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Given two families of loops on a closed smooth manifold, one can concatenate the loops at the intersections points of these families to obtain a new family of loops. This is the Chas–Sullivan product on the homology of the free loop space of a...

Harmonic functions on groups are connected to many properties of the groups: algebraic, geometric, analytic, and probabilistic.
For some groups (or spaces), it can be a challenge even to determine whether harmonic functions of certain types—such as...

In the 1930s, Einstein, Podolsky and Rosen devised the "EPR paradox", which shed light on a peculiar phenomenon in the mathematical modeling of quantum mechanics:  Very far apart particles can exhibit correlated behaviour, which seemed to suggest a...

The Shannon entropy of a discrete random variable quantifies the number of bits of information conveyed by sampling that variable. Although originally introduced in the context of information theory, techniques relying on Shannon entropy have been...

Property (T) was defined by Kazhdan in the 1960s, who used it to prove two conjectures of Selberg on lattices in high-rank Lie groups. Shortly after that, Margulis used it to construct expander graphs.

Property $\tau$ is a baby version of property (T...

In the 1850s, Kummer discovered some striking congruences mod powers of a prime number p between values of the Riemann zeta function at negative odd integers.  This was part of his attempt to understand structural aspects of certain algebraic...