideas for classroom activities!
Also read Tony Phillips' deeper dives into recent mathematical developments.
