Juggling for Geeks (abstract)

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I'll talk about the mathematics of juggling, introduce `siteswaps', and juggle a little. After that, I'll explain a classification of juggling patterns in terms of permutation groups. This classification helps us analyse juggling patterns, and create new and absurdly difficult juggling tricks. In fact, you'll get to see some juggling patterns performed that have probably never been attempted outside the walls of a mathematics department!

Hopefully I'll have time to demonstrate an application of juggling to mathematics, by enumerating juggling patterns in two different ways, in order to prove a non-trivial combinatorial result. I might even make some cryptic comments about affine Weyl groups and Poincar\'e series.

notes for the talk

Siteswaps - instructions for the juggler (things to juggle - for me, 423, 441, 51, 504, 12345, 52413, 531, for Allen, 534, 61616, ?)

How can we decide if a given list of numbers is a valid siteswap? What sort of things can go wrong?

Landing diagrams (what happens to 2s and 0s?)

• are juggle-able iff every point has one incoming and one outgoing arrow
• are periodic
• later we may want to remove the $\sigma(n) \geq n$ condition, to obtain landing diagrams for virtual juggling patterns using `antiballs'! (Explain and demonstrate?)

How to count balls? Prove the averaging theorem. (Describe an infinite-ball aperiodic pattern?)

Extract a permutation from a periodic landing diagram, and use this to prove the valid siteswap theorem.

Explain what the permutation tells us about the pattern

• cycle notation, and examples 5241, 633, 56414?

How do we go back from permutations to siteswaps?

• the zero ball virtual pattern
• the minimal ball `honest' pattern
• the structure of virtual juggling patterns, $S_n \semidirect \Z^n$
• examples, the family -101, 201, 501, 531, 504, 534, 561?
• a random example, juggled by Allen

Enumeration

• what exactly are we counting?
• the (b+1)^n result via cards
• the alternative calculation: sum over permutations to obtain \[ \sum_{m=0}^b D(n,m) \binomial{b-m+n-1}{n-1} = (b+1)^n - b^n \] and then calculate \[D(n,m) = \sum_{l=0}^m C_{n,m-l} ((l+1)^n - l^n),\] where \[C_{n,k} = \sum_{a_1 + \ldots + a_m = k} \Pi_{i=1}^m - \binomial{n+a_i-1}{a_i}.\]
• what about enumerating patterns if we count cyclic permutations as the same pattern (e.g. 534 and 453 are the same), and don't count repeated patterns (e.g. 5151 is just the shower, 51)?

Anything else?

• zero ball patterns and the affine Weyl group
• its Poincar\'e series, and the cohomology and Bruhat decomposition of a certain infinite dimensional flag manifold.
• the Banach-Tarski theorem