Pure Math Colloquium

Monday, February 22, 2021 4:00 pm - 4:00 pm EST (GMT -05:00)

Andrew Granville, Université de Montréal

"The Frobenius postage stamp problem and boundary turbulence"

The Frobenius postage stamp problem asks what exact postage one can make up out of (arbitrarily many) stamps of a given finite set of values; but how many stamps are really needed?   A knight on an infinite chess board can (eventually) get to any point on the board, but how many moves are really needed? These are examples of given finite sets A in Z(that is, the stamp values, or the feasible knight moves) for which we are asking to understand NA ={a1+...+aNa1,...,a∈ A}.  It turns out that the set NA is remarkably structured though boundary issues restrict what we can hope to prove. This is arguably surprising in light of Khovanskii's 1992 Theorem that there exists a polynomial f(t) such that |NA|=f(N) for N sufficiently large, but we will show that "sufficiently large" here can mean "not small". 

This problem appears in different guises in different areas, for example  in number theory, combinatorics, logic, algebraic geometry and the theory of dimensions of graded modules. This is joint work, via Zoom, with George Shakan and Aled Walker.

Zoom meeting: https://zoom.us/j/93240216981?pwd=aE0vbktRV1NvMTFzbFVaalVpb1pCdz09