Richard P. Stanley Seminar in Combinatorics
- When
- Wednesday, November 18, 20263:15 PM – 4:15 PM ESTAdd to calendarGoogle Calendar
- Where
- Harvard Science Center 232Open in Maps
- Hosted by
- Department of Mathematics
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About
Speaker: Jim Propp (UMass Lowell and Radcliffe Institute)
Title: Snake graphs and q-deformed real numbers
Abstract: I'll present a statistical-mechanical construction of q-deformed real numbers, inspired by the q-deformed rational numbers [x]_q of Morier-Genoud and Ovsienko. The new construction admits two equivalent formulations: one in terms of random filters in snake posets and the other in terms of random dimer covers of snake graphs.
I'll begin with a combinatorial approach to Stern's diatomic series and the Calkin–Wilf bijection via finite snake graphs, and then construct q-deformed rational numbers by introducing an activity parameter q into the model. For irrational values of x, the construction requires passing from finite to infinite snake graphs; the main technical step is to show that the associated limiting probability measures are well defined.
The resulting q-deformation satisfies [[x]]_1 = x for all x>0. There's evidence that, wherever both [x]_q and [[x]]_q are defined, [[x]]_q = q [x]_q. For more details, see arXiv:2607.14332.